ABSTRACT Title of Document: BAYESIAN ESTIMATION OF THE INBREEDING COEFFICIENT FOR SINGLE NUCLEOTIDE POLYMORPHISM USING COMPLEX SURVEY DATA Zhenyi Xue, Doctor of Philosophy, 2015 Directed By: Professor Partha Lahiri Associate Professor Yan Li Joint Program in Survey Methodology In genome-wide association studies (GWAS), single nucleotide polymorphism (SNP) is often used as a genetic marker to study gene-disease association. Some large scale health sample surveys have recently started collecting genetic data. There is now growing interest in developing statistical procedures using genetic survey data. This calls for innovative statistical methods that incorporate both genetic and statistical sampling. Under simple random sampling, the traditional estimator of the inbreeding coefficient is given by 1 - (number of observed heterozygotes) / (number of expected heterozygotes). Genetic data quality control reports published by the National Health and Nutrition Examination Survey (NHANES) and the Health and Retirement Study (HRS) use this simple estimator, which serves as a reasonable quality control tool to identify problems such as genotyping error. There is, however, a need to improve on this estimator by considering different features of the complex survey design. The main goal of this dissertation is to fill in this important research gap. First, a design-based estimator and its associated jackknife standard error estimator are proposed. Secondly, a hierarchical Bayesian methodology is developed using the effective sample size and genotype count. Lastly, a Bayesian pseudo-empirical likelihood estimator is proposed using the expected number of heterozygotes in the estimating equation as a constraint when maximizing the pseudo-empirical likelihood. One of the advantages of the proposed Bayesian methodology is that the prior distribution can be used to restrict the parameter space induced by the general inbreeding model. The proposed estimators are evaluated using Monte Carlo simulation studies. Moreover, the proposed estimates of the inbreeding coefficients of SNPs from APOC1 and BDNF genes are compared using the genetic data from the 2006 Health and Retirement Study. BAYESIAN ESTIMATION OF THE INBREEDING COEFFICIENT FOR SINGE NUCLEOTIDE POLYMORPHISM USING COMPLEX SURVEY DATA By Zhenyi Xue Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park, in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2015 Advisory Committee: Professor Frank Alt Professor Partha Lahiri, Chair Professor Yan Li, Co-Chair Professor Paul Smith Professor Jing Zhang © Copyright by Zhenyi Xue 2015 ii Acknowledgements First of all, I want to express my sincere gratitude to my academic advisor, Professor Partha Lahiri. Throughout many years of my part-time graduate study, he continuously guided me in my academic research through his insightful and inspiring thinking. I am also very thankful to my academic co-advisor, Professor Yan Li, for her dedicated support and guidance in my dissertation research. I greatly appreciate the enormous amount of time both of them spent with me, listening patiently and then advising using their profound knowledge. My special thanks go to Professor Paul Smith for his consistent encouragement and support during my entire PhD studies in the Department of Mathematics. I also want to extend my gratitude to Professor Frank Alt and Professor Jing Zhang for their time to read over my work and for their constructive comments and help on this dissertation. I also thank Professor Eric Slud, Professor Abram Kagan, and Professor Benjamin Kedem for their valuable teachings. I also thank the following in our Mathematics Department and Joint Program in Survey Methodology: Alverda McCoy, Haydee Hidalgo, Celeste Regalado, and Holly Rollins for their administrative help; Dr. Daniel Bonnery and Dr. Neung Soo Ha for their help and research discussion; and other fellow students in the Mathematics Department for their helpful discussions, encouragement, and friendship. To Ms. Rebecca Torguson, Dr. Ron Waksman at MedStar Health Research Institute, Mrs. Paula Johnson, Dr. Julie Berube and my senior colleagues at Becton, Dickinson iii and Company, I want to extend my gratitude for giving me the flexibility that allowed me to manage my work and study simultaneously during my part-time PhD studies. Last but not the least, I would like to thank my parents Guiyu Liang and Aofu Xue, my wife Chunhong, my daughter Hannah and my son Alex, for their love, patience and support. iv Table of Contents Acknowledgements ....................................................................................................... ii  Table of Contents ......................................................................................................... iv  List of Tables ............................................................................................................... vi  List of Figures .............................................................................................................. ix  Glossary of Technical Terms ....................................................................................... xi  Chapter 1: Introduction ................................................................................................. 1  1.1  Single Nucleotide Polymorphism (SNP) ...................................................... 1  1.2  Hardy-Weinberg Equilibrium (HWE) .......................................................... 2  1.3  Inbreeding Coefficient (f) ............................................................................. 6  1.4  Multinomial Model for the Genotype Distribution ..................................... 10  1.4.1  Existing Frequentist Estimator ............................................................... 12  1.4.2  Bayesian Method for HWE and Estimation ............................................ 17  1.5  Genetic Data from Complex Survey ........................................................... 23  1.5.1  National Health and Nutrition Examination Survey (NHANES) ............ 24  1.5.2  Health and Retirement Survey (HRS) ..................................................... 26  1.6  Empirical Likelihood .................................................................................. 28  1.7  Discussion and Overview of Dissertation ................................................... 29  Chapter 2: Estimation of the Inbreeding Coefficient under Simple Random Sampling ..................................................................................................................................... 31  2.1 Maximum Empirical Likelihood Estimator (MELE) ....................................... 31  2.2 Bayesian Pseudo-Empirical Likelihood Estimator (BPELE) ........................... 33  2.3 Simulation Study ............................................................................................... 35  2.3.1 Monte Carlo Simulation Error................................................................... 37  2.3.2 Compare Frequentist and Bayesian Estimators ........................................ 38  2.3.3 Use of Prior Knowledge in Analysis .......................................................... 45  Chapter 3: Estimation of the Inbreeding Coefficient under Unequal Probability Sampling ..................................................................................................................... 48  3.1  Direct Design-Based Estimator ................................................................... 48  3.2  Parametric Bayesian Estimator with Survey Weight .................................. 50  3.3  Bayesian Pseudo-Empirical Likelihood Estimator ..................................... 52  3.4  Simulation Study ......................................................................................... 58  3.4.1 Stratified Simple Random Sampling .......................................................... 59  3.4.2 Proportional to Population Size Sampling ................................................ 65  3.4.3 Model Misspecification .............................................................................. 71  Chapter 4: Estimation of the Inbreeding Coefficient with Family Level Correlation 73  4.1  Direct Design-Based Estimator ................................................................... 76  4.2  Parametric Bayesian Estimator with Survey Weight .................................. 77  4.3  Bayesian Pseudo-Empirical Likelihood Estimator ..................................... 79  4.4  Simulation Study ......................................................................................... 79  4.4.1 Stratified Simple Random Sampling .......................................................... 80  4.4.2 Proportional to Population Size Sampling ................................................ 85  Chapter 5: Estimation of the Inbreeding Coefficient Incorporating Between Subject Correlation .................................................................................................................. 90  5.1  Population Subdivision ............................................................................... 90  v 5.2  Bayesian Hierarchical Model ...................................................................... 92  5.3  Simulation Study ......................................................................................... 97  Chapter 6: Application to Health and Retirement Study ......................................... 100  Chapter 7: Summary and Future Research .............................................................. 108  Appendices: R Programs........................................................................................... 111  Bibliography ............................................................................................................. 118  vi List of Tables Table 1. Genotype Count from a Single Locus with k Alleles ..................................... 5  Table 2. Pattern of Union of Gametes in the Total Population ..................................... 8  Table 3. Expected Genotype Frequency under Allele Frequency p and Inbreeding Coefficient f ................................................................................................................ 36  Table 4. Comparison of Direct Sample Estimator, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Simple Random Sampling, n=50) ........................ 41  Table 5. Comparison of Direct Sample Estimator, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Simple Random Sampling, n=100) ...................... 42  Table 6. Comparison of Direct Sample Estimator, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Simple Random Sampling, n=200) ...................... 43  Table 7. Comparison of Priors in the Bayesian Estimation (Simple Random Sampling, n=50) .......................................................................................................... 47  Table 8. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling，n=20 per Strata) ...... 61  Table 9. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling，n=40 per Strata) ...... 62  Table 10 Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling，n=80 per Strata) ...... 63  Table 11. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators ................................................................................................. 67  Table 12. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators ................................................................................................. 68  Table 13. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators ................................................................................................. 69  Table 14. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Non-Homogenous Population, Stratified Simple Random Sampling，n=20 per Strata) ....................................................................................... 72  vii Table 15. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Non-Homogenous Population, Proportional to Size Sampling ，n=20) ....................................................................................................................... 72  Table 16. Child Genotype Probability Conditional on the Parents’ Genotypes (Mendel’s Law) ........................................................................................................... 74  Table 17. Family Genotype Distribution for Both Parents One Child Triads (2P1O) 74  Table 18. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling of Families，n=20 Families per Strata) ..................................................................................................... 81  Table 19. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling of Families，n=40 Families per Strata) ..................................................................................................... 82  Table 20. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling of Families，n=80 Families per Strata) ..................................................................................................... 83  Table 21. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators ................................................................................................. 86  Table 22. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators ................................................................................................. 87  Table 23. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators ................................................................................................. 88  Table 24. Comparison of Direct Sample Estimator and Parametric Bayesian Estimators (Single Common f Model) ........................................................................ 99  Table 25. Comparison of Direct Sample Estimator and Parametric Bayesian Estimators (Cluster Specific f Model) ........................................................................ 99  Table 26. Comparison of Design-Based Estimator and Parametric Bayesian Estimator (Cluster Specific f Model) .......................................................................................... 99  Table 27. Estimated Inbreeding Coefficients for Selected SNPs from APOC1 Gene (MAF>10%). ............................................................................................................. 104  Table 28. Estimated Inbreeding Coefficients for Selected SNPs from BDNF Gene (MAF>10%). ............................................................................................................. 105  viii Table 29. Estimated Inbreeding Coefficients for Selected SNPs from BDNF Gene (<1%MAF<5%). ....................................................................................................... 106  ix List of Figures Figure 1. Parameter Space for the Inbreeding Coefficient (f) Conditional on the Allele Frequency (p) Under the General Inbreeding Model. Dark gray area indicates negative f, while light gray area indicates positive f. ................................................... 9  Figure 2. MCMC Diagnostic Plots (History, Posterior Distribution, Autocorrelation and BGR Statistic). ..................................................................................................... 22  Figure 3. Simulation Results of DSE vs. MELE (p=0.1, f=0.05, sample size of 50 or 200). Red dash line is the lower limit of the parameter space for f when p=0.1. ....... 33  Figure 4. Monte Carlo Simulation Standard Error at Accumulative Simulation Iterations. BPELE=Bayesian Pseudo-Empirical Likelihood Estimator, DSE=Direct Sample Estimator, PBE=Parametric Bayesian Estimator. Allele frequency is p=0.3 and the inbreeding coefficient is f=0.05 in the simulation.......................................... 38  Figure 5 Density Plot of Simulated Estimators under Different Settings (Simple Random Sampling). BPELE=Bayesian Pseudo-Empirical Likelihood Estimator, DSE=Direct Sample Estimator, PBE=Parametric Bayesian Estimator. Parameter p is the allele frequency and f is the inbreeding coefficient in the simulation. Panel A, B and C are for sample size 50, 100 and 200 respectively. Red dashed vertical line is the true parameter f; red dotted vertical line is the lower bound of the parameter space for f conditional on allele frequency p. ............................................................................. 44  Figure 6. Density Plot of Simulated Estimators under Different Priors (Simple Random Sampling). BPELE1=Bayesian Pseudo-Empirical Likelihood Estimator with uniform prior (0,1) for f; BPELE2=Bayesian Pseudo-Empirical Likelihood Estimator with uniform prior (0,0.5) for f; DSE=Direct Sample Estimator; PBE1=Parametric Bayesian Estimator with uniform prior (0,1) for p and conditional uniform prior (- p/(1-p),1) for f; PBE2=Parametric Bayesian Estimator with beta prior (0.5,0.5) for p and conditional uniform prior (-p/(1-p),1) for f; PBE3=Parametric Bayesian Estimator with beta prior (0.5,0.5) for p and uniform prior (0,0.5) for f. .................................... 46  Figure 7. Density Plot of Simulated Estimators under Different Settings (Stratified Simple Random Sampling). BPELE=Bayesian Pseudo-Empirical Likelihood Estimator, DBE=Design-Based Estimator, PBE=Parametric Bayesian Estimator. Parameter p is the allele frequency and f is the inbreeding coefficient in the simulation. Panel A, B and C are for stratified sample size 20, 40 and 80 respectively. Red dashed vertical line is the true parameter f; red dotted vertical line is the lower bound of the parameter space for f conditional on allele frequency p. ....................... 64  Figure 8. Density Plot of Simulated Estimators under Different Settings (Proportional to Population Size Sampling). BPELE=Bayesian Pseudo-Empirical Likelihood x Estimator, DBE=Design-Based Estimator, PBE=Parametric Bayesian Estimator. Parameter p is the allele frequency and f is the inbreeding coefficient in the simulation. Panel A, B and C are for stratum sample size 20, 40 and 80 respectively. Red dashed vertical line is the true parameter f; red dotted vertical line is the lower bound of the parameter space for f conditional on allele frequency p. ....................... 70  Figure 9. Density Plot of Simulated Estimators under Different Settings (Stratified Simple Random Sampling). BPELE=Bayesian Pseudo-Empirical Likelihood Estimator, DBE=Design-Based Estimator, PBE=Parametric Bayesian Estimator. Parameter p is the allele frequency and f is the inbreeding coefficient in the simulation. Panel A, B and C are for stratum family sample size 20, 40 and 80 families respectively. Red dashed vertical line is the true parameter f; red dotted vertical line is the lower bound of the parameter space for f conditional on allele frequency p. ................................................................................................................. 84  Figure 10. Density Plot of Simulated Estimators under Different Settings (Proportional to Population Size Sampling). BPELE=Bayesian Pseudo-Empirical Likelihood Estimator, DBE=Design-Based Estimator, PBE=Parametric Bayesian Estimator. Parameter p is the allele frequency and f is the inbreeding coefficient in the simulation. Panel A, B and C are for stratum family sample size 20, 40 and 80 families respectively. Red dashed vertical line is the true parameter f; red dotted vertical line is the lower bound of the parameter space for f conditional on allele frequency p. ................................................................................................................. 89  Figure 11. Distribution of Estimated Inbreeding Coefficients for all Autosomal SNPs (“Figure 26” in Quality Control Report for Genotypic Data, 2012) ......................... 102  Figure 12. Comparison of Inbreeding Coefficient Estimators from HRS Genetic Data. BPELE=Bayesian Pseudo-Empirical Likelihood Estimator, DBE=Design-Based Estimator, PBE=Parametric Bayesian Estimator, SRS=Simple Random Sampling Estimator. Panel A, B and C are for APOC1 SNPs with allele frequency >10%, BDNF SNPs with allele frequency >10%, and BDNF SNPs with allele frequency between 1% and 5%, respectively. Solid triangles represent estimators with 95% CI do not cover zero (a significant deviation from HWE at 5% level). ......................... 107  xi Glossary of Technical Terms Allele A variant of similar DNA sequence located at a given locus Chromosome A packaged and organized structure containing most of the DNA of a living organism dbGap The Database of Genotypes and Phenotypes dbSNP The Database of Single Nucleotide Polymorphisms Diploid cell A cell that has two homologous copies of each chromosome DNA Deoxyribonucleic acid, a molecular that carries most of genetic information of living organism or viruses FIS The correlation between genes within individual relative to the genes from subpopulation FIT The correlation between genes within individual relative to the genes from population FST The correlation between genes within subpopulation relative to the genes from total population Gamete A cell that fuses with another cell during fertilization (conception) in organisms that sexually reproduce Gene A locus (or region) of DNA that encodes a functional RNA or protein product, and is the molecular unit of heredity Gene Drift The change in the frequency of a gene variant (allele) in a population due to random sampling of organisms Gene Flow Transfer of alleles or genes from one population to another Genotype A part of DNA sequence of a cell, and therefore of an organism or individual, which determines a specific characteristic (phenotype) of that cell/organism/individual GWAS Genome-wide association studies Heterozygote Organism’s genotype contains different alleles of a gene Homozygote Organism’s genotype contains same allele pair of a gene xii HWE Hardy-Weinberg Equilibrium, or Hardy-Weinberg Principal, named after Godfrey H. Hardy and Wilhelm Weinberg, states that genetic variation including allele and genotype frequency in a population will remain constant from one generation to the next generation in the absence of disturbing influences IBD Identity by descent, both alleles from two individuals come from the same allele copy in a common ancestor Inbreeding Coefficient (f) or Fixation Index FIS, first introduce by Wright (1921), measures the expected percentage of homozygosity based on a known pedigree (a fully documented genealogy for a fixed system of breeding) Locus (plural) Loci, specific location or position of a gene, DNA sequence, on a chromosome MAF Minor allele frequency, the lowest allele frequency at a locus in a population Mendel’s Law Allele pairs separate randomly, or segregate, from each other during the production of gametes: egg and sperm. Because allele pairs separate during gamete production, a sperm or egg carries only one allele for each inherited trait. When sperm and egg unite at fertilization, each contributes its allele, restoring the paired condition in the offspring. Mutation A permanent change of the nucleotide sequence of the genome of an organism, virus, or extrachromosomal DNA or other genetic elements Panmictic A panmictic population is one where all individuals are potential partners, e.g., no mating restrictions Panmixia or Panmixis, means random mating Pedigree A fully documented genealogy for a fixed system of breeding Phenotype The composite of an organism's observable characteristics or traits SNP Single nucleotide polymorphism, a DNA sequence variation occurring when a single nucleotide — adenine (A), thymine (T), cytosine (C), or guanine (G) — in the genome (or other shared sequence) differs between members of a species (or between paired chromosomes in an individual) *some references from https://en.wikipedia.org 1 Chapter 1: Introduction 1.1 Single Nucleotide Polymorphism (SNP) The human genome contains a complete set of genetic information, which is encoded as DNA sequences within twenty-three chromosome pairs. Most human cells have two versions of each chromosome, one inherited from the father and the other inherited from the mother. Therefore, at each specific location (locus) along the chromosome, there are two versions of the DNA sequence. A variant of the DNA sequence at a given locus is called an allele. In any particular diploid organism like human being, the genotype for each gene comprises of a pair of alleles present at that locus, which are the same in homozygotes and different in heterozygotes. Single nucleotide polymorphism (SNP, pronounced as “snip”) is a DNA sequence variation occurring when a single nucleotide — adenine (A), thymine (T), cytosine (C), or guanine (G) — in the genome (or other shared sequence) differs among members of a species (or between paired chromosomes in an individual). For example, a two-allele SNP may contain two sequenced DNA fragments ATTCCG and ATTTCG, which only differ in the fourth nucleotide, changing from C to T. In this case, we say that there are two alleles: C and T. Most commonly, these variations are often found in DNA non-coding regions between genes. Nevertheless, they may still have an effect on health and development. These are called linked (or indicative) SNPs. Other variations are within a gene or in a regulatory region of a gene. They can affect how human develop disease and/or respond to pathogens, drugs and vaccines, etc. Thus they are called causative SNPs. For example, a single base mutation in the APOE (apolipoprotein E) gene is associated with a higher risk for Alzheimer’s disease (Wolf, Caselli, Reiman, & Valla, 2013). These causative SNPs, therefore, can act as biological markers for early diagnosis. In practice, SNPs are currently used to screen genes that may be associated with disease. They can also be used to study the inheritance of disease genes within families and possible gene-disease associations for more complex diseases such as diabetes, heart disease and cancer, etc. For those cases, usually a group of SNPs work in coordination to manifest a disease condition. Not all single nucleotide changes are called SNPs. Rare polymorphisms that cause disease with high probability are often referred to as mutations. To be classified as a SNP, two or more alleles 2 must each be present in at least one percent (1%) of the population. The lowest allele frequency at a locus in a population is called a minor allele frequency (MAF). There are variations for the SNP frequency among human populations. A SNP that is common in one ethnic group or geographic region may be much rarer than in the other. In the National Center for Biotechnology Information (NCBI)’s Database of Single Nucleotide Polymorphisms (dbSNP), there are more than 10 million SNPs in the human genome (Wheeler, 2007). These SNPs occur throughout the 3-billion- nucleotide human genome with approximately one in every 300 nucleotide base pairs. SNPs have been widely used in genome-wide association studies (GWAS) as gene markers related to diseases or important traits. For example, a case-control study is often used to compare the distribution of SNPs between one healthy control group and another with disease. In order to rule out the effect of population subdivision, inbreeding or other evolutionary influence on the gene disease association and to confirm the independence within a population of an individual’s alleles at a locus, a test of Hardy-Weinberg Equilibrium is often performed for the study sample before any further screening of SNPs. 1.2 Hardy-Weinberg Equilibrium (HWE) If the union of gametes to produce the next generation is random, it can be shown that allele frequencies are constant from generation to generation. The Hardy-Weinberg Equilibrium (HWE), named after Godfrey H. Hardy and Wilhelm Weinberg, states that genetic variation including allele and genotype frequency in a population will remain constant from one generation to the next in the absence of disturbing influences. This principle is an ideal condition since one or more influences such as nonrandom mating, mutation, natural selection, gene flow (migration), and gene drift, are always present in a real population. Non-random mating, mostly caused by inbreeding, is one of the major sources of deviation from HWE by introducing more homozygotes into the population. Mutation, a change of the nucleotide sequence of the genome, introduces possible new alleles and changes allele frequencies in a population. The mutation probability is typically very small, an order of 10-6 or 10-5. Its influence on the population genetics alone is limited (Nei, 1987). Natural selection affects the allele frequencies and sometimes leads to the loss of all alleles except the favored one. Migration links 3 two or more populations together. This results in the transfer of alleles or genes from a more homogeneous population to the other. Migration into or out of a population may be responsible for a change in allele frequencies within such population. Sometimes, immigration introduces new alleles to the established gene pool of the existing population. Gene drift changes allele frequencies due to random sampling in a small population (Masel, 2011). Genes are inherited from parents. Therefore, the alleles in the offspring can be considered as a sample of those in the parents. In a small population, chance plays a great role in deciding each individual’s survival and the ability to reproduce. This may cause certain gene variants to disappear completely. Gene flow transfers alleles or genes from one population to the other. Furthermore, in genetic studies, sampling stratification and small sample size of the study may also statistically alter HWE from observed data. Lastly, even with today’s advanced technology in DNA sequencing, SNPs with genotype error are often found to have substantial excesses or deficiencies of heterozygosity which ultimately results in outlying departure from HWE (Weir B. , 2010). Throughout this dissertation, a single digit subscript is used to represent the allele property and a double digit subscript is for the genotype property unless otherwise specified. To be consistent with the genetics literature, the term frequency is used instead of proportion. Suppose at a single locus from a large monoecious diploid population, there are k alleles A1, A2, …, Ak with allele frequencies ݌ଵ, ݌ଶ, … , ݌௞ (∑ ݌௜ ൌ 1௞௜ୀଵ ). This diploid organism is homozygous at this locus when its cells contain the same allele pair of a gene, denoted as AiAi or (Ai, Ai), i=1,…,k. The organism’s genotype is a homozygote and can be represented as Aii (i=1,…,k). This diploid organism is heterozygous at this locus when its cells contain different alleles of a gene as AiAj, or (Ai, Aj), i, j=1,…,k, i≠j. The organism’s genotype is a heterozygote and can be denoted as Aij (i=1,…,k; j=i+1,…, k). Each organism produces the same number of male and female gametes and has two alleles at this locus. The copies of a gene separate so that each gamete receives only one allele. Each parent randomly contributes a single allele copy to the offspring. With current widely used gene sequence data, we cannot differentiate the allele pairs (Ai, Aj) and (Aj, Ai) in the offspring and both pairs contain the same genetic information, thus the same genotype Aij (j≥i). Therefore, these k alleles result in a total of k(k+1)/2 distinct genotypes A11, A12,…, A1k, A22, …, A2k,…, Akk with corresponding genotype frequencies݌ଵଵ, ݌ଵଶ, … , ݌ଵ௞, ݌ଶଶ, … , ݌ଶ௞, … , ݌௞௞ (∑ ∑ ݌௜௝௞௝ୀ௜ ൌ 1௞௜ୀଵ ). 4 In a sample of size n, there are 2n alleles at a single locus. Let ݊௜௝ be the number of genotype ܣ௜௝, ݅ ൌ 1,… , ݇; ݆ ൌ ݅, … , ݇. Let ݊ ൌ ∑ ∑ ݊௜௝௞௝ୀ௜௞௜ୀଵ be the total sample size, and ݊௟ be the number of allele Al (l=1, …, k) in the sample. Table 1 shows the genotype counts and allele counts from a single locus with k alleles. In order to count the allele Al, we need to count all genotypes containing Al (both vertically and horizontally in Table 1), ݊௟ ൌ ∑ ݊௜௟௟௜ୀଵ ൅ ∑ ݊௟௝௞௝ୀ௟ and ∑ ݊௟௞௟ୀଵ ൌ 2݊. The last equation is true because each subject in the sample has two alleles at the same locus. From Table 1, the sample estimator of genotype frequency and allele frequency are defined as follows:   ̂݌௜௝ ൌ ݊௜௝݊ , ݅ ൌ 1,… , ݇; ݆ ൌ ݅, . . , ݇; and ̂݌௜ ൌ ݊௜ 2݊ , ݅ ൌ 1,… , ݇; ̂݌௟ ൌ ݊௟2݊ ൌ ∑ ݊௜௟௟௜ୀଵ ൅ ∑ ݊௟௝௞௝ୀ௟ 2݊ ൌ 1 2ቆ෍ ̂݌௜௟ ௟ ௜ୀଵ ൅෍ ̂݌௟௝ ௞ ௝ୀ௟ ቇ ൌ ̂݌௟௟ ൅ 12 ቆ෍ ̂݌௜௟ ௟ିଵ ௜ୀଵ ൅෍ ̂݌௟௝ ௞ ௝ୀ௟ାଵ ቇ  (1) Under random mating, organisms produce offspring by the random union of gametes. As defined earlier, ݌௜௜ ൌ ܲሺܣ௜௜ሻ ൌ ܲሺܣ௜ܣ௜ሻ ൌ ܲሺܣ௜ሻܲሺܣ௜ሻ, where the last equality is due to independence of receiving one copy of an allele from parents. Similarly, ݌௜௝ ൌ ܲ൫ܣ௜௝൯ ൌ ܲ൫ܣ௜ܣ௝ ⋃ܣ௝ܣ௜൯ ൌ ܲ൫ܣ௜ܣ௝൯ ൅ ܲ൫ܣ௝ܣ௜൯ ൌ ܲሺܣ௜ሻܲ൫ܣ௝൯ ൅ ܲ൫ܣ௝൯ܲሺܣ௜ሻ ൌ 2ܲሺܣ௜ሻܲ൫ܣ௝൯, where the third equality sign is due to mutually exclusive events and the fourth equality sign is due to independence of receiving one copy of an allele from parents. Therefore, the relationship between genotype frequency ݌௜௝ and allele frequency ݌௜ can be written as   ݌௜௜ ൌ ݌௜ଶ, ݅ ൌ 1,… , ݇;  ݌௜௝ ൌ 2݌௜݌௝, ݅ ൌ 1,… , ݇; ݆ ൌ ݅ ൅ 1,… , ݇.  (2) From above relationship, it can be easily verified that both total genotype frequencies and total allele frequencies equal one, e.g., ∑ ∑ ݌௜௝௞௝ୀ௜ ൌ௞௜ୀଵ ∑ ݌௜ଶ௞௜ୀଵ ൅ ∑ ∑ 2݌௜݌௝௞௝ୀ௜ାଵ௞௜ୀଵ ൌ ሺ∑ ݌௜௞௜ୀଵ ሻଶ ൌ 1. These frequencies (2) define the Hardy-Weinberg Equilibrium. The genotype frequencies of the next generation depend only on the allele frequencies of the current population. Denoting ݌௟ሺ௧ାଵሻ to represent the allele Al frequency at generation t+1, under HWE we have: 5 ݌௟ሺ௧ାଵሻ ൌ ݌௟௟ሺ௧ାଵሻ ൅ 12ቆ෍ ݌௜௟ ሺ௧ାଵሻ௟ିଵ ௜ୀଵ ൅෍ ݌௟௝ሺ௧ାଵሻ ௞ ௝ୀ௟ାଵ ቇ ൌ ൫݌௟ሺ௧ሻ൯ଶ ൅ ݌௟ሺ௧ሻ ቆ෍ ݌௜ሺ௧ሻ ௟ିଵ ௜ୀଵ ൅෍ ݌௝ሺ௧ሻ ௞ ௝ୀ௟ାଵ ቇ ൌ ݌௟ሺ௧ሻ෍ ݌௜ሺ௧ሻ ௞ ௜ୀଵ ൌ ݌௟ሺ௧ሻ. Table 1. Genotype Count from a Single Locus with k Alleles        ♂  ♀  A1  A2  A3  …Al…  Ak  Total  A1  n11  n12  n13  …n1l…  n1k  ෍ ݊ଵ௝ ௞ ௝ୀଵ   A2  n21  n22  n23  …n2l…  n2k  ෍ ݊ଶ௝ ௞ ௝ୀଶ   A3  n31  n32  n33  …n3l…  n3k  ෍ ݊ଷ௝ ௞ ௝ୀଷ   …  Al  …  …  nl1  …  …  nl2  …  …  nl3  …  …  nll  …  …  nlk  …    ෍ ݊௟௝ ௞ ௝ୀ௟   Ak  nk1  nk2  nk3  …nkl…  nkk  ෍ ݊௞௝ ௞ ௝ୀ௞   Total  ෍ ݊௜ଵ ௞ ௜ୀଵ   ෍ ݊௜ଶ ௞ ௜ୀଶ   ෍ ݊௜ଷ ௞ ௜ୀଷ   ෍ ݊௜௟ ௞ ௜ୀ௟   ෍ ݊௜௞ ௞ ௜ୀ௞   n  Note: The greyed upper triangle is the observed genotype count; the lower triangle is a reflection of the observed count under slightly different notation such that ௝݊௜ ൌ ݊௜௝, ݆ ൐ ݅. 6 Thus, the allele frequencies remain constant from generation to generation. The first equality sign can be similarly derived as the second equation in (1). 1.3 Inbreeding Coefficient (f) Among various reasons for the deviation from HWE, our research is motivated by the fact that inbreeding causes the production of offspring from the mating of genetically related individuals. Populations usually do not constitute a single panmictic unit without any mating restriction. They are rather further divided into smaller regions geographically. The inbreeding coefficient f, or fixation index FIS, first introduced by Wright (1921), measures the expected percentage of homozygosity based on a known pedigree (a fully documented genealogy for a fixed system of breeding). In this dissertation, we follow the genetics literature by using f to denote the inbreeding coefficient. This should be differentiated from the traditional use of f in statistics as a density function. The inbreeding coefficient is also defined as the probability that two homologous alleles are identity by descent (IBD), which is, both coming from the same allele copy in a common ancestor. Pedigrees are, however, usually not available for individuals within the study population. Instead, the inbreeding coefficient is an expected value (or a statistical value) derived from individual’s pedigree. It is not directly measureable by looking at the individual’s genome. In another words, its definition exists even if such individual’s genome does not actually contains such gene. When studying a population, f measures the degree of deviation from random pairing of genes, that is, the association between pairs of uniting gametes. A positive f means excessive homozygotes, while negative value means excessive heterozygotes in the system. Although an inbreeding coefficient can be parameterized as genotype specific, such as ௜݂௝ , ݅ ൌ 1, … , ݇; ݆ ൌ ݅, … , ݇, it is often assumed in practice that all pairs of allele frequencies are equally influenced and, therefore, have a common f. Under this general inbreeding model, the genotype frequency ݌௜௝ is a function of the allele frequency ݌௜, ݌௝, and the inbreeding coefficient f. Cavalli-Sforza and Bodmer (1971) explained that the population can be separated into one fully inbred part with relative proportion f and the other bred at random with relative proportion 1 - f. For the inbred population, the individual homozygotes Aii genotype frequency by descent therefore is f times the allele Ai frequency ݌௜, ∑ ݂݌௜௄௜ୀଵ ൌ ݂. For the bred at random population, the genotype frequencies are (1 – 7 f) times the genotype frequencies under HWE in equation (2). When f=1, there is a fully inbred population and f=0 means individuals mate completely at random. Therefore, the total genotype frequencies are the sum of these two fractions.   ݌௜௜ ൌ ሺ1 െ ݂ሻ݌௜ଶ ൅ ݂݌௜ ൌ ݌௜ଶ ൅ ݌௜ሺ1 െ ݌௜ሻ݂, ݅ ൌ 1,… , ݇. ݌௜௝ ൌ 2ሺ1 െ ݂ሻ݌௜݌௝, ݅ ൌ 1,… , ݇; ݆ ൌ ݅ ൅ 1,… , ݇.  (3) From the above model, the total probability ∑ ∑ ݌௜௝௞௝ୀ௜௞௜ୀଵ is still one, as demonstrated below: ෍ ෍ ݌௜௝ ௞ ௝ୀ௜ ൌ ௞ ௜ୀଵ ෍ ෍ ݌௜௝ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ ൅෍ ݌௜௜ ௞ ௜ୀଵ ൌ෍ ෍ ݌௜௝ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ ൅෍ ሾሺ1 െ ݂ሻ݌௜ଶ ൅ ݂݌௜ ௞ ௜ୀଵ ሿ ൌ ෍ ෍ ݌௜௝ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ ൅ ሺ1 െ ݂ሻ෍ ݌௜ଶ ௞ ௜ୀଵ ൅ ݂ ൌ 1. Simplifying the last expression above to solve for f yields:   ݂ ൌ 1 െ ∑ ∑ ݌௜௝ ௞௝ୀ௜ାଵ௞௜ୀଵ 1 െ ∑ ݌௜ଶ௞௜ୀଵ ൌ 1 െ ∑ ∑ ݌௜௝௞௝ୀ௜ାଵ௞௜ୀଵ 2∑ ∑ ݌௜݌௝௞௝ୀ௜ାଵ௞௜ୀଵ ൌ 1 െ ܪைܪா. (4) In the above equation, ܪை ൌ ∑ ݌௜௝௞௜ୀଵ,௝வ௜ is the heterozygotes frequency and ܪா ൌ 2∑ ∑ ݌௜݌௝௞௝ୀ௜ାଵ௞௜ୀଵ ൌ 1 െ ∑ ݌௜ଶ௞௜ୀଵ is the expected heterozygotes frequency with observed allele frequency under HWE. Therefore, f can be expressed as one minus the ratio of the observed proportion of heterozygotes to the expected proportion of heterozygotes under HWE. When k = 2, ݂ ൌ 0 is the sufficient and necessary condition for HWE, in which case ݌௜௝ ൌ 2݌௜݌௝; ݅ ൌ 1, ݆ ൌ 2. However, when k > 2, ݂ ൌ 0 is only a sufficient condition for ݌௜௝ ൌ 2݌௜݌௝; ݅ ൌ 1,… , ݇; ݆ ൌ ݅ ൅ 1,… , ݇ since it only implies ∑ ∑ ݌௜௝௞௝ୀ௜ାଵ௞௜ୀଵ ൌ 2∑ ∑ ݌௜݌௝௞௝ୀ௜ାଵ௞௜ୀଵ . For a simple illustration, considering a simple bi-allelic case k=2 with alleles A and a, the alleles frequency are ݌ଵ ൌ ݌ and ݌ଶ ൌ 1 െ ݌. So the inbreeding coefficient can be expressed as   ݂ ൌ 1 െ ݌ଵଶ2݌ሺ1 െ ݌ሻ  (5) 8 To understand why the inbreeding coefficient is a measure of correlation, suppose we draw two gametes (G1 and G2) at random from this population with allele A frequency p. Assign indicator variable Y=1 if a gamete is of allele A and Y=0 if it is of allele type a. Therefore we see immediately that over replicates of this sampling process, ܧሺݕሻ ൌ 1 ൈ ݌ ൅ 0 ൈ ሺ1 െ ݌ሻ ൌ ݌, ܧሺݕଶሻ ൌ ݌ , ܸܽݎሺݕሻ ൌ ܧሺݕଶሻ െ ሺܧሺݕሻሻଶ ൌ ݌ െ ݌ଶ ൌ ݌ሺ1 െ ݌ሻ and ݒ݋ܥሺݕଵ, ݕଶሻ ൌ ܧሺݕଵݕଶሻ െ ܧሺݕଵሻܧሺݕଶሻ ൌ ݌ െ ௣భమଶ െ ݌ଶ ൌ ݌ሺ1 െ ݌ሻ െ ݌ଵଶ/2 as shown in Table 2. The correlation between two homologous genes in uniting gametes can be expressed as ஼௢௩ሺ௬భ,௬మሻඥ௏௔௥ሺ௬భሻ௏௔௥ሺ௬మሻ ൌ ௣ሺଵି௣ሻି೛భమమ ௣ሺଵି௣ሻ ൌ 1 െ ௣భమ ଶ௣ሺଵି௣ሻ, where ݌ଵଶ is the observed heterozygote frequency (HO) and 2݌ሺ1 െ ݌ሻ is the expected heterozygosity (HE) under HWE (Balding, Bishop, & Cannings, 2007). This is exactly the same as was shown in (5). Table 2. Pattern of Union of Gametes in the Total Population  Gamete 2 A a Total Gamete 1 A ݌ െ ݌ଵଶ/2 ݌ଵଶ/2 p a ݌ଵଶ/2 1 െ ݌ െ ݌ଵଶ/2 1-p Total p 1-p 1 Notice that all genotype frequencies range from 0 to 1 under the above inbreeding model. There is a natural constraint for f: 0 ൑ ݌௜௜ ൌ ሺ1 െ ݂ሻ݌௜ଶ ൅ ݂݌௜ ൑ 1 ⇒ െ݌௜1 െ ݌௜ ൑ ݂ ൑ 1 ൅ 1 ݌௜ , 0 ൑ ݌௜௝ ൌ 2ሺ1 െ ݂ሻ݌௜݌௝ ൑ 1 ⇒ ݂ ൑ 1. Therefore, conditional on the allele frequencies, we have a constraint for f: െ݌௠௜௡ 1 െ ݌௠௜௡ ൑ ݂ ൑ 1,where ݌௠௜௡ ൌ ݉݅݊ሺ݌ଵ, … , ݌௞ሻ . Although the inbreeding coefficient f is considered as a measure of correlation, its range is not [- 1,1] as it is for the product moment correlation coefficient. The lower limit is dependent on the 9 distribution of actual population allele frequency. For k=2, Figure 1 shows that the parameter space for the inbreeding coefficient f is conditional on the allele frequency p. When p=0.5, the range of f is fully [-1, 1]. The range slowly decreases to [0, 1] when the allele frequency moves toward either end (0 or 1). For the special case when the MAF is at the cutoff for the definition of SNP, e.g., p=0.01, the parameter space for f is [-0.0101, 1]. Figure 1. Parameter Space for the Inbreeding Coefficient (f) Conditional on the Allele Frequency  (p) Under the General Inbreeding Model. Dark gray area indicates negative f, while light gray area  indicates positive f.  When the inbreeding coefficient is estimated from a single sample, it measures the deviation from random pairing of the genes. Sampling from a whole population may be considered as a random sample from an infinitely large pool of zygotes (Curie-Cohen, 1982). Suppose a sample of size n subjects are taken from the target population, and we count the number of subjects with each genotype. By substituting sample values in equation (4), a Direct Sample Estimator (DSE) ݂ሺ௦ሻ can be derived as a function of observed and expected number of heterozygotes under HWE (Li & Horvitz, 1953):   ݂ሺ௦ሻ ൌ 1 െ 2݊∑ ∑ ݊௜௝ ௞௝ୀ௜ାଵ௞௜ୀଵ ∑ ∑ ݊௜ ௝݊௞௝ୀ௜ାଵ௞௜ୀଵ .  (6) The estimator compares this sample with an infinitely large population with no mutation, migration, natural selection and with random mating except for a fixed number of inbred mating. 10 The above general inbreeding model reduces to HWE when f=0. Therefore, the HWE model is nested under this general model. Instead of a chi-square test based on the expected and the observed counts, the testing of HWE can be constructed as the testing of the null hypothesis H0: f=0 versus the alternative hypothesis H1: f≠0. In GWAS, researchers are more interested in testing the HWE than in estimating the inbreeding coefficient f. Genes that deviate from HWE are, therefore, screened out in the gene-disease association studies. However, a statistical test based on the frequentist p-value does not always have a direct interpretability to geneticists. The test does not directly measure the effect size f, i.e., the magnitude of deviation. A relatively large f with a non-significant p-value does not prove that a population is in HWE. On the other hand, when the sample size increases toward infinity, a highly significant p-value does not necessarily warrant a final conclusion of departure from HWE when the estimated f is often only barely worth mentioning. For studies with multiple SNPs screening, a simple Bonferroni correction is often performed to adjust the final p-values of tests (Weir, Hill, & Cardon, 2004). It is not clear whether the estimated f with its associated confidence interval should also be adjusted by the multiple estimations in a similar way. For human genetics, there is another challenge because the amount of inbreeding usually is extremely small. Table 7.3 of Cavalli-Sforza & Bodmer (1971) gives estimates of the inbreeding coefficients for a wide range of populations. The average of the inbreeding coefficients in human populations is generally less than 0.0001; an inbreeding coefficient larger than 0.01 is considered exceptional. 1.4 Multinomial Model for the Genotype Distribution For a total sample size of n subjects with observed numbers of genotypes ݊௜௝ (∑ ∑ ݊௜௝௞௝ୀ௜௞௜ୀଵ ൌ ݊) and genotype frequency ݌௜௝ , i=1,…, k, j ≥ i, the k(k+1)/2 dimensional vector ࢔ ൌ ሺ݊ଵଵ, ݊ଵଶ, … , ݊ଵ௞, ݊ଶଶ, … , ݊ଶ௞, … , ݊௞௞ሻ follows a multinomial distribution with parameter ࢖෥ ൌ ሺ݌ଵଵ, ݌ଵଶ, … , ݌ଵ௞, ݌ଶଶ, … , ݌ଶ௞, … , ݌௞௞ሻ: ܲሺ࢔|࢖෥ሻ ൌ ݊!∏ ∏ ݊௜௝!௞௝ୀ௜௞௜ୀଵ ෑ ෑ ݌௜௝ ௡೔ೕ ௞ ௝ୀ௜ ௞ ௜ୀଵ . We use ࢖෥ temporarily here for the genotype frequencies, to distinguish it from the notation for the allele frequencies. The marginal distribution of ݊௜௝ is a binomial with parameter ݌௜௝: 11 ܲ൫݊௜௝ห݌௜௝൯ ൌ ݊!݊௜௝! ൫݊ െ ݊௜௝൯! ݌௜௝ ௡೔ೕሺ1 െ ݌௜௝ሻ௡ି௡೔ೕ. Therefore, the moment estimators of the genotype frequency and allele frequency can be estimated as sample proportions: ̂݌௜௝ ൌ ௡೔ೕ௡ and ̂݌௜ ൌ ଶ௡೔೔ାሺ௡ି௡೔೔ሻ ଶ௡ ൌ ଶ௡೔೔ା∑ ௡೔ೕೕ:ೕಯ೔ ଶ௡ , respectively. They are both unbiased estimators since ܧሺ̂݌௜௝ሻ ൌ ܧ ቀ݊௜௝݊ ቁ ൌ ݌௜௝, and ܧሺ̂݌௜ሻ ൌ ܧ ቆ 2݊௜௜ ൅ ∑ ݊௜௝௝:௝ஷ௜ 2݊ ቇ ൌ ݌௜௜ ൅ 1 2෍ ݌௜௝ ൌ௝:௝ஷ௜ ݌௜. However, the direct sample estimator of the inbreeding coefficient, ݂ሺ௦ሻ ൌ 1 െ 2݊∑ ∑ ݆݊݅ ݆݇ൌ݅൅1݇݅ൌ1 ∑ ∑ ݆݆݊݅݊݇ൌ݅൅1݇݅ൌ1 ൌ 1 െ ∑ ∑ ̂݌௜௝௞௝ୀ௜ାଵ௞௜ୀଵ 1 െ ∑ ̂݌௜ଶ௞௜ୀଵ , is consistent but biased (Gorroochurn & Hodge, 2006). Using the delta method, Cohen (1982) showed that the bias is of the order of ݊ିଵ. The multinomial model can be further re-parameterized as: ܲሺ࢖|࢔, ݂ሻ ൌ ݊!∏ ∏ ݊௜௝!௞௝ୀ௜௞௜ୀଵ ෑ ሺ ሺ1 െ ݂ሻ݌௜ଶ ൅ ݂݌௜ሻ௡೔೔ ௞ ௜ୀଵ ෑ ෑ ሺ2ሺ1 െ ݂ሻ݌௜݌௝ሻ௡೔ೕ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ , where ࢖ ൌ ሺ݌ଵ, ݌ଶ, … , ݌௞ሻ is a ݇ ൈ 1 dimensional vector with ݌௞ ൌ 1 െ ∑ ݌௜௞ିଵ௜ୀଵ . Therefore, the log-likelihood is ݈ሺ࢖, ݂ሻ ൌ log ݊!∏ ∏ ݊௜௝!௞௝ୀ௜௞௜ୀଵ ൅෍ ݊௜௜ log൫ ሺ1 െ ݂ሻ݌௜ଶ ൅ ݂݌௜൯ ௞ ௜ୀଵ ൅෍ ෍ ݊௜௝ log൫2ሺ1 െ ݂ሻ݌௜݌௝൯ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ ൌ log ݊!∏ ∏ ݊௜௝!௞௝ୀ௜௞௜ୀଵ ൅෍ ݊௜௜ log൫ ሺ1 െ ݂ሻ݌௜ ൅ ݂൯ ௞ ௜ୀଵ ൅෍ ݊௜௜ ௞ ௜ୀଵ logሺ݌௜ሻ ൅෍ ෍ ݊௜௝ log 2 ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ ൅෍ ෍ ݊௜௝ logሺ1 െ ݂ሻ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ ൅෍ ෍ ݊௜௝logሺ݌௜ሻ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ ൅෍ ෍ ݊௜௝logሺ݌௝ሻ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ 12 ൌ ܥ ൅෍ ݊௜௜ log൫ሺ1 െ ݂ሻ݌௜ ൅ ݂൯ ௞ ௜ୀଵ ൅෍ ෍ ݊௜௝ logሺ1 െ ݂ሻ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ ൅෍ ෍ ݊௜௝logሺ݌௜ሻ ௞ ௝ୀ௜ ௞ ௜ୀଵ ൅෍ ෍ ݊௜௝logሺ݌௝ሻ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ ൌ ܥ ൅෍ ݊௜௜ log൫ሺ1 െ ݂ሻ݌௜ ൅ ݂൯ ௞ ௜ୀଵ ൅෍ ෍ ݊௜௝logሺ1 െ ݂ሻ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ ൅෍ ሺ݊௜ െ ݊௜௜ሻlogሺ݌௜ሻ ௞ ௜ୀଵ ൌ ܥ ൅෍ ݊௜௜ log൫ሺ1 െ ݂ሻ݌௜ ൅ ݂൯ ௞ିଵ ௜ୀଵ ൅෍ ෍ ݊௜௝logሺ1 െ ݂ሻ ௞ ௝ୀ௜ାଵ ௞ ௜ୀଵ ൅෍ ሺ݊௜ െ ݊௜௜ሻ logሺ݌௜ሻ ௞ିଵ ௜ୀଵ ൅ ݊௞௞ log ൭ሺ1 െ ݂ሻሺ1 െ෍ ݌௜ ௞ିଵ ௜ୀଵ ሻ ൅ ݂൱ ൅ ሺ݊௞ െ ݊௞௞ሻlogሺ1 െ෍ ݌௜ ௞ିଵ ௜ୀଵ ሻ , where ܥ ൌ log ௡!∏ ∏ ௡೔ೕ!ೖೕస೔ೖ೔సభ ൅ ∑ ∑ ݊௜௝ log 2 ௞௝ୀ௜ାଵ௞௜ୀଵ and ݊௜ ൌ ∑ ݊௜௟௞௟ୀ௜ ൅ ∑ ݊௠௜௜௠ୀଵ . 1.4.1 Existing Frequentist Estimator Differentiating the above log-likelihood with respect to ݌௜ and f and setting these equal to 0 yields: ߲݈ ߲݌௜ ൌ ݊௜௜ሺ1 െ ݂ሻ ሺ1 െ ݂ሻ݌௜ ൅ ݂ ൅ ݊௜ െ ݊௜௜ ݌௜ െ ݊௞௞ሺ1 െ ݂ሻ ሺ1 െ ݂ሻ݌௞ ൅ ݂ െ ݊௞ െ ݊௞௞ ݌௞ ൌ 0, ߲݈ ߲݂ ൌ෍ ݊௜௜ሺ1 െ ݌௜ሻ ሺ1 െ ݂ሻ݌௜ ൅ ݂ ௞ ௜ୀଵ െ෍ ݊௜௝1 െ ݂ ௞ ௜ୀଵ,௝வ௜ ൌ 0. By solving the above equations numerically using the Newton’s method (Monohan, 2011), the estimators of ̂݌௜ and መ݂ can be derived. The Hessian matrix, including the following second partial derivatives of the log-likelihood, can also be estimated ߲ଶ݈ ߲݌௜ଶ ൌ െ ݊௜௜ሺ1 െ ݂ሻଶ ൫ሺ1 െ ݂ሻ݌௜ ൅ ݂൯ଶ െ ݊௜ െ ݊௜௜݌௜ଶ ൅ ݊௞௞ሺ1 െ ݂ሻଶ ൫ሺ1 െ ݂ሻ݌௞ ൅ ݂൯ଶ ൅ ݊௞ െ ݊௞௞݌௞ଶ , and ߲ଶ݈ ߲݂ଶ ൌ െ෍ ݊௜௜ሺ1 െ ݌௜ሻଶ ൫ሺ1 െ ݂ሻ݌௜ ൅ ݂൯ଶ ൅෍ ݊௜௝ሺ1 െ ݂ሻଶ ௞ ௜ୀଵ,௝வ௜ ௞ ௜ୀଵ . 13 The variance of the estimator can be calculated as ݎܽݒ൫ መ݂൯ ൌ െܧሺ߲2݈߲݂2ሻ and ݎܽݒሺ̂݌௜ሻ ൌ െܧሺ ߲2݈ ߲2݅݌ ሻ using the information matrix. When k=2, let p be the population proportion of allele A, and q be the proportion of allele a, where p + q =1. Under the general inbreeding model, the genotype frequencies can be reparametrized as ቐ ݌ଵଵ ൌ ݌ଶ ൅ ݂ݍ݌ ൌ ሺ1 െ ݂ሻ݌ଶ ൅ ݂݌, ݌ଵଶ ൌ 2ݍ݌ሺ1 െ ݂ሻ, ݌ଶଶ ൌ ݍଶ ൅ ݂ݍ݌ ൌ ሺ1 െ ݂ሻݍଶ ൅ ݂ݍ. The log-likelihood is given by ݈ሺ݌, ݂ሻ ൌ ݈݃݋ ൜ ݊!݊ଵଵ! ݊ଵଶ! ݊ଶଶ! ൫ሺ1 െ ݂ሻ݌ ଶ ൅ ݂݌൯௡భభሺ2ሺ1 െ ݂ሻݍ݌ሻ௡భమ൫ሺ1 െ ݂ሻݍଶ ൅ ݂ݍ൯௡మమൠ ൌ ݈݃݋ ݊!݊ଵଵ! ݊ଵଶ! ݊ଶଶ! ൅ ݊ଵଵ ݈݃݋൫ሺ1 െ ݂ሻ݌ ଶ ൅ ݂݌൯ ൅ ݊ଵଶ ݈݃݋ሺ2ሺ1 െ ݂ሻݍ݌ሻ ൅ ݊ଶଶ݈݋ ݃൫ሺ1 െ ݂ሻݍଶ ൅ ݂ݍ൯ . Letting q = 1 – p and taking the first derivatives with respective to p and f and setting these equal to 0, we have ۖە ۔ ۖ݌߲݈߲ۓ ൌ ݊ଵଵሺ2݌ሺ1 െ ݂ሻ ൅ ݂ሻ ሺ1 െ ݂ሻ݌ଶ ൅ ݂݌ ൅ ݊ଵଶሺ2ሺ1 െ ݂ሻሺ1 െ 2݌ሻሻ 2ሺ1 െ ݂ሻݍ݌ ൅ ݊ଶଶሺሺ2݌ െ 2ሻሺ1 െ ݂ሻ െ ݂ሻ ሺ1 െ ݂ሻݍଶ ൅ ݂ݍ ൌ 0, ߲݈ ߲݂ ൌ ݊ଵଵሺ݌ െ ݌ଶሻ ሺ1 െ ݂ሻ݌ଶ ൅ ݂݌ ൅ ݊ଵଶሺെ2ݍ݌ሻ 2ሺ1 െ ݂ሻݍ݌ ൅ ݊ଶଶሺݍ െ ݍଶሻ ሺ1 െ ݂ሻݍଶ ൅ ݂ݍ ൌ 0. In order to simplify the presentation, let ݎ ൌ ݂/ሺ1 െ ݂ሻ; then ݂ ൌ ݎ/ሺݎ ൅ 1ሻ. Substituting f into the above equations gives ە ۔ ݊ۓଵଵሺ2݌ ൅ ݎሻ݌ሺ݌ ൅ ݎሻ ൅ ݊ଵଶሺ1 െ 2݌ሻ ݍ݌ ൅ െ݊ଶଶሺ2ݍ ൅ ݎሻ ݍሺݍ ൅ ݎሻ ൌ 0, ݊ଵଵݍ ݌ ൅ ݎ െ ݊ଵଶ ൅ ݊ଶଶ݌ ݍ ൅ ݎ ൌ 0. Multiplying by ሺ1 െ 2݌ሻ/ݍ݌ in the second equation and adding these two equations together to eliminate the middle term yields ݊ଵଵሺ2݌ ൅ ݎሻ ݌ሺ݌ ൅ ݎሻ ൅ ݊ଵଵݍ ݌ ൅ ݎ ∗ 1 െ 2݌ ݍ݌ ൌ ݊ଶଶሺ2ݍ ൅ ݎሻ ݍሺݍ ൅ ݎሻ െ ݊ଶଶ݌ ݍ ൅ ݎ ∗ 1 െ 2݌ ݍ݌ ⟺ 14 ݊ଵଵሺ2ݍ݌ ൅ ݎݍ ൅ ݍ െ 2ݍ݌ሻ ݍ݌ሺ݌ ൅ ݎሻ ൌ ݊ଶଶሺ2ݍ݌ ൅ ݎ݌ െ ݌ ൅ 2݌ଶሻ ݍ݌ሺݍ ൅ ݎሻ ⟺ ݊ଵଵሺݍ ൅ ݎݍሻ ݌ ൅ ݎ ൌ ݊ଶଶሺ݌ ൅ ݎ݌ሻ ݍ ൅ ݎ ⟺ ݊ଵଵݍ ݌ ൅ ݎ ൌ ݊ଶଶ݌ ݍ ൅ ݎ . Combining above result with earlier equation ௡భభ௤௣ା௥ െ ݊ଵଶ ൅ ௡మమ௣ ௤ା௥ ൌ 0 gives ൞ ݊ଵଵݍ ݌ ൅ ݎ ൌ ݊ଵଶ 2 , ݊ଶଶ݌ ݍ ൅ ݎ ൌ ݊ଵଶ 2 . Rewriting the above equation only with respect to p and r gives ൞ ݊ଵଵሺ1 െ ݌ሻ ൌ ݊ଵଶ2 ݌ ൅ ݊ଵଶ 2 ݎ, ݊ଶଶ݌ ൌ ݊ଵଶ2 ሺ1 െ ݌ሻ ൅ ݊ଵଶ 2 ݎ. Solving the above equation for p, we have estimator ̂݌ ൌ ଶ௡భభା௡భమଶሺ௡భభା௡భమା௡మమሻ. This estimator is also valid when f=1 (݊ଵଶ ൌ 0 in such case). Furthermore, solving the above equation for r, we have ̂ݎ ൌ ଶ௡భభሺଵି௣ොሻ௡భమ െ ̂݌. Therefore, መ݂ ൌ ̂1ݎ ൅ ̂ݎ ൌ 1 െ 1 1 ൅ ̂ݎ ൌ 1 െ ݊ଵଶ 2݊ଵଵሺ1 െ ̂݌ሻ െ ݊ଵଶ̂݌ ൅ ݊ଵଶ ൌ 1 െ ݊ଵଶሺ2݊ଵଵ ൅ ݊ଵଶሻሺ1 െ ̂݌ሻ ൌ 1 െ 2݊ ∗ ݊ଵଶ ሺ2݊ଵଵ ൅ ݊ଵଶሻሺ2݊ଶଶ ൅ ݊ଵଶሻ , which is a special case of ݂ሺ௦ሻ ൌ 1 െ ଶ௡∑ ∑ ௡೔ೕ ೖೕస೔శభೖ೔సభ ∑ ∑ ௡೔௡ೕೖೕస೔శభೖ೔సభ when k=2. Further simplification yields   መ݂ ൌ 1 െ 2݊ ∗ ݊ଵଶሺ2݊ଵଵ ൅ ݊ଵଶሻሺ2݊ଶଶ ൅ ݊ଵଶሻ ൌ 1 െ ݊ଵଶ 2݊̂݌ሺ1 െ ̂݌ሻ,   (7) 15 where ݊ଵଶ is the observed number of heterozygotes, and 2݊̂݌ሺ1 െ ̂݌ሻ is the expected number of heterozygotes under HWE (f=0). Therefore, መ݂ can be treated as a measure from the deviation in heterozygosity from the expected under random mating. The Hessian matrix can be calculated by taking the second partial derivatives of the log-likelihood. However, even for k=2, many terms are involved to simply the second derivatves. Without showing that at least one second partial derivative is negative and that the Jacobian of the second partial derivatives is positive, the solution derived in (7) may not be the MLE. From equation (7), it is obvious that መ݂ ൑ 1. The equality sign holds when there is no observed heterozygotes, e.g., ݊ଵଶ ൌ 0. When the population allele A frequency p is known, from above equation ݊ଵଵሺ1 െ ݌ሻ ൌ ௡భమଶ ݌ ൅ ௡భమ ଶ ݎ , we have መ݂ ൌ 1 െ ௡భమ ሺଶ௡భభା௡భమሻሺଵି௣ሻ ൌ ଶ௡భభିሺଶ௡భభା௡భమሻ௣ ሺଶ௡భభା௡భమሻሺଵି௣ሻ ൌ ି௣ା మ೙భభమ೙భభశ೙భమ ଵି௣ ൐ ି௣ ଵି௣ . Similarly, from equation ݊ଶଶ݌ ൌ ௡భమ ଶ ሺ1 െ ݌ሻ ൅ ௡భమ ଶ ݎ and conditional on population allele a frequency q, we have መ݂ ൌ ି௤ା మ೙మమ మ೙మమశ೙భమ ଵି௤ ൐ ି௤ ଵି௤. Therefore, for this special case of k=2, conditional on the allele frequency p (or equivalently, q), the estimator መ݂ satisfies the constraint ݉ܽݔ ሺ ି௣ଵି௣ , ି௤ ଵି௤ሻ ൌ ି௠௜௡ ሺ௣,௤ሻ ଵି௠௜௡ ሺ௣,௤ሻ ൑ መ݂ ൑ 1. However, in most genetic studies, the population allele frequency is unknown and needs to be estimated from the data. The estimator ̂݌ is included in the calculation of መ݂. Therefore, መ݂ may be outside of the parameter space induced from the inbreeding model. When the allele frequency p is close to 0.5, this is not likely to be a serious issue since the parameter space for f is in the [-1, 1] interval. However, when the minor allele frequency (MAF) approaches zero, the lower limit of f shrinks toward zero (Figure 1). The estimator may be seriously outside of parameter space, especially when the sample size is small. When the sample size goes to infinity, the consistent estimator ̂݌ converges to the parameter p. Under this large sample size condition, asymptotically the estimator መ݂ resides within its parameter space for the reason we just outlined earlier. Using the property of binomial distribution and Taylor expansion, Cohen (1982) showed that ܧ൫ መ݂൯ ൌ ܧ ൬1 െ ݊ଵଶ2ݍ̂݌ො݊൰ ൌ 1 െ ܧ ൬ ݊ଵଶ 2ݍ̂݌ො݊൰ 16 ൌ 1 െ 2ሺ1 െ ݂ሻݍ݌2݊݊ݍ݌ െ ݍ݌ሺ݂ ൅ 1ሻ ൅ 2݊ݍ݌ሺ݌ െ ݍሻଶሺ1 െ ݂ሻ ൫2݊ݍ݌ െ ݍ݌ሺ݂ ൅ 1ሻ൯ଶ െ 2݊ݍ݌ሺ݌ െ ݍሻଶሺ1 ൅ ݂ሻ ∗ 2ሺ1 െ ݂ሻ݊ݍ݌ ൫2݊ݍ݌ െ ݍ݌ሺ݂ ൅ 1ሻ൯ଷ ൅ ܱ ൬1݊൰ ൌ ݂ ൅ ܱ ൬1݊൰. Thus, መ݂ has a bias of the order of ݊ିଵ . Moreover, Cohen (1982) showed that the asymptotic variance of the estimator መ݂ is given by ܸܽݎ൫ መ݂൯ ൌ ܸܽݎ ൬1 െ ݊ଵଶ2ݍ̂݌ො݊൰ ൌ ܸܽݎ ൬ ݊ଵଶ 2ݍ̂݌ො݊൰ ൌ 2ሺ1 െ ݂ሻ݊ݍ݌ሾ1 െ 2ሺ1 െ ݂ሻݍ݌ሿ൫2݊ݍ݌ െ ݍ݌ሺ݂ ൅ 1ሻ൯ଶ ൅ ሺ2ሺ1 െ ݂ሻ݊ݍ݌ሻ ଶ2݊ݍ݌ሺ݌ െ ݍሻଶሺ1 ൅ ݂ሻ െ 4ሺ1 െ ݂ሻ݊ݍ݌൫2݊ݍ݌ െ ݍ݌ሺ݂ ൅ 1ሻ൯2݊ݍ݌ሺ݌ െ ݍሻଶሺ1 െ ݂ሻ ൫2݊ݍ݌ െ ݍ݌ሺ݂ ൅ 1ሻ൯ସ ൅ ܱ ൬ 1݊ଶ൰ ൌ ݊ ଷሺ1 െ ݂ሻሺ2ݍ݌ሻଶሾ2ݍ݌ െ ሺ1 െ ݂ሻሺ2ݍ݌ሻଶ െ ሺ1 െ ݂ሻଶ2ݍ݌ሺ݌ െ ݍሻଶሿ ൫2݊ݍ݌ െ ݍ݌ሺ݂ ൅ 1ሻ൯ସ ൅ ܱ ൬ 1 ݊ଶ൰ ൎ ݊ ଷሺ1 െ ݂ሻሺ2ݍ݌ሻଶ2ݍ݌ሾ1 െ 2ݍ݌ሺ1 െ ݂ሻ െ ሺ݌ െ ݍሻଶሺ1 െ ݂ሻଶሿ ሺ2݊ݍ݌ሻସ ൌ ሺ1 െ ݂ሻሾ1 െ 2ݍ݌ሺ1 െ ݂ሻ െ ሺ݌ െ ݍሻ ଶሺ1 െ ݂ሻଶሿ 2݊ݍ݌ .  (8) The asymptotic sample variance of the estimator መ݂ is dependent on p and f. Since ݍ݌ ൌ ݌ሺ1 െ ݌ሻ ൌ െቀ݌ െ ଵଶቁ ଶ ൅ ଵସ, pq is an increase function in 0 ൏ ݌ ൑ ଵ ଶ and reaches its maximum value at p=q=1/2. At the same time, ሺ݌ െ ݍሻଶ reaches its minimum value of zero at p=q=1/2. Several observations for the asymptotic variance estimator can be made from equation (8): 1.) The asymptotic variance can be further simplified as ܸܽݎ൫ መ݂൯ ൌ ௙ሺଶି௙ሻଶ௡௣௤ ൅ ሺ௙ିଵሻሺଶ௙ିଵሻ ௡ . When p=1/2, the variance of መ݂ towards ሺଵି௙ሻሺଵା௙ሻ௡ and it reaches its minimal value; 17 2.) When allele frequency is very small close to zero, the variance of መ݂ towards ሺଵି௙ሻିሺଵି௙ሻయ ଶ௡௣௤ . Therefore, small increase of f may have large increase in its variance when the allele frequency is small (Cohen, 1982). 3.) When the inbreeding coefficient is f=0, the variance of መ݂ becomes ଵିଶ௣௤ିሺ௣ି௤ሻమଶ௡௣௤ ൌ ଵିሺ௣ା௤ሻమାଶ௣௤ ଶ௡௣௤ ൌ ଵ ௡. It does not depend on the allele frequency, but only depends on the sample size (Cohen, 1982). Under the null hypothesis of f=0, a Z-test statistic can be calculated as ܼ ൌ ௙መି଴ଵ/√௡ ൌ √݊ መ݂ . The Z-test of HWE is strongly related to the sample size and observed inbreeding coefficient. For example, when the observed inbreeding coefficient is as moderate as 0.196, the test will reach a statistical significance at 0.05 level when the sample size is 100 or larger. 4.) For the estimator of sample variance of the estimated inbreeding coefficient ܸܽݎ෢ ൫ መ݂൯, an estimator of ̂݌ and መ݂ can be used in the equation (8) instead. Except for the above simple bi-allelic model (k=2), generally the estimator መ݂ is not equal to the direct sample estimator ݂ሺ௦ሻ for k≥3. This is not surprising since homozygotes with alleles that are identity by decent (IBD) only provide partial information than the alleles that are not IBD under the influence of inbreeding (Gorroochurn & Hodge, 2006). Although the solution of f cannot be explicitly written when k≥3, it is proven to be biased when k=2 (Li & Horvitz, 1953). Unfortunately, all eight estimators for f proposed by Li & Horvitz (1953) including the above two ሺ መ݂, ݂ሺ௦ሻሻ, are biased estimators, even though all of them are essentially identical when k=2. 1.4.2 Bayesian Method for HWE and Estimation In parallel with the recent increased interest in testing HWE among human genetics studies, Bayesian methods have found enormous applications in GWAS where there are large amount of single nucleotide polymorphism (SNPs) examined from the same or different genes. The rationale for controlling of the family-wise error rate is not obvious in a genome-wide context when it is practically not reasonable to expect all nulls to be true. Therefore, traditional frequentist methods such as Bonferroni correction to control the family level type I error may pose a major risk that true genetic association might be discarded. Even if we accept the concept that type I error need 18 be controlled, the choice of a specific threshold for all SNPs in a study is difficult (Stephens 2009). In Health and Retirement Study genotype Quality Control report (University of Washington, 2012), a p-value cutoff of 0.0001 is recommended for testing HWE. However, other literature propose a more strict cutoff as low as 5×10-7 (Pongpanich, Sullivan, & Tzeng, 2010). Bayesian method, on the other hand, can provide measure of evidence that can be interpreted directly and even be compared among different SNPs without worrying about the interpretation of the p-value and a fixed error probability α resulted in multiple testing. The main advantage of Bayesian method is its flexibility to incorporate non-sampling information into analysis. For example, under HWE assumption, dominant or co-dominant genetic model as well as prior belief of allele frequency distributions can be considered altogether into the estimation of the allele frequency from nominal measure of the phenotype (Gunel & Wearden, 1995). Using the multinomial distribution, under both HWE and general model, conjugate Dirichlet priors are often applied to the allele or genotype frequencies to derive the Bayes Factor for model selection (Wakefield, 2010). Conssoni etc. (2008) discussed various compatible priors, including Kullback-Leibler Conjugate Approximation (KLCA) and Jeffreys’ Conditioning (JC) priors, for the testing of HWE. Later they further introduced a class of objective intrinsic priors for the same testing of HWE model and carried out a sensitivity analysis for the prior (Consonni, Morenob, & Venturinic, 2011). When there are potential genotype error associated with the gene polymorphism, small sample size, or the existence of population stratification in the sampling design, the study cohort may not be exactly in HWE. On the other hand, a statistical significant test with point estimate close to zero does not likely to provide convincing evidence of departure from HWE. As an alternative to the hypothesis testing, estimation provides more useful information in understanding of the magnitude of departure from HWE. The Bayesian method uses a prior distribution for the effect-size parameter while not categorizes it into zero and non-zero, and then focus on estimating the parameter instead of testing whether it is equal to zero. With the introduction of Markov Chain Monte Carlo (MCMC), the posterior density is available so that the uncertainty of the estimation could be displayed visually. Bayesian credible interval could be constructed from the posterior distribution. The posterior probability of parameter f within interval suggested by the National Research Council (1996) therefore could be calculated. This helps scientists to incorporate 19 background information and reduce data dependence when investigating specific research question. One of the other advantages of the Bayesian estimation method is that it naturally incorporates prior information about parameter f learned from previous researches. For example, studies from isozyme loci and pedigrees suggest that the inbreeding coefficient f is small, usually less than 0.03 and often less than 0.01 (Chakraborty & Jin, 1992; Doeder, Escobar, Kadane, & Balazs, 1998). Under the multinomial model with conjugate beta prior, the prior is weighted against small or large departure from HWE. A step prior, which gives equal weight to both small and large departures from HWE, could be constructed as the conditional distribution of φሺ݂|݌ሻ or the joint distribution φሺ݂, ݌ሻ uniformly distributed over the range of f|p (Shoemaker, Painter, & Weir, 1998). Using dominant phenotype data from seven different populations of an endangered orchid specie, Holsinger applied a hierarchical binomial model with non-informative beta priors for the parameters f an FST. The posterior distribution for the parameters were approximated by a beta distribution with parameters estimated by its posterior mean and variance from MCMC (Holsinger & Wallace, 2004). The prior beta distribution can be chosen such that the median fits the plausible value under our knowledge for the parameter. Such prior places positive probability on a range of values that more than adequately covers all possible values for the parameter. When genotype data is available from different populations, a hierarchical multinomial model can, therefore, be constructed without further consideration of the genetic model in data analysis. Based on the multinomial likelihood, a hierarchical model was built to incorporate the hypo- parameters for the genotype frequency (Shoemaker, Painter, & Weir, 1998). At the first level, the genotype count ࢔ ൌ ሺ݊ଵଵ, ݊ଵଶ, ݊ଶଶሻ follows a multinomial distribution: ܲݎሺ࢖|࢔෥ሻ ൌ ܲݎ൫ሺ݊ଵଵ, ݊ଵଶ, ݊ଶଶሻหሺ݌ଵଵ, ݌ଵଶ, ݌ଶଶሻ൯ ൌ ݊!݊ଵଵ! ݊ଵଶ! ݊ଶଶ! ݌ଵଵ ௡భభ݌ଵଶ௡భమ݌ଶଶ௡మమ. The genotype frequency ࢖෥ ൌ ሺ݌ଵଵ, ݌ଵଶ, ݌ଶଶሻ can be re-parameterized as the following: ቐ ݌ଵଵ ൌ ݌ଶ ൅ ݂ݍ݌ ݌ଵଶ ൌ 2ݍ݌ሺ1 െ ݂ሻ ݌ଶଶ ൌ ݍଶ ൅ ݂ݍ݌ . 20 Under this multinomial model, the likelihood function is given by ݈ሺ݌, ݂ሻ ൌ ܲݎሺ݌|࢔, ݂ሻ ൌ ݊!݊ଵଵ! ݊ଵଶ! ݊ଶଶ! ሺሺ1 െ ݂ሻ݌ ଶ ൅ ݂݌ሻ௡భభሺ2ሺ1 െ ݂ሻݍ݌ሻ௡భమሺሺ1 െ ݂ሻݍଶ ൅ ݂ݍሻ௡మమ. At the second level, a uniform prior ߮ሺ݌ሻ for allele frequency p is applied. Conditionally on p, further assume a uniform prior distribution߮ሺ݂|݌ሻ for f with constrained parameter space on ି௠௜௡ ሺ௣,ଵି௣ሻ ଵି௠௜௡ ሺ௣,ଵି௣ሻ ൑ ݂ ൑ 1 ሺWakefield, 2010ሻ . Therefore, the joint prior distribution is a non- informative ߮ሺ݌, ݂ሻ ൌ ߮ሺ݌ሻ߮ሺ݂|݌ሻ ൌ 1, 0 ൏ ݌ ൏ 1, ି௠௜௡ ሺ௣,ଵି௣ሻଵି௠௜௡ ሺ௣,ଵି௣ሻ ൑ ݂ ൑ 1 . The posterior distribution is given by ܲݎሺ݌, ݂|࢔ሻ ൌ ܲݎሺ݌|࢔, ݂ሻ߮௣ሺ݌ሻ߮௙|௣ሺ݂|݌ሻݎܲ׬ሺ݌|࢔, ݂ሻ߮௣ሺ݌ሻ߮௙|௣ሺ݂|݌ሻ݂݀݀݌ . It is not difficult to verify that denominator is a finite quantity as shown below: නܲݎሺ݌|࢔, ݂ሻ߮௣ሺ݌ሻ߮௙|௣ሺ݂|݌ሻ݂݀݀݌ ൌ න න ݈ሺ݌, ݂ሻ߮ሺ݌, ݂ሻ݂݀݀݌ ଵ ି௣೘೔೙ଵି௣೘೔೙ ଵ ଴ ൌ න න ݊!݊ଵଵ! ݊ଵଶ! ݊ଶଶ! ൫ሺ1 െ ݂ሻ݌ ଶ ൅ ݂݌൯௡భభሺ2ሺ1 െ ݂ሻݍ݌ሻ௡భమ൫ሺ1 െ ݂ሻݍଶ ൅ ݂ݍ൯௡మమ݂݀݀݌ ଵ ି௠௜௡ ሺ௣,௤ሻ ଵି௠௜௡ ሺ௣,௤ሻ ଵ ଴ ൏ න න ܥሺ݊ሻ݂݀݀݌ ଵ ି௠௜௡ሺ௣,௤ሻ ଵି௠௜௡ሺ௣,௤ሻ ଵ ଴ ൌ ܥሺ݊ሻන ቆ1 ൅ ݉݅݊ሺ݌, ݍሻ1 െ ݉݅݊ሺ݌, ݍሻቇ݀݌ ଵ ଴ ൏ 2ܥሺ݊ሻ, where ܥሺ݊ሻ ൌ ௡!௡భభ!௡భమ!௡మమ!, a constant depended on data. The first inequality is due to the fact that all the genotype frequencies are equal to or less than one. Therefore, the posterior distribution is proper and has a closed form when k=2. The posterior mean of f can be obtained as 21 න න ݂ ଵ ି௠௜௡ሺ௣,௤ሻ ଵି௠௜௡ሺ௣,௤ሻ ଵ ଴ ܲݎሺ݌, ݂|࢔ሻ݂݀݀݌ ൌ ׬ ׬ ݂ଵି௠௜௡ሺ௣,௤ሻ ଵି௠௜௡ሺ௣,௤ሻ ଵ ଴ ܲݎሺ݌|࢔, ݂ሻ߮௣ሺ݌ሻ߮௙|௣ሺ݂|݌ሻ݂݀݀݌ ׬ ׬ ܲݎሺ݌|࢔, ݂ሻ߮௣ሺ݌ሻ߮௙|௣ሺ݂|݌ሻ݂݀݀݌ଵି௠௜௡ሺ௣,௤ሻ ଵି௠௜௡ሺ௣,௤ሻ ଵ ଴ ൌ ׬ ׬ ݂ ݊!݊ଵଵ! ݊ଵଶ! ݊ଶଶ! ൫ሺ1 െ ݂ሻ݌ ଶ ൅ ݂݌൯௡భభሺ2ሺ1 െ ݂ሻݍ݌ሻ௡భమ൫ሺ1 െ ݂ሻݍଶ ൅ ݂ݍ൯௡మమ݂݀݀݌ଵି௠௜௡ሺ௣,௤ሻ ଵି௠௜௡ሺ௣,௤ሻ ଵ ଴ ׬ ׬ ݊!݊ଵଵ! ݊ଵଶ! ݊ଶଶ! ൫ሺ1 െ ݂ሻ݌ ଶ ൅ ݂݌൯௡భభሺ2ሺ1 െ ݂ሻݍ݌ሻ௡భమ൫ሺ1 െ ݂ሻݍଶ ൅ ݂ݍ൯௡మమ݂݀݀݌ଵି௠௜௡ሺ௣,௤ሻ ଵି௠௜௡ሺ௣,௤ሻ ଵ ଴ ൌ ׬ ׬ ݂൫ሺ1 െ ݂ሻ݌ଶ ൅ ݂݌൯௡భభሺ2ሺ1 െ ݂ሻݍ݌ሻ௡భమ൫ሺ1 െ ݂ሻݍଶ ൅ ݂ݍ൯௡మమ݂݀݀݌ଵି௠௜௡ሺ௣,௤ሻ ଵି௠௜௡ሺ௣,௤ሻ ଵ ଴ ׬ ׬ ൫ሺ1 െ ݂ሻ݌ଶ ൅ ݂݌൯௡భభሺ2ሺ1 െ ݂ሻݍ݌ሻ௡భమ൫ሺ1 െ ݂ሻݍଶ ൅ ݂ݍ൯௡మమ݂݀݀݌ଵି௠௜௡ሺ௣,௤ሻ ଵି௠௜௡ሺ௣,௤ሻ ଵ ଴ , which is not a standard integral function, but can be obtained using the numerical integration method. Markov Chain Monte Carlo (MCMC) could also be used to derive the posterior mean and variance. In the following simulation study, we choose to use MCMC from OPENBUGS to derive the posterior mean and its 95% credible interval for the estimator. The model for this MCMC is listed in below. Level 1: ݈ܽ݅݉݋݊݅ݐ݈ݑܯ~࢔ሺ݊, ࢖෥ሻ, ࢔ ൌ ሺ݊ଵଵ, ݊ଵଶ, ݊ଶଶሻ, ࢖෥ ൌ ሺ݌ଵଵ, ݌ଵଶ, ݌ଶଶሻ, ቐ ݌ଵଵ ൌ ݌ଶ ൅ ݂ݍ݌ ݌ଵଶ ൌ 2ݍ݌ሺ1 െ ݂ሻ ݌ଶଶ ൌ ݍଶ ൅ ݂ݍ݌ Level 2: ݉ݎ݋݂ܷ݅݊~݌ሾ0，1ሿ; ݂|݉ݎ݋݂ܷ݅݊~݌ሾି୫୧୬ሺ௣,ଵି௣ሻଵି୫୧୬ሺ௣,ଵି௣ሻ , 1ሿ; Since ି௫ ଵି௫ ൌ 1 െ ଵ ଵି௫ is a decreasing function in x, this is same as ݂|݉ݎ݋݂ܷ݅݊~݌ሾmax ሺ ି௣ଵି௣ , ିሺଵି௣ሻ ௣ ሻ, 1ሿ; model { q[1] <- (1 - f) * p * p + f * p q[2] <- 2 * (1 - f) * p * (1 - p) q[3] <- (1 - f) * (1 - p) * (1 - p) + f * (1 - p) y[1:3] ~ dmulti(q[], n) 22 p ~ dunif(0, 1) f <- w * (1 - f.min) + f.min f.min <- max(-p/(1 - p), -(1 - p)/p) w ~ dunif(0, 1) n <- sum(y[]) } After MCMC, standard diagnostic plot can be used to check the simulation history, posterior distribution of the parameter, autocorrelation function, and Brooks-Gelman-Rubin convergence statistic (Brooks & Gelman, 1998). Figure 2 shows examples of these plots when the observed genotype data is 3, 32, and 165 (simulated counts for a sample size of 200, the allele frequency p=0.1 and the inbreeding coefficient f=0.05). The skewed posterior distribution of f indicates that the confidence interval of the DSE under asymptotical normality assumption may not be satisfactory. Figure  2.  MCMC  Diagnostic  Plots  (History,  Posterior  Distribution,  Autocorrelation  and  BGR  Statistic).   Current Bayesian application in the estimation of inbreeding coefficient is restricted mainly to the simple random sampling from either single population or independent populations (Wakefield, 2010). When sampling design includes unequal selection probabilities or clustering, Bayesian analysis needs to consider the sampling structure so that effective sample size or possible correlations due to clustering may be incorporated in the hierarchical model. However, the final obtained analysis dataset may not contain detailed sample design information. Even when all the 23 variables for sampling design are available, exact hierarchical modeling may result in a substantial number of hyper-parameters in the model. 1.5 Genetic Data from Complex Survey In the past few years, population based national surveys, such as the National Health and Nutrition Examination Survey (NHANES) and Health and Retirement Survey (HRS), started collecting a subsample of their participants’ DNA specimens. These surveys usually employ a complex sampling design, which often involves stratification, clustering and multi-stage sampling. In order to produce reliable statistics for minority represented subgroup of survey participants, these surveys typically over-sample such subpopulation. Therefore, within a cluster of the population, individuals are often related to one another, due to same ethnicity, some shared ancestry or finite within cluster sample size. On the other hand, complex sampling structure introduces correlations among genes from different clusters. This may introduce a local level inbreeding, which results in an excessive homozygotes and affects our interpretation of the gene polymorphism if it is not appropriately considered in analysis. Statistical tests based on allelic distributions or levels of variability often depend on sample drawn from sub-divided population, and therefore, can be significant in absence of departing from HWE (Nielsen, 2001). For sample designs with unequal probabilities of inclusion, a generalized weighted quadratic test statistic is used in design-based approaches to conduct hypothesis testing of HWE. Under certain regulatory conditions, this test statistic converges to linear combinations of i.i.d chi-square random variables (Li & Graubard, 2009). Similarly, testing HWE is equivalent to testing the null hypothesis of disequilibrium coefficient equals to zero. Under the null hypothesis, the test statistic can be proven to be a chi-square random variable corrected by a simple design correction factor which is a function of design effect estimates of the genotype frequency (Moonesinghe, et al., 2010). In population-based household survey such as NHANES, genetic related individuals are sampled from the same household. To accommodate the correlation induced by genetic relatedness among individuals within a household, as well as the correlation due to multi-stage clustering sampling, an estimation equation under an inbreeding model was used to derive a quasi-score test statistic. Under the null hypothesis, this test statistic converges to a chi-square random variable (Li, Y.; Graubard, B.I., 2011). 24 The extension of testing HWE for data coming from complex survey provides great opportunities for scientists to screen genes to identify potential ones for gene-disease association study. However, the lack of robust estimator limits the interpretation of these findings in a scientific content. Researchers may be interested in a moderate estimate of the departure from HWE even when the data suggests otherwise. On the other hand, statistical significance from above test with a negligible f does not give researchers much confidence to screen out these potential important genes. Furthermore, although Bayesian method is increasingly used in the testing of HWE or the estimation of the inbreeding coefficient (Shoemaker, Painter, & Weir, 1998; Wakefield, 2010; Consonni, Morenob, & Venturinic, 2011; Holsinger & Wallace, 2004; Ayres & Balding, 1998), we have not found such application to the genetic data from complex sampling design thus far. When the inbreeding coefficient is estimated from a sample with unequal selection probabilities, it still measures the deviation from random pairing of the genes. Since researchers often devote their time to use the genotype data in the context of population survey, proper estimation of the inbreeding coefficient for the study population may provide an initial screening tool for the potential genes. Extending the Bayesian methods to the genetic data from complex survey may build a bridge between the hypothesis testing and the estimation by using the posterior distribution. A hypothesis testing could be formulized through widely used Bayes Factor or through the credible interval in the Bayesian frame work. 1.5.1 National Health and Nutrition Examination Survey (NHANES) The National Health and Nutrition Examination Survey (NHANES) is a program designed to assess the health and nutritional status of adults and children in the United States by interviews and physical examinations. The sample for the survey is selected to represent the U.S. population of all ages. NHANES is a complex sample survey. The sample weights reflect the unequal probabilities of selection, non-response adjustments and adjustments to independent population controls. Primary Sampling Units (PSUs) are generally single counties. In order to meet a minimum population size, small counties are sometimes combined as one PSU. Within a sampled PSU, clusters of households are selected. Each person in a selected household is screened for demographic characteristics. One 25 or more persons per household are selected for the final sample. Data are sometimes collected on subsamples of the full design for NHANES. Therefore, each subsample involves another stage of selection and separate sample weights that account for that stage of selection and additional non- response, etc. In order to produce reliable statistics, NHANES over-samples persons 60 and older, African Americans, and Hispanics (Ezzati, Massey, Waksberg, Chu, & Maurer, 1992). NHANES collects data for chronic conditions, previously undiagnosed conditions, as well as those known to and reported by respondents in the survey. The studied disease or medical conditions includes anemia, cardiovascular disease, diabetes, environmental exposures, eye diseases, hearing loss, infectious diseases, kidney disease, nutrition, obesity, oral health, osteoporosis, physical fitness and physical functioning, reproductive history and sexual behavior, respiratory disease (asthma, chronic bronchitis, emphysema), sexually transmitted diseases, vision, etc. NHANES also collects risk factors, including person’s lifestyle, constitution, heredity, environment, smoking, alcohol consumption, sexual practices, drug use, physical fitness and activity, weight, and dietary intake, etc. Starting from NHANES III, DNA specimens have been collected from participant whose age is 20 or more. The genetic data adds to the extensive amount of information collected for the purpose of describing the health of the population. Genotype data is obtained from DNA samples extracted from cell lines (NHANES III) or blood (NHANES 1999-2002) and can be analyzed along with survey data such as NHANES 1999-2000 or NHANES 2001-2002 or all four years combined (NHANES 1999-2002). The quality control of the genetic data is performed by assessing the deviation from Hardy-Weinberg Equilibrium for each of the three main race/ethnicity groups (non- Hispanic white, non-Hispanic black, and Mexican-American) and by assessing the discordant pairs between duplicated samples and the complete set (NHANES, 2010). Unlike mutation, the genetic data collected in NHANES is mainly for common disease. Each polymorphism contributes only a small amount towards overall disease risk. Most often, a participant’s life style, characteristics such as body mass index, and/or social-economic status may be more important risk factors for disease prediction. Nevertheless, NHANES is the first US survey with a national probability sampling design to make unbiased estimates about population genetics and associations between disease and genetic variants (Chang, Lindegren, Butler, & et.al., 26 2009). Linkage of the NHANES III or NHANES 1999-2002 phenotype data with the genetic information provides an opportunity to investigate the association of a wide variety of health factors with regard to genetic variation. From the release of NHANES genetic data component, public health scientists have been using the data for screening of genetic marker, studying genetic variation among US population, pre-dispositioning to chronic disease, and identifying of risk factors (Fesinmeyer & et.al, 2013; Zhang & et.al, 2013; Yang & et.al, 2010; Crawford & et.al, 2006; Steinberg & et.al, 2001). 1.5.2 Health and Retirement Survey (HRS) The Health and Retirement Study (HRS), conducted by the University of Michigan, is a longitudinal panel study of representative samples of Americans over 50 every two years. The target population includes all adults in the contiguous United States born during the years 1931 - 1941 who reside in households. In addition to the core sample, HRS cohort is supplemented by over sampling African Americans, Hispanics as well as respondents who are residents of the state of Florida. The core cohort was augmented in following years to include additional AHEAD cohort (The Study of Assets and Health Dynamics Among the Oldest Old, born before 1924), CODA cohort (Children of Depression, born 1924-30), War Baby (born 1942-47), Early Baby Boomers (born 1948-53), and Mid Boomers (born 1954–59). The current sample includes over 26,000 persons in 17,000 households. The study collects information about participants’ income, work, assets, pension plans, health insurance, disability, physical health and functioning, cognitive functioning, and health care expenditures. It is supplemented with administrative linkages to Medicare claims files providing diagnostic and medication utilization information, to the National Death Index, and to the Social Security (Health and Retirement Study Survey Design, 2008). The HRS sample is selected under a four-stage area probability sampling design. Each multi-stage component of the HRS area probability sample is consistent with the general sample design framework and sampling procedures of the SRC National Sample (Heeringa, Connor, & Darrah, 1986). First, the primary stage of sampling units (PSUs) involve probability proportional to size (PPS) selection of U.S. Metropolitan Statistical Areas (MSAs) and non-MSA counties. The National Sample PSUs are assigned to 84 explicit strata based on MSA/non-MSA status, PSU size and geographic location. Sixteen of the 84 strata contain only a single self-representing (SR) PSU, 27 each of which is included with certainty in the primary stage of sample selection. The remaining 68 nonself-representing (NSR) strata contain more than one PSU and one PSU is sampled with PPS. To reduce the between-PSU component of sample variation, a controlled selection with "one- per-stratum" design allocation is used to select PSUs. The full 84 strata are 2/3 partitioned into sampling error computation strata. Despite the expected improvement in sample precision, the expected consequence of collapsing design strata into sampling error computation strata is the overestimation of the true sampling error. With the exception of New York, Los Angeles and Chicago MSAs, which have two sampling error computation strata collapsed from similar NSR design strata, each SR design stratum is represented by one sampling error computation stratum. In 1992 survey, there were a total of 61 sampling error computation strata, including 16 self- representing MSA PSUs and a stratified subsampling of 45 of the 68 nonself-representing PSUs (HRS 1992 (Wave 1) Documentation, 1992). Secondly, sampling of area segments (SSUs) is conducted within each selected PSU. In order to estimate the sampling error using the Balanced Repeated Replication (BRR) method or the approximate Taylor Series method, the half sample units are created by dividing sample cases into random halves based on SSU number order to preserve the stratification and second stage clustering properties within each computation stratum. The half samples to each NSR computation stratum has a one-to-one correspondence to sample design NSR PSUs. Thirdly, a systematic selection of the housing units (HUs) is conducted from a complete list of all HUs that are physically located within the bounds of the selected SSU. The household financial unit must contain at least one age-eligible member from the cohort. This includes 1) a single unmarried age-eligible person; 2) a married couple in which both persons are age-eligible; or 3) a married couple in which only one spouse is age-eligible. The fourth and last stage is the selection of an age-eligible person within a sampled HU. If selected age-eligible person has a spouse, the spouse is automatically selected even if he or she is not age- eligible. In 2006 and 2008, HRS genotyped 12,507 respondents who provided saliva DNA samples and signed consent forms. Additional 6,000 more samples were expected to be added with the inclusion 28 of a substantial expansion of the minority samples. The genotyping was performed by the NIH Center for Inherited Disease Research using the Illumina Human Omni-2.5 Quad beadchip (Weir D. R., 2012). This technology has capability to cover more than 2.5 million SNPs. These restricted genotype data and a limited set of phenotype measures are deposited in the NIH GWAS repository dbGap (http://www.ncbi.nlm.nih.gov/gap) for approved researchers to use. Researchers may link the genotype data with the HRS main survey data by applying the access of the HRS-dbGaP Cross- Reference File (http://hrsonline.isr.umich.edu/gwas). In addition, dbGap provides genotypes for exonic DNA variants on approximate 16,000 samples. Furthermore, it maintains the imputation of genotypes for approximately 21 million DNA variants from the 1000 Genomes Project (http://www.1000genomes.org). 1.6 Empirical Likelihood The Empirical Likelihood (EL), first introduced by Owen (1988), is a robust semiparametric alternative to the classical likelihood approach. Assume yi, i=1,…, n, is a random variable from a distribution ܲሺߠሻ with parameter ߠ . Instead of formalizing the likelihood through the density function as ݈ሺ࢟|ߠሻ ൌ ܲሺߠ|࢟ሻ , the empirical likelihood is defined as ݈ா௅ሺ࢟|ߠሻ ൌ ∏ ߨ௜௡௜ୀଵ or ∑ log ሺߨ௜ሻ௡௜ୀଵ , ࣊ is in the set ሼ࣊ ∈ ሾ0, 1ሿ௡, ߨ௜ ∈ ሾ0,1ሿ, ∑ ߨ௜௡௜ୀଵ ൌ 1, and∑ ߨ௜௡௜ୀଵ ݄ሺݕ௜|ߠሻ ൌ 0ሽ . The last constraint is an estimation equation such that ܧሾ݄ሺܻ, ߠሻሿ ൌ 0. Without this last constraint, the empirical likelihood has a nonparametric empirical cumulative distribution. Extensive discussion and application of this topic can be found in Owen (2001). The concept of empirical likelihood in survey sampling was first used by Hartley & Rao (1968) through their “scale load” approach. To obtain an empirical likelihood under general sampling design with unequal probabilities of selection and clustering, Chen & Sitter (1999) proposed a pseudo-empirical likelihood function. Wu & Rao (2006), Rao & Wu (2010), Yang & Qin & Qin (2011) extended this method to stratified complex sampling for the estimation of confidence intervals. The proposed method has been proved to have design-based frequentist asymptotical property. The formal application of Bayesian approach to the empirical likelihood started with the discussion in Monahan & Boos (1992). They proposed that empirical likelihood is appropriate for the Bayesian inference if the associated posterior credible intervals has correct specified coverage. Using simulation approach, Lazar (2003) showed that empirical likelihood has the correct posterior 29 coverage for moderate sample size. Using higher order asymptotics, Fang & Mukerjee first theoretically investigated the coverage probability of posterior credible interval for empirical type likelihood admitting a probability-matching prior (2005; 2006). Chang & Mukerjee (2008) later extended to a general class of empirical-type likelihoods for the population mean and studied its existence of a confidence interval that has approximately correct posterior as well as frequentist coverage for any given prior. Rao & Wu (2010) applied the Bayesian empirical likelihood to estimate the posterior mean and coverage interval in a finite population sampling setting. Chaudhuri & Ghosh (2011) considered Bayesian empirical likelihood in the context of small area estimation, which handles discrete and continuous data in a unified manner and does not require a parametric likelihood or any linearity assumptions. In this dissertation, the genotype data is discrete. The inbreeding coefficient is a function of the genotype frequencies which can be considered as a correlation measure. Extension of the Bayesian empirical likelihood approach to the estimation of a function of population proportions such as the inbreeding coefficient f is one of our main interests. 1.7 Discussion and Overview of Dissertation In this chapter, we have presented a broad overview of the estimation methods of the inbreeding coefficient, including their advantages and disadvantages. We have introduced two national level complex surveys that collect genotypic data. Lastly, we have briefly reviewed empirical likelihood with the Bayesian approach and its current application in the complex survey. Such discussion motivates our research in the application of these methods to the population genetics based on national level complex surveys. The more and more genetic data coming from these types of surveys make us believe that a wide range of applications of the Bayesian methods including a nonparametric approach could benefit future research. As an initial gene screening tool for GWAS focusing on the measured magnitude of the deviation from HWE, we believe that it is necessary to take the complex survey sampling design into consideration. This dissertation is organized as follows. In Chapter 2, we first illustrate the maximum empirical likelihood estimator and the parametric Bayesian estimator of the inbreeding coefficient. We further propose the nonparametric Bayesian estimator with the empirical likelihood under a simple 30 random sampling design. We compare those estimators in simulation studies. In Chapter 3, we extend above methods to estimate the finite population inbreeding coefficient based on samples with unequal selection probabilities. We have developed the Bayesian pseudo-empirical likelihood estimator with the sampling weight. The proposed estimator is compared with the parametric Bayesian estimator using the effective sample size, and with the design-based estimator in simulation studies. In Chapter 4, we further extend the proposed methods to family level genetic data which includes within family genetic correlation. In Chapter 5, we explore the analysis approach incorporating between subject correlations for clustered genetic data with population subdivision. In Chapter 6, we apply the proposed Bayesian methods to genotype data from the 2006 Health and Retirement Study. Finally in Chapter 7, we provide a summary of this dissertation and discuss directions for future research. 31 Chapter 2: Estimation of the Inbreeding Coefficient under Simple Random Sampling In this chapter, to lay out the framework for our research, we first focus on the estimation of the inbreeding coefficient under simple random sampling design. The existing frequentist estimator and parametric Bayesian estimator from posterior distribution is compared. In order to simplify our illustration, we used a single locus with two alleles (k=2) throughout discussion. As a benchmark for comparison, the frequentist Direct Sample Estimator (DSE, 6) or the estimator derived in equation (7) by setting the first derivative of the log likelihood to zero is used. A Bayesian model described in Section 1.4.2 is used as our parametric Bayesian estimator. 2.1 Maximum Empirical Likelihood Estimator (MELE) Let Y be the number of allele A for each subject in the population. We define Y=2 if the observed genotype is AA, Y=1 if the genotype is Aa, and Y=0 if the genotype is aa. Under the general inbreeding model, the distribution of Y is ܲሺܻ ൌ ݕሻ ൌ ቐ ሺ1 െ ݌ሻଶ ൅ ݌ሺ1 െ ݌ሻ݂ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻ ݌ଶ ൅ ݌ሺ1 െ ݌ሻ݂ ݕ ൌ 0 ݂݅ ݕ ൌ 1 ݕ ൌ 2 . Let ݕଵ, ݕଶ, … , ݕ௡ be a random sample from the distribution of Y. Further, define a binary indicator variable ܸ ൌ ܻሺ2 െ ܻሻ for the homozygotes AA and aa (Y=0 or 2, therefore V=0) or heterozygote Aa (Y=1, therefore V=1), ݒ௜ ൌ ݕ௜ሺ2 െ ݕ௜ሻ. The indicator variable V has the following distribution: ܲሺܸ ൌ ݒሻ ൌ ൜1 െ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻ2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻ ݂݅ ݒ ൌ 0 ݒ ൌ 1. It should be recognized that V can be derived from Y, but not the reverse. Based on their distributions, ܧሺܻሻ ൌ 2݌ and ܧሺܸሻ ൌ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻ . Therefore, a nonparametric way to analyze the data is to maximize the empirical log likelihood ݈ா௅ሺ݂ሻ ൌ ∑ log ሺߨ௜ሻ௡௜ୀଵ with respect to the constraints ߨ௜ ൐ 0, ∑ ߨ௜௡௜ୀଵ ൌ 1, ∑ ߨ௜௡௜ୀଵ ݕ௜ ൌ 2݌, and ∑ ߨ௜௡௜ୀଵ ݒ௜ ൌ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻ. The objective function ∑ log ሺߨ௜ሻ௡௜ୀଵ is a strictly concave function on a convex set of weight vectors. 32 Therefore a unique global maximum exists (Owen, 2001). Similar as below case with a single estimating equation constraint in the next section, we use the Lagrange multiplier method by first setting up ܩ ൌ ∑ log ሺߨ௜ሻ௡௜ୀଵ െ ݊ߣଵ ∑ ߨ௜൫ݒ௜ െ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻ൯௡௜ୀଵ െ ݊ߣଶ ∑ ߨ௜ሺݕ௜ െ 2݌ሻ௡௜ୀଵ െ ߛሺ∑ ߨ௜௡௜ୀଵ െ 1ሻ. If there is only one constraint for ∑ ߨ௜௡௜ୀଵ ൌ 1, we know that empirical likelihood ∑ log ሺߨ௜ሻ௡௜ୀଵ ൌ log ሺ∏ ߨ௜௡௜ୀଵ ሻ is maximized at ߨො௜ ൌ 1/݊ (Owen, 2001). At this maximized empirical likelihood, we have ∑ ߨො௜௡௜ୀଵ ݕ௜ ൌ ଵ௡ ∑ ݕ௜௡௜ୀଵ ൌ ݕത and ∑ ߨො௜௡௜ୀଵ ݒ௜ ൌ ଵ ௡ ∑ ݒ௜௡௜ୀଵ ൌ ̅ݒ . Therefore, with additional constraints based on estimating equations for the means ܧሺܻ െ 2݌ሻ ൌ 0 and ܧሺܸ െ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻሻ ൌ 0, we expect to see that the MELEs are similar to DSEs for both ̂݌ and መ݂. Furthermore, if we use DSE ̂݌ to replace p and derive a profiled likelihood, the MELE is expected to remain the same as DSE for መ݂. Such dimension reduction can significantly decrease the calculation time. As Table 3 shown, genotype frequency may be rare when both the allele frequency p and the inbreeding coefficient f are small. With small sample size and conditioning on p, these parameter settings are more likely to produce estimators that are outside of the parameter space for f. To illustrate that MELE has a similar performance as DSE, a simulation study (250 runs) was conducted for the allele frequency p=0.1 and the inbreeding coefficient f=0.05 under a sample size of 50. To derive MELE, an EM algorithm implemented in the R/emplik package (Zhou, 2005) is used to maximize the empirical log likelihood function with above constraints. Figure 3 shows that DSE and MELE have similar values in each simulation iteration so that the data pairs stay along the diagonal line. Furthermore, MELE has the same problem as DSE. At the sample size of 50, 14.4% of the simulated estimators are outside of the lower limit of the parameter space for f under p=0.1 and thus -p/(1-p) = -0.1111. There are still 0.4% of the simulated estimators that are outside of the range when the sample size increases to 200. The simulation is in line with what we discussed earlier that DSE f(s) is a biased (in the order of n-1) but consistent estimator. The estimator moves toward true value f when sample size increases toward infinity. 33 Figure 3. Simulation Results of DSE vs. MELE (p=0.1, f=0.05, sample size of 50 or 200). Red dash  line is the lower limit of the parameter space for f when p=0.1.  2.2 Bayesian Pseudo-Empirical Likelihood Estimator (BPELE) The above computation burden to derive MELE is not small. In order to search estimators of p and f so that they maximize the empirical likelihood with two constraints based on estimating functions, we need first calculate the empirical likelihood for each pair of p and f. Since we are not interested in the unknown population nuisance parameter p, a profile likelihood method can be used to replace p with its MLE ̂݌ ൌ ௒തଶ. As we discussed in Chapter 1, the estimator ̂݌ of the allele frequency is unbiased. A general estimation equation therefore can be formulized as ܧሾܸ െ 2ሺ1 െ ݂ሻ̂݌ሺ1 െ ̂݌ሻሿ ൌ 0. We want to maximize the empirical likelihood ݈ா௅ሺ݂ሻ ൌ ∑ log ሺߨ௜ሻ௡௜ୀଵ subject to ߨ௜ ൐ 0, ∑ ߨ௜௡௜ୀଵ ൌ 1 and ∑ ߨ௜ሺݒ௜ െ 2ሺ1 െ ݂ሻ̂݌ሺ1 െ ̂݌ሻሻ௡௜ୀଵ ൌ 0. A Lagrange multiplier method can be used to find መ݈ா௅ሺ መ݂ሻ. First, we set up ܩ ൌ෍ log ሺߨ௜ሻ ௡ ௜ୀଵ െ ݊ߣ෍ ߨ௜൫ݒ௜ െ 2ሺ1 െ ݂ሻ̂݌ሺ1 െ ̂݌ሻ൯ ௡ ௜ୀଵ െ ߛሺ෍ ߨ௜ ௡ ௜ୀଵ െ 1ሻ. Taking the partial derivative with respect to ߨ௜, this equals 0 yields: 1 ߨ௜ െ ݊ߣ൫ݒ௜ െ 2ሺ1 െ ݂ሻ̂݌ሺ1 െ ̂݌ሻ൯ െ ߛ ൌ 0. 34 Summing the above equation over i=1 to n, it yields ݊ െ ݊ߣ∑ ߨ௜൫ݒ௜ െ 2ሺ1 െ ݂ሻ̂݌ሺ1 െ ̂݌ሻ൯௡௜ୀଵ ൌ ߛ ∑ ߨ௜௡௜ୀଵ ൌ ߛ ,ߛ ൌ ݊. Solving this equation for ߨ௜, it yields ߨ௜ ൌ ଵ௡ఒ൫௩೔ିଶሺଵି௙ሻ௣ොሺଵି௣ොሻ൯ା௡ . To find the solution of ߣ ൌ ߣሺ݂ሻ, it needs solve the equation ∑ ௩೔ିଶሺଵି௙ሻ௣ොሺଵି௣ොሻଵାఒ൫௩೔ିଶሺଵି௙ሻ௣ොሺଵି௣ොሻ൯ ௡௜ୀଵ ൌ 0. This usually can be done through the Newton algorithm. The empirical likelihood is then derived as ݈ா௅ሺ݂ሻ ൌ෍ logሺߨ௜ሻ ௡ ௜ୀଵ ൌ െ݊logሺ݊ሻ െ෍ log ቀ1 ൅ ߣሺ݂ሻ൫ݒ௜ െ 2ሺ1 െ ݂ሻ̂݌ሺ1 െ ̂݌ሻ൯ቁ ௡ ௜ୀଵ . As we showed in previous section, MELE is expect to be approximately the same as DSE f(s). When the sample size is small and the allele frequency is small, it is likely to produce estimators that lie outside of the parameter space. Borrowing the same idea from the parametric Bayesian analysis, Bayesian approach can also be applied to the empirical likelihood setting. With our inbreeding model, we assume a uniform prior ߮ሺ݂ሻ ∝ 1 on the parameter space [ ି୫୧୬ ሺ௣ො,ଵି௣ොሻଵି୫୧୬ ሺ௣ො,ଵି௣ොሻ, 1]. The posterior distribution therefore is ߮ሺ݂|ݕሻ ൌ ܿሺݕሻ exp ൤െ෍ log൛1 ൅ ߣሺ݂ሻ൫ݒ௜ െ 2ሺ1 െ ݂ሻ̂݌ሺ1 െ ̂݌ሻ൯ൟ ௡ ௜ୀଵ ൨ , where ܿሺݕሻ is a normalizing constant such that ߮׬ሺ݂|ݕሻ ݂݀ ൌ 1. Since the uniform prior is proper and the empirical likelihood is bounded, the posterior distribution is therefore proper (Chaudhuri & Ghosh, 2011). We can write the posterior distribution as ߮ሺ݂|ݕሻ ൌ ܲݎሺ݂|ݕଵ, … , ݕ௡ሻ ∝ ߮ሺ݂ሻ݈ா௅ሺ݂ሻ ൌ exp൫log൫߮ሺ݂ሻ൯ ൅ log൫݈ா௅ሺ݂ሻ൯൯ . Without loss of generality, let f0 maximizes the prior ߮ሺ݂ሻ and መ݂௡ maximizes the empirical likelihood ݈ா௅ሺ݂ሻ. We expand the prior and the empirical likelihood by the Taylor linearization up to the second derivative. To simplify the notation, we ignore the remainder term after the second derivatives. log൫ߨሺ݂ሻ൯ ൌ log൫߮ሺ ଴݂ሻ൯ ൅ ሺ݂ െ ଴݂ሻ߮ᇱሺ ଴݂ሻ െ 12 ሺ݂ െ ଴݂ሻ ଶ߮ᇱᇱሺ ଴݂ሻ ൌ log൫߮ሺ ଴݂ሻ൯ െ 12 ሺ݂ െ ଴݂ሻ ଶ߮ᇱᇱሺ ଴݂ሻ 35 log൫݈ா௅ሺ݂ሻ൯ ൌ log ቀ݈ா௅൫ መ݂௡൯ቁ ൅ ൫݂ െ መ݂௡൯݈ா௅ᇱ൫ መ݂௡൯ െ 12 ൫݂ െ መ݂௡൯ ଶ݈ா௅ᇱᇱ൫ መ݂௡൯ ൌ ݈݃݋ ቀ݈ா௅൫ መ݂௡൯ቁ െ 12 ൫݂ െ መ݂௡൯ ଶ݈ா௅ᇱᇱ൫ መ݂௡൯. Substituting above equations into ߮ሺ݂|ݕሻ, the posterior distribution therefore is ߮ሺ݂|ݕሻ ൌ exp ൜log൫߮ሺ ଴݂ሻ൯ െ 12 ሺ݂ െ ଴݂ሻ ଶߨᇱᇱሺ ଴݂ሻ ൅ log ቀ݈ா௅൫ መ݂௡൯ቁ െ 12 ൫݂ െ መ݂௡൯ ଶ݈ா௅ᇱᇱ൫ መ݂௡൯ൠ ∝ exp ൜െ12 ሺ݂ െ ଴݂ሻ ଶ߮ᇱᇱሺ ଴݂ሻ െ 12 ൫݂ െ መ݂௡൯ ଶ݈ா௅ᇱᇱ൫ መ݂௡൯ൠ ൌ exp ൝െ12ቆ݂ െ ଴݂߮ᇱᇱሺ ଴݂ሻ ൅ መ݂௡݈ா௅ᇱᇱ൫ መ݂௡൯ ߮ᇱᇱሺ ଴݂ሻ ൅ ݈ா௅ᇱᇱ൫ መ݂௡൯ ቇ ଶ ቀ߮ᇱᇱሺ ଴݂ሻ ൅ ݈ா௅ᇱᇱ൫ መ݂௡൯ቁൡ. Under general regularity condition, as ݊ → ∞, the posterior distribution ߮ሺ݂|ݕሻ converges to a normal distribution with mean ௙బ߮ᇲᇲሺ௙బሻା௙መ೙௟ಶಽᇲᇲ൫௙መ೙൯߮ᇲᇲሺ௙బሻା௟ಶಽᇲᇲ൫௙መ೙൯ and variance ߮ ᇱᇱሺ ଴݂ሻ ൅ ݈ா௅ᇱᇱ൫ መ݂௡൯ (Owen, 2001). In our case, we select a non-informative uniform prior ߮ሺ݂ሻ ∝ 1, therefore ߮ᇱᇱሺ ଴݂ሻ ൌ 0 . The posterior converges to a normal distribution with mean መ݂௡ and variance ݈ா௅ᇱᇱ൫ መ݂௡൯ as ݊ → ∞. 2.3 Simulation Study A simulation study is conducted for a single locus with two alleles (A, a) to compare the frequentist estimator and the Bayesian estimators of f. At each one of R=1000 iterations (The rationale of the choice of 1000 simulation runs is evaluated below), a sample size of n=50, 100 or 200 genotype records are generated randomly from the general multinomial inbreeding model with parameter p and f: ܲݎሺ݌|࢔, ݂ሻ ൌ ݊!∏ ݊௜௝!ଶ௜ୀଵ,௝ஹ௜ ෑ ݌௜௝ ௡೔ೕ ଶ ௜ୀଵ,௝ஹ௜ ൌ ݊!݊ଵଵ! ݊ଵଶ! ݊ଶଶ! ሾ݌ሺ݌ ൅ ݂ െ ݂݌ሻሿ ௡భభሾ2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻሿ௡భమሾሺ1 െ ݌ሻሺ1 െ ݌ ൅ ݂݌ሻሿ௡మమ. The parameters for the simulation are the allele A frequency of p=0.1, 0.3, or 0.5 and the inbreeding coefficient of f=-0.1, 0, 0.05, 0.1, or 0.2. Under those parameter setting, Table 3 shows the expected 36 genotype frequencies. When the allele frequency is small (p=0.1), the expected genotype AA frequency is around 0.01~0.03 range depending on the value of f. If the sample size is small, the likelihood to have samples without any observed genotype AA is high. When this happens, the sample estimator of ݌ଵଵ ൌ ݌ଶ ൅ ݌ሺ1 െ ݌ሻ݂ tends to get zero, which results in the estimator of f approaches lower bound of parameter space, -p/(1-p). Table 3. Expected Genotype Frequency under Allele Frequency p and Inbreeding Coefficient f    p    f  Expected Genotype Frequency  p(aa)  p(Aa)  p(AA)  0.1  ‐0.1  0.801  0.198  0.001  0  0.81  0.18  0.01  0.05  0.8145  0.171  0.0145  0.1  0.819  0.162  0.019  0.2  0.828  0.144  0.028  0.3  ‐0.1  0.469  0.462  0.069  0  0.49  0.42  0.09  0.05  0.5005  0.399  0.1005  0.1  0.511  0.378  0.111  0.2  0.532  0.336  0.132  0.5  ‐0.1  0.225  0.55  0.225  0  0.25  0.5  0.25  0.05  0.2625  0.475  0.2625  0.1  0.275  0.45  0.275  0.2  0.3  0.4  0.3  For the realized data, estimators of the inbreeding coefficient are calculated by both the frequentist and the Bayesian methods, respectively. The bias, variance and mean square error (MSE) of the estimator መ݂ are calculated as Bias௙൫ መ݂൯ ൌ ܧ௙൫ መ݂൯ െ ݂ ൌ 1ܴ෍ ൫ መ݂ ሺ௥ሻ െ ݂൯ ோ ௥ୀଵ ; Var൫ መ݂൯ ൌ 1ܴ෍ ሺ መ݂ ሺ௥ሻ െ መ݂ሻ̅ଶ ோ ௥ୀଵ ; MSE൫ መ݂൯ ൌ ܧሾሺ መ݂ െ ݂ሻଶሿ ൌ 1ܴ෍ ቀሺ መ݂ ሺ௥ሻ െ መ݂ሻ̅ ൅ ሺ መ݂̅ െ ݂ሻቁଶ ோ ௥ୀଵ ൌ Bias௙ሺ መ݂ሻଶ ൅ ܸܽݎ൫ መ݂൯, 37 where መ݂ሺ௥ሻ is the corresponding estimator at the r-th simulation run and መ݂ ̅ ൌ ଵோ ∑ መ݂ሺ௥ሻோ௥ୀଵ . The ratio of the root mean square error (RRMSE) may also be calculated to compare Bayesian estimators with DSE. Furthermore, 95% confidence interval ሾ ଴݂.଴ଶହୈୗ୉ሺ௥ሻ, ଴݂.ଽ଻ହୈୗ୉ሺ௥ሻሿ for the frequentist estimators is calculated as መ݂ሺ௥ሻ േ 1.96 ൈ ݎݎ݁݀ݐݏሺ መ݂ሺ௥ሻሻ . For Bayesian estimators, 95% credible intervals ሾ ଴݂.଴ଶହ୔୆୉ሺ௥ሻ, ଴݂.ଽ଻ହ୔୆୉ሺ௥ሻሿ and ሾ ଴݂.଴ଶହ୆୔୉୐୉ሺ௥ሻ, ଴݂.ଽ଻ହ୆୔୉୐୉ሺ௥ሻሿ are calculated from the posterior distribution. The interval length is calculated as the difference between the upper bound and the lower bound of each confidence (or credible) interval for the estimator. The average interval length therefore is calculated and compared among all three methods. The coverage probability, defined as the proportion of simulations that the confidence intervals or the credible intervals contain the true value (simulation parameter), is estimated as coverage probabity ൌ 100%ൈ ଵோ ∑ ܫ௥ோ௥ୀଵ , where ܫ௥ ൌ ൜1, if the confidence or credible interval covers ݂ 0, otherwise , ݎ ൌ 1,… , ܴ. 2.3.1 Monte Carlo Simulation Error To evaluate the impact of the number of simulation runs to the Monte Carlo Simulation Standard Error (MCSE), we perform R=5,000 simulation runs for each inbreeding coefficient f=-0.1, 0, 0.05, 0.1 or 0.2 under a moderate allele frequency p=0.3. At each simulation, a sample size of 50 genotypes are generated from the general inbreeding model. Since ݂ሺ௦ሻ or መ݂ is a consistent estimator, we choose the sample size of 50 to evaluate the performance of the proposed estimators under small sample size situation. Let መ݂ሺ௥ሻ be the Monte Carlo estimator of f from a simulation with r replicates. At each cumulative simulation run r (=1, …, 5000), the Monte Carlo Simulation Standard Error, MCSEሺ መ݂ሺ௥ሻሻ ൌ ටVarሺ መ݂ሺ௥ሻሻ, is calculated as if the simulations are stopped at that iteration (Koehler, Brown, & Haneuse, 2009). The results are plotted against the simulation iterations in Figure 4 (f=0.05, other parameter settings not shown since all have the similar pattern). Similar to Wu & Rao’s (2006) simulation studies specifically for the empirical likelihood and many other studies discussed in the review literature (Koehler, Brown, & Haneuse, 2009), Figure 4 suggests that a total of 1000 simulation iterations provide a reasonable compromise between 38 computational burden and the control of simulation error. For simulation runs at 5000, 2000, 1000 and 500, the corresponding MCSEs for BPELE are 0.0018, 0.0029, 0.0042 and 0.0059 respectively; the corresponding MCSEs for DSE are 0.0020, 0.0032, 0.0047 and 0.0066 respectively. The MCSEs for BPE are similar to the ones for BPELE at all simulation iterations. Considering both the simulation time and MCSE, we decide to use a total of 1000 runs in most of our simulation studies in this dissertation. Figure 4. Monte Carlo Simulation Standard Error at Accumulative Simulation Iterations. BPELE=Bayesian  Pseudo‐Empirical Likelihood Estimator, DSE=Direct Sample Estimator, PBE=Parametric Bayesian  Estimator. Allele frequency is p=0.3 and the inbreeding coefficient is f=0.05 in the simulation.  2.3.2 Compare Frequentist and Bayesian Estimators Generally as expected, when sample size increases, the bias, the MSE as well as the average confidence interval length decrease for both the DSE and the Bayesian estimators. As Cohen (1982) shown previously that the DSE is a consistent estimator, the simulations show that the bias diminishes to zero as the sample size increase to infinity. Table 4, Table 5, and Table 6 present the simulation results in comparing the DSE and two Bayesian estimators at the sample size of 50, 100 or 200, respectively. The parametric Bayesian method tends to have a positive bias, while the frequentist method often has a negative bias. The Bayesian empirical likelihood method has a positive bias when the allele frequency is small (p=0.1), while shows a negative bias when the allele frequency is moderate (p=0.3, 0.5). When the allele frequency is small (p=0.1), the Bayesian 39 methods have larger bias especially when the inbreeding coefficient is small (f=-0.1, 0, 0.05). A similar observation can be made for the MSE. When the allele frequency or the inbreeding coefficient gets bigger, both Bayesian methods have better performance in terms of the MSE. In all simulations except the first case (p=0.1, f=-0.1), the coverage of the Bayesian empirical likelihood method is able to maintain the nominal level and tends to show better value than the frequentist method and the parametric Bayesian method. This advantage becomes more obvious when the allele frequency is small. When the allele frequency is small (p=0.1), the DSE does not maintain the nominal coverage probability of 95%. As previously specified, in the multinomial model with re-parameterized genotype frequencies, the inbreeding coefficient f has a natural constraint ି௠௜௡ ሺ௣,ଵି௣ሻଵି௠௜௡ ሺ௣,ଵି௣ሻ ൑ ݂ ൑ 1 resulted from the law of total probability. This constraint is embedded in the Bayesian method we use. In practice, the allele frequency is often an unknown parameter and need be estimated itself from the data. Therefore, it is practically not feasible to check whether the estimator is within the model specified range or not. This can be easily checked in our simulation studies since the parameter inputs are all known to generate the data. Simulation studies show that a large percentage of the DSEs are outside of the constraint when the allele frequency is small (p=0.1). The problem gets worse if the actual inbreeding coefficient is also small. For example, when the sample size is 50, p=0.1 and f=0, 17.4% of all DSEs are outside of the parameter space of [-0.1111, 1] (Figure 5, Panel A). In human genomic data analysis, the minor allele frequency and the inbreeding coefficient are often small. Ignoring parameter constraint, therefore, results in the DSE resides outside of the parameter space induced by the model. On the other hand, the Bayesian estimator naturally satisfies the f constraint through a proper prior distribution. When the allele frequency is moderate or large (p=0.3, 0.5), all of the DSEs are within the constraint induced by the model. The simulation also shows that both Bayesian methods have coverage problem when the allele frequency is small (p=0.1) and the inbreeding coefficient is close to the lower boundary (f=-0.1). It should be recognized that under this parameter setting, the expected genotype frequencies are (0.801, 0.198, 0.001) and the lower bound of the expected inbreeding coefficient is -0.11. Given the small sample size of 50~200 in our simulation, the simulated number of genotype AA count maybe sparse. This resulted in a non-stable parametric Bayesian estimator. At the same time, the 40 empirical likelihood calculation is not stable at the boundary of the parameter space, thus it results in not optimal coverage for the estimator when the true value is close to the boundary. Since DSE has problem of being outside of parameter space, we argue that this situation need be carefully evaluated since none of three methods produce satisfactory solution when considering all four criteria including the coverage, the bias, the MSE, and the parameter space. Figure 5 plots the kernel densities for all three estimators from those 1000 simulations. When the allele frequency is large (p=0.5), all three density functions overlap with each other. These data is not plotted in this dissertation to simplify the output. Under all three sample sizes, the BPELE always has a single mode that is close to the true parameter input; while both the DSE and the PBE may have two modes in some simulation settings. In all three panels, the DSE is closer to the parameter input when both the allele frequency and the inbreeding coefficient are small (p=0.1, f=-0.1). However, it is also clear that a large proportion of the estimators are outside of the parameter’s lower boundary. In summary, when the allele frequency is moderate or large (p=0.3, 0.5), the PBE has smaller bias and better MSE. It also has larger coverage probability and shorter interval length than the frequentist DSE. The BPELE is comparable to the DSE. When the allele frequency is small (p=0.1), the Bayesian estimators have larger coverage probability and smaller MSE in the exchange of larger bias as well as longer interval length. On the other hand, the DSE is often outside of the model specified parameter range for f. 41 Table 4. Comparison of Direct Sample Estimator, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Simple Random Sampling, n=50) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Direct Sample Estimator  Parametric Bayesian Estimator   Bayesian Pseudo‐Empirical  Likelihood Estimator  Bias  MSE  CP  AL  ORP  Bias  MSE  CP  AL  Bias  MSE  CP  AL  0.1    ‐0.1  ‐0.0014  0.0033  93.4 0.1622 38.3 0.1416 0.0253  77 0.5067 0.2065 0.0463 55.2 0.5583 0  ‐0.0066  0.0187  40.6 0.3621 17.4 0.1132 0.0241  96.2 0.5558 0.1458 0.0266 100 0.606 0.05  ‐0.0046  0.0267  52.8 0.4507 10.6 0.1017 0.0252  93.7 0.5759 0.1192 0.0215 99.9 0.6296 0.1  ‐0.0102  0.0312  62.4 0.5191 7.8 0.0836 0.0236  95.6 0.5902 0.0896 0.0167 99.5 0.6489 0.2  ‐0.0298  0.0402  72.2 0.6037 2.2 0.0478 0.0215  97 0.6168 0.0382 0.0129 100 0.6849 0.3  ‐0.1  ‐0.0026  0.0173  93.2 0.5055 0 0.031 0.0152  96 0.4888 0.0186 0.0123 99.1 0.5599 0  ‐0.0134  0.0199  91.8 0.5362 0 0.0116 0.0164  95.1 0.51 ‐0.0073 0.015 98.5 0.5854 0.05  ‐0.0109  0.0207  92.1 0.5481 0 0.0096 0.0171  95.2 0.5172 ‐0.0105 0.0165 98 0.5931 0.1  ‐0.0007  0.0217  92.6 0.5574 0 0.0146 0.0183  95.7 0.5229 ‐0.0061 0.018 97.8 0.5974 0.2  ‐0.0145  0.0211  94.2 0.5645 0 ‐0.0063 0.0176  96.7 0.5279 ‐0.0266 0.0189 98 0.5973 0.5  ‐0.1  ‐0.0106  0.0214  93 0.5431 0 0.007 0.0191  96.4 0.5171 ‐0.0066 0.0197 96 0.5329 0  ‐0.0102  0.0208  93.2 0.548 0 0.0019 0.0183  95.1 0.5199 ‐0.0102 0.0192 94.9 0.5364 0.05  ‐0.0104  0.0197  94.5 0.5485 0 ‐0.0014 0.0173  95.5 0.52 ‐0.0124 0.0182 95.9 0.5367 0.1  ‐0.0046  0.0201  94.5 0.5469 0 0.0011 0.0177  96.1 0.5188 ‐0.0087 0.0186 95.6 0.5352 0.2  ‐0.0051  0.019  94.8 0.5399 0 ‐0.0053 0.017  96 0.514 ‐0.0131 0.0177 95.2 0.5274 42 Table 5. Comparison of Direct Sample Estimator, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Simple Random Sampling, n=100) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Direct Sample Estimator  Parametric Bayesian Estimator   Bayesian Pseudo‐Empirical  Likelihood Estimator  Bias  MSE  CP  AL  ORP  Bias  MSE  CP  AL  Bias  MSE  CP  AL  0.1    ‐0.1  ‐0.0004  0.0016  94.4 0.1241 36.8 0.0816 0.0088  84.6 0.3286 0.1522 0.0248 54.6 0.4271 0  ‐0.0066  0.0098  62.1 0.3113 10.7 0.06 0.0108  95.8 0.3919 0.0924 0.0116 99.9 0.4797 0.05  ‐0.0052  0.0132  75 0.378 3.1 0.0546 0.012  95.9 0.4215 0.0707 0.0092 99.9 0.5093 0.1  ‐0.0063  0.0157  83.7 0.4364 1.6 0.0452 0.0129  96.2 0.443 0.0474 0.0079 100 0.5342 0.2  ‐0.0135  0.02  88.3 0.5083 0.8 0.0248 0.0144  95.8 0.4777 0.0081 0.009 99.4 0.5769 0.3  ‐0.1  ‐0.002  0.0095  92.4 0.3653 0 0.0159 0.0088  93.6 0.3575 0.0021 0.008 98.3 0.4363 0  ‐0.0042  0.0096  93.9 0.3865 0 0.0086 0.0087  95.6 0.3745 ‐0.0062 0.0088 98 0.4472 0.05  ‐0.0064  0.01  93.8 0.3935 0 0.0041 0.0091  95.1 0.3802 ‐0.0103 0.0095 97.7 0.4473 0.1  ‐0.0107  0.0102  94.3 0.3982 0 ‐0.0024 0.0091  96.2 0.3847 ‐0.0159 0.0098 98 0.4456 0.2  ‐0.0047  0.0109  94.3 0.4018 0 ‐0.0014 0.01  95.4 0.3873 ‐0.0123 0.0106 96.9 0.4337 0.5  ‐0.1  ‐0.0063  0.0099  94.6 0.3873 0 0.0024 0.0093  94.8 0.3778 ‐0.0043 0.0095 94.9 0.3834 0  0.0018  0.01  94.3 0.39 0 0.0071 0.0094  94.6 0.38 0.0016 0.0096 94.7 0.3856 0.05  ‐0.0034  0.0099  95 0.3897 0 0.0004 0.0093  94.6 0.38 ‐0.0044 0.0095 95.2 0.3851 0.1  ‐0.0038  0.0108  92.8 0.3883 0 ‐0.0014 0.0101  93.5 0.3787 ‐0.0058 0.0104 93.3 0.3837 0.2  ‐0.0112  0.0092  94.2 0.3837 0 ‐0.0114 0.0087  95.1 0.3746 ‐0.0151 0.009 94.3 0.379 43 Table 6. Comparison of Direct Sample Estimator, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Simple Random Sampling, n=200) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Direct Sample Estimator  Parametric Bayesian Estimator   Bayesian Pseudo‐Empirical  Likelihood Estimator  Bias  MSE  CP  AL  ORP  Bias  MSE  CP  AL  Bias  MSE  CP  AL  0.1    ‐0.1  ‐0.0012  0.0008  95.2 0.0939 37.4 0.0432 0.0028  89.3 0.1989 0.1081 0.0123 61.4 0.3176 0  ‐0.0029  0.0053  83.3 0.249 3.7 0.0326 0.0055  95.6 0.2751 0.0544 0.0048 99.9 0.3792 0.05  ‐0.0054  0.0064  88.5 0.2958 0.6 0.026 0.0059  96.4 0.3023 0.0326 0.0036 100 0.4077 0.1  ‐0.0095  0.0083  88.5 0.3293 0.3 0.0181 0.0071  94.7 0.3243 0.0137 0.0042 99.9 0.4326 0.2  0.0003  0.0101  92.6 0.3804 0 0.0186 0.0088  94.8 0.3605 ‐0.0018 0.007 99.6 0.4734 0.3  ‐0.1  ‐0.002  0.0047  93.8 0.2609 0 0.0069 0.0045  94.8 0.2576 ‐0.0025 0.0045 98 0.3244 0  0.0029  0.0051  94 0.2758 0 0.0091 0.0049  95 0.271 0.001 0.005 97.8 0.323 0.05  ‐0.0039  0.0053  94.8 0.2798 0 0.0013 0.0051  95.1 0.2747 ‐0.0062 0.0053 98 0.321 0.1  ‐0.0015  0.0052  94.4 0.2833 0 0.0024 0.0049  95 0.2783 ‐0.0045 0.0051 97 0.3177 0.2  ‐0.0026  0.0058  92.9 0.2858 0 ‐0.001 0.0056  93.7 0.2806 ‐0.0065 0.0058 95.7 0.3107 0.5  ‐0.1  ‐0.0012  0.0054  93.8 0.2749 0 0.0029 0.0052  94.4 0.2713 ‐0.0002 0.0052 94 0.2734 0  ‐0.0028  0.005  94.9 0.2765 0 0.0001 0.0049  94.9 0.2729 ‐0.0028 0.0049 95.2 0.2749 0.05  ‐0.0055  0.005  95 0.2763 0 ‐0.0034 0.0049  95.1 0.2727 ‐0.006 0.0049 95.1 0.2746 0.1  0.0002  0.0052  93.1 0.2752 0 0.0013 0.005  93.7 0.2718 ‐0.0009 0.0051 93.6 0.2735 0.2  ‐0.0043  0.005  94.6 0.2713 0 ‐0.0044 0.0049  94.4 0.2676 ‐0.0062 0.0049 94.6 0.2697 44 A. n=50 B. n=100 C. n=200 Figure  5  Density  Plot  of  Simulated  Estimators  under  Different  Settings  (Simple  Random  Sampling).  BPELE=Bayesian  Pseudo‐Empirical  Likelihood  Estimator,  DSE=Direct  Sample  Estimator,  PBE=Parametric  Bayesian Estimator. Parameter p is the allele frequency and f is the inbreeding coefficient in the simulation.  Panel A, B and C are  for  sample size 50, 100 and 200 respectively. Red dashed vertical  line  is  the  true  parameter f; red dotted vertical line is the lower bound of the parameter space for f conditional on allele  frequency p.  45 2.3.3 Use of Prior Knowledge in Analysis One of the advantages of Bayesian method is to incorporate prior knowledge of the parameter of interest into modeling. With much advanced genetic sequencing technology in recent years, genomic data approaches exascale into exabyte range. Simply ignoring these accumulative information is a pitiful waste of data. For example, for the important SNPs of interest, the allele frequencies in the population are often well studied. Quite often, such information is also available for a subgroup of the population with certain characteristics. Furthermore, the official definition of the SNP requires that the MAF is equal to or greater than 1%. Using a non-informative uniform distribution on [0, 1] as a prior distribution for the allele frequency is practically convenient but not scientifically well justified. To investigate the impact of priors to the estimation of the inbreeding coefficient, we performed a simulation study as shown in Table 7. Beside the uniform priors on the whole parameter space that we used in previous simulations, we included a more restricted parameter range for the inbreeding coefficient. For the parametric Bayesian method, instead of a uniform prior for the allele frequency p, a beta(0.5, 0.5) is used to give more weights for the small allele frequencies. Conditioning on the p, a uniform prior for f in the range of (0, 0.5) is also studied to justify the situation when we have learned from previous knowledge of the parameter. In all these cases, using previous information produces estimators with less MSE, better coverage, and much less average coverage interval length. Under studied allele frequency p=0.1 and 0.3, compared with a non-informative prior as most literatures do, using previous information for the inbreeding coefficient results in estimators that are more concentrated around the true value as shown in Figure 6. This trend is most obvious for the Bayesian pseudo-empirical likelihood estimator (BPELE2) with a uniform prior (0, 0.5) for f under p=0.3. 46 Figure  6.  Density  Plot  of  Simulated  Estimators  under  Different  Priors  (Simple  Random  Sampling).  BPELE1=Bayesian Pseudo‐Empirical Likelihood Estimator with uniform prior (0,1) for f; BPELE2=Bayesian  Pseudo‐Empirical  Likelihood  Estimator  with  uniform  prior  (0,0.5)  for  f;  DSE=Direct  Sample  Estimator;  PBE1=Parametric Bayesian Estimator with uniform prior (0,1) for p and conditional uniform prior (‐p/(1‐ p),1) for f; PBE2=Parametric Bayesian Estimator with beta prior  (0.5,0.5) for p and conditional uniform  prior (‐p/(1‐p),1) for f; PBE3=Parametric Bayesian Estimator with beta prior  (0.5,0.5) for p and uniform  prior (0,0.5) for f.    47 Table 7. Comparison of Priors in the Bayesian Estimation (Simple Random Sampling, n=50)      DSE  PBE1 PBE2 PBE3 BPELE1  BPELE2 p=0.1  f=0.05  Bias  ‐0.0061  0.0993 0.0978 0.1318 0.1196  0.1505  MSE  0.0266  0.0248 0.0241 0.021 0.0217  0.0242  CP  51.9  95.2 95.3 94.5 99.8  99.7  AL  0.4455  0.5738 0.566 0.4175 0.6297  0.445  p=0.1  f=0.1  Bias  ‐0.0183  0.0782 0.0759 0.0962 0.0885  0.1098  MSE  0.0297  0.0217 0.0208 0.0132 0.0155  0.0136  CP  61  97.6 97.5 98.6 99.7  100  AL  0.504  0.5887 0.5805 0.4251 0.6478  0.4501  p=0.1  f=0.2  Bias  ‐0.0304  0.0449 0.0414 0.0296 0.0354  0.0313  MSE  0.0382  0.0215 0.0206 0.0057 0.0124  0.003  CP  75  97.7 98.1 99.8 99.5  100  AL  0.6141  0.6159 0.6091 0.4306 0.684  0.4554  p=0.3  f=0.05  Bias  ‐0.0185  0.0033 0.0037 0.0939 ‐0.0177  0.0985  MSE  0.0213  0.0175 0.0174 0.0128 0.0173  0.013  CP  91.3  95.5 95.3 95 97.8  97.3  AL  0.547  0.5155 0.5138 0.3363 0.593  0.3508  p=0.3  f=0.1  Bias  ‐0.0107  0.006 0.0061 0.0698 ‐0.0153  0.0718  MSE  0.0218  0.0181 0.0179 0.0098 0.0181  0.0092  CP  92.6  95.7 96 97.9 97.4  99.2  AL  0.5563  0.5224 0.5205 0.3614 0.5973  0.3744  p=0.3  f=0.2  Bias  ‐0.0082  ‐0.0002 ‐0.0007 0.0209 ‐0.0204  0.0188  MSE  0.0222  0.0184 0.0183 0.0068 0.0199  0.0058  CP  92.4  94.8 95.1 98.4 96.6  99.2  AL  0.563  0.5256 0.525 0.3885 0.5944  0.4014  * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length DSE=Direct Sample Estimator; PBE1=Parametric Bayesian Estimator with uniform prior (0,1) for p and conditional uniform prior (-p/(1-p),1) for f; PBE2=Parametric Bayesian Estimator with beta prior (0.5,0.5) for p and conditional uniform prior (-p/(1-p),1) for f; PBE3=Parametric Bayesian Estimator with beta prior (0.5,0.5) for p and uniform prior (0,0.5) for f. BPELE1=Bayesian Pseudo-Empirical Likelihood Estimator with uniform prior (0,1) for f; BPELE2=Bayesian Pseudo-Empirical Likelihood Estimator with uniform prior (0,0.5) for f; 48 Chapter 3: Estimation of the Inbreeding Coefficient under Unequal Probability Sampling The analysis of genetic data need not only consider the variation associated with statistical sampling, but also take the evolutionary genetic sampling into account. The former can be controlled by increasing the number of sampled units within populations and/or the number of sampled populations. However, the latter is an intrinsic property of the stochastic process (evolution) and therefore cannot be controlled by the sample size. Even if we could take the sampled populations back to a previous time point and run the evolution process under the exact same biological conditions, the genotype frequencies in the new evolved populations would differ from what has been already observed. Increasing the number of sampled individuals or the number of sampled populations will not decrease the variation associated with this evolutionary process. Holsinger & Weir (2009) suggested to characterize such genetic sampling by F-statistics. With the collection of genetic data from complex survey, it poses more challenges to data analysis because of its non-simple random sampling design. In both the NHANES and the HRS genetic data quality control report, the test of HWE is conducted as if the data is from a simple random sampling design (NHANES, 2015; Weir D. R., 2012). Furthermore, the estimated inbreeding coefficient from the HRS study is also under a simple random sampling assumption (Weir D. R., 2012). We believe that ignoring the statistical sampling in analysis may introduce some bias to the estimation of the inbreeding coefficient for the study population. In this chapter, we are trying to estimate the target population inbreeding coefficient, however based on samples with unequal selection probabilities. Previously developed multinomial model under simple random sampling need be extended to such sampling design with unequal probability of selection of units. 3.1 Direct Design-Based Estimator The design-based and model-assisted frequentist approach have been widely used in survey practice due to its capability to handle complex survey sampling design. The estimators of finite population parameters are proved to be design consistent for large samples. Let’s again look at a 49 biallelic case with alleles A and a (k=2). Suppose a finite population of size N, let yi (i=1, 2, …, N) be the number of allele A for each subject in the population. Individual with genotype aa has yi=0; individual with genotype Aa has yi=1; and individual with genotype AA has yi=2. The population total of each genotype count can be calculated from indicator function as ଴ܰ ൌ ෍ ܫሾ௬೔ୀ଴ሿ ே ௜ୀଵ , ଵܰ ൌ ෍ ܫሾ௬೔ୀଵሿ ே ௜ୀଵ , ଶܰ ൌ ෍ ܫሾ௬೔ୀଶሿ ே ௜ୀଵ , where ܫሾ௒ୀ௬ሿ is an indicator function with value of 1 if Y=y and 0 otherwise. If this finite population genotype is assumed to be selected from a super population with general inbreeding model described in (3), then the finite population inbreeding coefficient can be derived as ݂ ൌ 1 െ ଵܰ ܰ⁄ 2 ∗ 2 ଶܰ ൅ ଵܰ2ܰ ∗ 2 ଴ܰ ൅ ଵܰ2ܰ . It can also be expressed as 1 - (number of observed heterozygotes) / (number of expected heterozygotes) at the finite population level. Finite population parameter f can be viewed as a function of population’s genotype frequencies ଴ܰ ܰ, ଵܰ ܰ⁄⁄ , and ଶܰ ܰ⁄ . Similarly as we discussed in Chapter 1, this can be interpreted as a correlation coefficient between two homologous genes in uniting gametes from this finite population. Under unequal probability sampling of n subjects from this finite population, each sampled subject has a design weight ݀௜, which is the inverse of sampling unit inclusion probability with possible nonresponse and post stratification adjustments. Therefore, Horvitz–Thompson type estimators of the population total and three genotype count totals are ෡ܰ ൌ ෍ ݀௜ ௡ ௜ୀଵ , ଴ܰ෢ ൌ෍ ݀௜ ௡ ௜ୀଵ ܫሾ௬೔ୀ଴ሿ, ଵܰ෢ ൌ෍ ݀௜ ௡ ௜ୀଵ ܫሾ௬೔ୀଵሿ, and ଶܰ෢ ൌ෍ ݀௜ ௡ ௜ୀଶ ܫሾ௬೔ୀଶሿ, respectively. The design-based direct survey estimator for the finite population inbreeding coefficient therefore can be expressed as a function of the estimators of the population genotype frequencies, e.g., 50 መ݂ ൌ 1 െ ෡ܰଵ ෡ܰ⁄ 2 ∗ 2 ෡ܰଶ ൅ ෡ܰଵ2 ෡ܰ ∗ 2 ෡ܰ଴ ൅ ෡ܰଵ 2 ෡ܰ . The analytical form of its standard error using the Taylor linearization is not trivial and is outside of the scope of this dissertation. Instead, the variance can be easily derived by using the resampling technique (Rust & Rao, 1996). We use jackknife method to estimate the variance of the survey deign-based estimator መ݂. 3.2 Parametric Bayesian Estimator with Survey Weight As we see from Chapter 2, the Bayesian method can restrict the estimator to its parameter space through the prior distribution. With the introduction of MCMC, the Bayesian method is increasingly used for multi-level modeling. However, the full Bayesian model under the complex sampling design is not feasible to include all the sampling design information (Gelman, 2007). On the other hand, not all survey sampling design information is generally included in the public released data in order to prevent disclosure of participants’ social, economic and personal health information. Typical national surveys such as the NHANES and the HRS include sampling design information including the analysis stratum, the primary sampling units (PSUs) as well as the final adjusted sampling weight. Rao (2011) had an extensive review of the application of the Bayesian method in the complex survey practice, focusing on the descriptive finite population parameters. Among all three challenges (appropriate likelihood, proper prior and posterior) to the Bayesian method, formalizing the likelihood function is the biggest hurdle to the survey statistician. For multipurpose complex survey, the parametric Bayesian method with distribution assumptions cannot be easily generalized due to the difficulties in validating its assumptions. For this dissertation, we are interested in screening SNPs through the inbreeding coefficients to measure their magnitude of deviation from the HWE. Let ݀ప෩ ൌ ௗ೔∑ ௗ೔೙೔సభ be the normalized design weight so that ∑ ݀ప෩ ௡௜ୀଵ ൌ 1. Assume that the weights ݀ప෩ are independent of yi, i=1, …, n, therefore they can be considered as fixed numbers given the sample. Follow Skinner (1989) and Ghosh & Maiti (2004), the weighted likelihood is ݈ௗሺ݌, ݂ሻ ൌ ∏ ܲݎௗഢ෪ሺݕ௜|݌, ݂ሻ௡௜ୀଵ ൌ ܥሺ݀, ݕሻ݌଴∑ ௗ෨೔ ೙೔సభ ூൣ೤೔సబ൧݌ଵ∑ ௗ෨೔ ೙೔సభ ூൣ೤೔సభ൧݌ଶ∑ ௗ෨೔ ೙೔సభ ூൣ೤೔సమ൧, 51 where ܥሺ݀, ݕሻ is a constant depending on the data and the sampling weights. It can be seen from the above likelihood that it still follows a multinomial distribution with the parameter ࢖ ൌ ሺ݌଴, ݌ଵ, ݌ଶሻ and the normalized weighted counts for genotypes. We use ݌଴, ݌ଵ, ݌ଶ to represent the previous genotype frequency ݌ଶଶ, ݌ଵଶ, ݌ଵଵ when y=0, 1, or 2, respectively. ݈ܽ݅݉݋݊݅ݐ݈ݑܯ~࢖|∗࢔ሺ࢔∗, ࢖ሻ, or, ܲݎሺ࢖|∗࢔ሻ ൌ ܲݎ൫ሺ݊଴∗, ݊ଵ∗, ݊ଶ∗ሻหሺ݌଴, ݌ଵ, ݌ଶሻ൯ ൌ ݊ ∗! ݊଴∗! ݊ଵ∗! ݊ଶ∗! ݌଴ ௡బ∗݌ଵ௡భ∗݌ଶ௡మ∗. In our Bayesian framework, we use 2p to denote a finite population parameter തܻ ൌ ଵே∑ ݕ௜ே௜ . The design-based variance is calculated as Varୢୣୱ୧୥୬ሺ̂݌ሻ ൌ ଵସ Varୢୣୱ୧୥୬ሺ തܻ෠ሻ . Under simple random sampling without replacement, the variance is Varୱ୰ୱ୵୭୰ሺ̂݌ሻ ൌ ଵସ Varୱ୰ୱ୵୭୰ሺ തܻ෠ሻ. The design effect is Deff ൌ Varୢୣୱ୧୥୬ሺ̂݌ሻ/Varୱ୰ୱ୵୭୰ሺ̂݌ሻ and therefore the effective sample size is ݊∗ ൌ ݊/Deff. We incorporate the complex sampling design through the effective sample size ݊∗ and effective genotype count ࢔∗ ൌ ሺ݊଴∗, ݊ଵ∗, ݊ଶ∗ሻ in our Bayesian multinomial model, in which ݊଴∗ ൌ ݊∗෍ ሚ݀௜ ௡ ௜ୀଵ ܫሾ௬೔ୀ଴ሿ, ݊ଵ∗ ൌ ݊∗෍ ሚ݀௜ ௡ ௜ୀଵ ܫሾ௬೔ୀଵሿ, ݊ଶ∗ ൌ ݊∗෍ ሚ݀௜ ௡ ௜ୀଵ ܫሾ௬೔ୀଶሿ. To be included in the multinomial model, all above effective sample size and effective genotype counts are rounded to their nearest integers. The Bayesian multinomial model using the above effective sample size and effective genotype counts can therefore be specified as follows: Level 1: ࢔∗~Multinomialሺ݊∗, ࢖ሻ, ࢔∗ ൌ ሺ݊଴∗, ݊ଵ∗, ݊ଶ∗ሻ, ࢖ ൌ ሺ݌଴, ݌ଵ, ݌ଶሻ, ݊∗ ൌ ݊଴∗ ൅ ݊ଵ∗ ൅ ݊ଶ∗, ቐ ݌଴ ൌ ݍଶ ൅ ݂ݍ݌, ݌ଵ ൌ 2ݍ݌ሺ1 െ ݂ሻ ݌ଶ ൌ ݌ଶ ൅ ݂ݍ݌. , Level 2: ݌~Uniformሾ0，1ሿ ݂|݌~Uniform ቂି୫୧୬ሺ௣,ଵି௣ሻଵି୫୧୬ሺ௣,ଵି௣ሻ , 1ቃ 52 To model the categorical observations, Jiang & Lahiri proposed generalized linear mix model in the small area estimation context (2006). The survey design may be informative and need be considered in the model (Chen & Lahiri, 2012). Our using of effective sample size in the hierarchical Bayesian model with complex survey data is further inspired by a similar usage in the Bayesian method with pseudo-empirical likelihood (Rao & Wu, 2010) that we will discuss in the next section. Such application of effective sample size is not new for survey practitioners. A recent example can be found in the context of small area estimation from Chen, Wakefield and Lumely (2014). To take the sampling design into account, other alternative method is to model the distribution of the weights for the non-sampled units in the population and simultaneously include them as predictors in a nonparametric Gaussian process regression (Si, Pillai, & Gelman, 2015). 3.3 Bayesian Pseudo-Empirical Likelihood Estimator The Empirical likelihood (EL), introduced by Owen (1988), is a powerful nonparametric method for statistical inference and recently gets more and more attention. Although in this dissertation the proposed method is mainly for estimating the inbreeding coefficient from survey sample, it can be a useful tool to handle auxiliary variables such as environmental information in the gene-disease association study. For example, disease such as cancer is often age dependent. In the national level complex survey, the population average of age is known from the Census. By incorporating age into the Bayesian model with the empirical likelihood, an age-adjusted genotypic effect can be estimated from complex survey sample. The proposed method is also flexible in terms of incorporating the prior information. For information available on the parameter of interest, either from independent researches or from related measures within the same study, it can be incorporated into the analysis through proper usage of the prior distribution. Considering the finite population a random sample from an infinite superpopulation, Chen & Sitter (1999) first proposed a pseudo empirical log-likelihood for a general sampling design as ∑ ݀௜log ሺߨ௜ሻ௜∈௦ , a Horvitz-Thompson estimator of the “census” log-likelihood ݈ேሺπሻ ൌ ∑ log ሺߨ௜ሻே௜ୀଵ . Wu and Rao (2006) proposed the pseudo-empirical likelihood in the form of ݈௉ா௅ ൌ ݊∑ ݀ప෩ log ሺߨ௜ሻ௜∈௦ , where ݀ప෩ ൌ ௗ೔∑ ௗ೔೔∈ೞ are the normalized design weights. If the design weights ݀௜ are all equal, then ݊∑ ݀ప෩ log ሺߨ௜ሻ௜∈௦ is reduced to the original definition of the empirical likelihood 53 ∑ log ሺߨ௜ሻ௜∈௦ . The sample size n in the above empirical likelihood could also be replaced by the effective sample size, which is defined as sample size n divided by the design effect. By using the effective sample size, this method combines the survey design information and Bayesian models in the estimation of population mean (Rao & Wu, 2010). Maximizing those proposed pseudo empirical likelihood functions with the same constraints on those ߨ௜’s are essentially the same. We need to maximize the likelihood ݈௉ா௅ with the following restrictions on the total probability and unbiased estimating functions: ߨ௜ ൐ 0,෍ ߨ௜௜∈௦ ൌ 1,෍ ߨ௜ሾݕ௜ሺ2 െ ݕ௜ሻ௜∈௦ െ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻሿ ൌ 0. Although it is possible to include another constraint based on estimating equation ܧሺܻ െ 2݌ሻ ൌ 0 by using Lagrange method as we discussed in Section 2.3, the benefit is significantly impacted by the computation burden imposed by the calculation of the Lagrange parameter ߣ. Lagrange method is used to derive the ߨො௜ to maximize the pseudo empirical log-likelihood by first setting ܩ ൌ ݊෍ ݀ప෩ log ሺߨ௜ሻ௜∈௦ െ ݊ߣ෍ ߨ௜ሾݕ௜ሺ2 െ ݕ௜ሻ௜∈௦ െ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻሿ െ ߛሺ෍ ߨ௜௜∈௦ െ 1ሻ. Taking partial derivative of G with respect to ߨ௜ and setting to 0, we have: ߲ܩ ߲ߨ௜ ൌ ݊ ሚ݀௜ ߨ௜ െ ݊ߣሾݕ௜ሺ2 െ ݕ௜ሻ െ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻሿ െ ߛ ൌ 0. The above equation gives solution ߨො௜ ൌ ሚ݀௜ 1 ൅ ߣሾݕ௜ሺ2 െ ݕ௜ሻ െ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻሿ , where ߣ need be solved by equation ෍ ሚ݀௜ሾݕ௜ሺ2 െ ݕ௜ሻ െ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻሿ1 ൅ ߣሾݕ௜ሺ2 െ ݕ௜ሻ െ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻሿ௜∈௦ ൌ 0. Unlike method developed in Qin & Lawless (1994) where the constraint based on the second moment is a known function of the first moment parameter, constraint in our case depends on the unknown population nuisance parameter p as well as the parameter f we are interested in. 54 Therefore, it is not a smooth function. Sitter & Wu (2002) suggested that the estimation based on the second-order population function such as quadratic function requires knowledge of the second- order inclusion probability induced from a synthetic finite population as well as the knowledge of the second-order auxiliary variable information. Without an obvious known auxiliary variable which is closely related to the genotype, to improve the efficiency of computation, an unbiased sample estimator of the allele frequency ̂݌ is used to replace p. To simplify our notation, we define V=Y(Y-2), an indicator variable for the heterozygote. Let ߠ ൌ 2ሺ1 െ ݂ሻ̂݌ሺ1 െ ̂݌ሻ. Therefore, we have ݒ௜ ൌ ݕ௜ሺ2 െ ݕ௜ሻ and ݑ௜ ൌ ݕ௜ሺ2 െ ݕ௜ሻ െ 2ሺ1 െ ݂ሻ̂݌ሺ1 െ ̂݌ሻ ൌ ݒ௜ െ ߠ. We want to maximize the profile pseudo-empirical log-likelihood with new constraints as: ߨ௜ ൐ 0,෍ ߨ௜௜∈௦ ൌ 1,෍ ߨ௜௜∈௦ ݑ௜ ൌ 0. Using the Lagrange method by taking the partial derivative of ܩ ൌ ݊∑ ݀ప෩ log ሺߨ௜ሻ௜∈௦ െ ݊ߣ∑ ߨ௜௜∈௦ ݑ௜ െ ߛሺ∑ ߨ௜௜∈௦ െ 1ሻ with respect to ߨ௜ and setting this to 0, we have: ߲ܩ ߲ߨ௜ ൌ ݊ ݀ప෩ ߨ௜ െ ݊ݑߣ௜ െ ߛ ൌ 0. The above equation gives the solution ߨො௜ ൌ ݀ప ෩ 1 ൅ ݑߣ௜ . The empirical likelihood can be estimated as ݈௉ா௅ ൌ ݊෍ ݀ప෩ log ሺ ݀ప ෩ 1 ൅ ݑߣ௜ሻ௜∈௦ ൌ ݊෍ ݀ప ෩ log ሺ݀ప෩ ሻ௜∈௦ െ ݊෍ ݀ప ෩ logሺ1 ൅ ݑߣ௜ሻ௜∈௦ , where ߣ ൌ ߣሺ݂ሻ is the solution to the equation ∑ ௗഢ෪௨೔ଵାఒ௨೔௜∈௦ ൌ 0. Chen, Sitter & Wu (2002) proposed an algorithm based on the modified Newton’s method and it is theoretically guaranteed to be convergent. Multiplying 1 ൅ ݑߣ௜ െ ݑߣ௜ to ௗഢ෪௨೔ଵାఒ௨೔, it can be verified that 55 ෍ ݀ప෩ݑ௜1 ൅ ݑߣ௜௜∈௦ ൌ෍ ݀ప෩ݑ௜ሺ1 ൅ ݑߣ௜ െ ݑߣ௜ሻ 1 ൅ ݑߣ௜௜∈௦ ൌ෍ െ݀ߣప෩ݑ௜ଶ ൅ ݀ప෩ݑ௜ሺ1 ൅ ݑߣ௜ሻ 1 ൅ ݑߣ௜௜∈௦ ൌ െߣ෍ ݀ప෩ݑ௜ ଶ 1 ൅ ݑߣ௜௜∈௦ ൅෍ ݀ప ෩ݑ௜௜∈௦ ൌ 0. Therefore ߣ ∑ ௗഢ෪௨೔మଵାఒ௨೔௜∈௦ ൌ ∑ ݀ప෩ݑ௜௜∈௦ . Let ܷ∗ ൌ ݉ܽݔ௜∈௦ݑ௜ , which is bounded since yi is the count for the allele A and yi(2- yi) is an indicator for the heterozygote, we have |ఒ|ଵା|ఒ|௎∗ ∑ ݀ప෩ݑ௜ଶ௜∈௦ ൑ |∑ ݀ప෩ݑ௜௜∈௦ |. Under the regulatory assumption that the Horvitz-Thompson type estimator is asymptotically normally distributed, we have ∑ ݀௜௜∈௦ ݒ௜ ൌ ܸ ൅ గܱሺ݊ିଵ/ଶሻ , where ܸ is the finite population heterozygote frequency. Furthermore, assuming the sampling design π(s) satisfies ∑ ௗ೔೔∈ೞே ൌ 1 ൅ గܱሺ݊ିଵ/ଶሻ, we have ෍ ሚ݀௜௜∈௦ ݒ௜ ൌ ∑ ݀௜௜∈௦ ݒ௜ ∑ ݀௜௜∈௦ /ܰ ൌ ܸ ൅ గܱ ൬݊ ିଵଶ൰. Since ∑ ݀ప෩ݑ௜ଶ ൌ ∑ ݀ప෩ ሺݒ௜ െ ߠሻଶ௜∈௦௜∈௦ is a Hajek estimator of ܵ௩ଶ, ∑ ݀ప෩ݑ௜ଶ ൌ௜∈௦ ܵ௩ଶ ൅ ܱሺ1ሻ. It can be derived that ߣ ൌ ∑ ௗഢ෪௨೔೔∈ೞ ∑ ௗഢ෪௨೔మ೔∈ೞ ൅ గܱሺ݊ ିଵ/ଶሻ. Using the Taylor series expansion of log(1+x) at ݔ ൌ ݑߣ௜ ൌ ߣሺݒ௜ െ ߠሻ up to the second order, suppose we have a non-informative but proper uniform prior ߮ሺ݂ሻ ∝ 1, the posterior therefor is ߮ሺ݂|ݕሻ ൌ ݁௟ುಶಽ ൈ ߮ሺ݂ሻ ൌ exp ሼlogሾ߮ሺ݂ሻሿ ൅ ݊෍ ݀ప෩ log ሺߨො௜ሻ௜∈௦ ሽ ∝ exp ሼെ݊෍ ݀ప෩ log ሺ1 ൅ ݑߣ௜ሻሽ௜∈௦ ൌ exp ሼെ݊෍ ݀ప෩ ሺݑߣ௜ െ ሺݑߣ௜ሻଶ/2ሻሽ ௜∈௦ ൌ exp ሼെ݊ߣሾ෍ ݀ప෩ ሺݒ௜ െ ߠሻ െ ∑ ݀ప෩ ሺݒ௜ െ ߠሻ௜∈௦ ∑ ݀ప෩ ሺݒ௜ െ ߠሻଶ௜∈௦ ൈ ∑ ݀ప෩ ሺݒ௜ െ ߠሻଶ௜∈௦ 2 ሿሽ ௜∈௦ 56 ൌ exp ቊെ݊ ∑ ݀ప෩ ሺݒ௜ െ ߠሻ௜∈௦ ∑ ݀ప෩ ሺݒ௜ െ ߠሻଶ௜∈௦ ൈ ∑ ݀ప෩ ሺݒ௜ െ ߠሻ௜∈௦ 2 ቋ ൌ exp ൝െ݊൫∑ ݀ప෩௜∈௦ ݒ௜ െ ߠ൯ ଶ 2∑ ݀ప෩ ሺݒ௜ െ ߠሻଶ௜∈௦ ൡ ൌ exp ۖە ۔ ۖۓ െ ሾ݂ െ ቆ1 െ ∑ ݀ప෩௜∈௦ ݒ௜2̂݌ሺ1 െ ̂݌ሻቇሿଶ 2ܵ௩ଶ/݊ ۙۖ ۘ ۖۗ sinceܧగ൫∑ ݀ప෩௜∈௦ ݒ௜൯ ≐ തܸ , ܧగሼ∑ ݀ప෩ ሺݒ௜ െ ߠሻଶ௜∈௦ ሽ ≐ ܵ௩ଶ and ௌೡ మ ௡ ൌ గܸሺ∑ ݀ప෩௜∈௦ ݒ௜ሻ.Therefore similarly as concluded in Rao & Wu (2010), the Bayesian posterior pseudo-empirical maximum likelihood estimator of the inbreeding coefficient is asymptotically normal. Unlike a non-informative improper prior for the location parameter in Rao & Wu’s (2010) method, in our case the non-informative prior ߮ሺ݂ሻ is a uniform distribution. The empirical likelihood is bounded above by one. It can be verified that ׬ ݁௟ುಶಽ߮ሺ݂ሻ݂݀ ൑ ߮׬ሺ݂ሻ݂݀ ൌ 1, which implies that the posterior is proper (Chaudhuri & Ghosh, 2011). For more complicate improper priors, Monahan & Boos (1992) discussed that the empirical likelihood is proper if the resulting credible intervals have correct coverages for all absolutely continuous prior. Lazar (2003) first studied this property in a simulation study. Fang & Mukerjee (2005; 2006) studied the coverage of the posterior credible intervals using the empirical likelihood and proved that the posterior one-sided credible interval have o(m-1) margin of error. In Chaudhuri & Ghosh (2011)’s simulation study, the coverage probabilities were compared to their nominal values to justify the use of the empirical likelihood in the Bayesian inference. Due to the nonparametric nature of the empirical likelihood, the posterior distribution is not in a closed form therefore is not tractable. Approximate Bayesian Computation (ABC) method can be used for the Bayesian inference when the posterior is not tractable but the distribution for the prior and the likelihood function can be both simulated. However, the empirical likelihood makes it difficult for computation based on a complex statistical model. To bypass the explicit computation 57 of the posterior distribution, the importance sampling method can be performed with obvious two reasons. First, the posterior empirical likelihood is convergent and therefore proper for the Bayesian inference as the number of observations increase to infinity. Second, without requirement of simulating replications of data having many tuning parameters from the sampling model, it substantially reduces the computation time. Using the empirical likelihood as the basis for the Bayesian inference (Lazar, 2003), we can calculate a finite expectation of the form ܧఝ൫݂หݕ൯൫ݑሺܨሻ൯ ൌ ݑ׬ሺ݂ሻ߮ሺ݂|ݕሻ݂݀ with respect to the posterior distribution as නݑሺ݂ሻ߮ሺ݂|ݕሻ߮ሺ݂ሻ ߮ሺ݂ሻ݂݀ ൌ නݑሺ݂ሻ ݁௟ುಶಽ ൈ ߮ሺ݂ሻ ߮ሺ݂ሻ ߮ሺ݂ሻ݂݀ ൌ නሼݑሺ݂ሻݓሺ݂ሻሽ ߮ሺ݂ሻ݂݀ ൌ ܧఝሺ௙ሻሼݑሺܨሻݓሺܨሻሽ. To keep the notation consistent, here F represents the random variable for inbreeding coefficient f and ݓሺ∙ሻ ൌ ఝ൫∙หݕ൯ఝሺ∙ሻ is the weighting function. If we are interested in estimating the posterior mean, then ݑሺܨሻ ൌ ܨ. A basic sampling algorithm could be developed as following two steps to generate an importance sample ቀ ௜݂, ݁௟ುಶಽ൫ ௜݂ห࢟൯ቁ , ݅ ൌ 1,… ,ܯ: Step 1: for i=1 to M, sampling fi from ߮ሺ݂ሻ; Step 2: use Lagrange method and Newton algorithm to calculate ݁௟ುಶಽ൫ ௜݂ห࢟൯, which is a natural distance and importance weight (Mengersen, Pudlo, & Robert, 2013). Bayesian inference therefore can be generated using the estimated posterior distribution based on the importance sampling. With the above prove that the posterior empirical likelihood is proper, we used a grid search method to find the posterior mean as well as 95% credible interval from the posterior distribution, instead of a rigorous simulation. In order to derive the posterior distribution ߮ሺ݂|ݕሻ , we need find the normalizing constant function c(y) such that ׬ ߮ሺ݂|ݕሻ ݂݀ ൌ ׬ ܿሺݕሻ݁௟ುಶಽ߮ሺ݂ሻ ݂݀ ൌ 1 . Under our concavity assumption for the posterior, the grid search method is applied to calculate this normalizing factor. The theoretical parameter support for the posterior density is ି௣೘೔೙ଵି௣೘೔೙ ൑ ݂ ൑ 1, ݌௠௜௡ ൌ ݉݅݊ሺ̂݌, 1 െ ̂݌ሻ. Computation of the pseudo posterior 58 density near the endpoint of the parameter support is problematic since the non-existence of the solution at those endpoints. A non-zero tolerance ϵ (for example, 0.00001) is used such that the practical support (c1, c2) is used with ି௣೘೔೙ଵି௣೘೔೙ ൏ ܿଵ ൏ መ݂௉ா௅ ൏ ܿଶ ൏ 1, ݄ሺܿଵሻ ൏ ϵ and ݄ሺܿଶሻ ൏ ϵ, where ݄ሺ݂ሻ ൌ ݁௟ುಶಽሺ௙|௬ሻ ൌ exp ሼ݊෍ ݀ప෩ log൫ߨො௜ሺ݂ሻ൯ሽ௜∈௦ ൌ exp ሼ݊෍ ݀ప ෩ log ሺ1 ൅ ߣሺ݂ሻݑ௜ሻሽ௜∈௦ . First, we divide the un-normalized posterior into m grids, each with width w, starting from c1, which is a value that is at the far left of parameter space. We then evaluate the un-normalized posterior at each grid point at c1+w, c1+2w, …, c2=c1+mw, where the last grid point is at the far right of the practical support. At each grid point m, fm, ߣሺ ௠݂ሻ, ߨො௠and the height at un-normalized posterior ݁௟ುಶಽ௠ can be calculated. The normalizing constant function ܿሺݕሻ ൌ ሼ׬ ݄ሺ݂ሻ݂݀௖మ௖భ ሽିଵ is therefore estimated as the inverse of the area under the un-normalized posterior, which is the sum of the areas of rectangles calculated as the width w times the height ݁௟ುಶಽ௠ at the un-normalized posterior ordinates. Finally, the posterior mean and 95% credible interval ሾ ଴݂.଴ଶହ୆୔୉୐୉, ଴݂.ଽ଻ହ୆୔୉୐୉ሿ can be derived from the estimated posterior distribution. 3.4 Simulation Study A simulation study for a single locus with two alleles (A, a) is conducted to evaluate the performance of the above three estimators. Let a finite population be of size N= 4,800 individuals consisting of two stratum with sub-population size ratio as 5:1. The genotype records are generated independently from the multinomial model with genotype frequencies of ܲሺܽܽሻ ൌ ሺ1 െ ݌ሻଶ ൅ ݌ሺ1 െ ݌ሻ݂, ܲሺܽܣሻ ൌ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻ, and ܲሺܣܣሻ ൌ ݌ଶ ൅ ݌ሺ1 െ ݌ሻ݂. The parameters for the simulation are the allele A frequency p=0.1, 0.3, or 0.5 and the inbreeding coefficient f=-0.1, 0, 0.05, 0.1, or 0.2. For the realized data, estimators of the inbreeding coefficient are calculated by methods described above. The bias and mean square error (MSE) of the estimator are calculated similarly as in Chapter 2, except that the parameter is based on the finite population of size N=4,800 instead of the super population parameter input. 59 3.4.1 Stratified Simple Random Sampling At each one of 1000 iterations in our first simulation design, a sample size of twenty (20), forty (40) or eighty (80) samples were randomly selected from each stratum. Therefore, the ratio of the sample selection probability is 1:5, an inverse of stratum population size. The weights from this stratified simple random sampling design is non-informative. Table 8, Table 9, and Table 10 report the simulation results in comparing design-based estimator and two Bayesian estimators for the bias, MSE, coverage probability (CP), out of range probability (ORP), and average length (AL). Similarly as the results from simple random sampling in Chapter 2, the parametric Bayesian method tends to have positive bias, while the frequentist method often has negative bias. The Bayesian empirical likelihood method has a positive bias when the allele frequency is small (p=0.1), while shows negative bias when the allele frequency is moderate (p=0,3, 0.5). When the allele frequency is small (p=0.1), the Bayesian methods have larger bias especially when the inbreeding coefficient is also small (f=-0.1, 0, 0.05). When the allele frequency is moderate (p=0.3, 0.5), the Bayesian methods tend to have smaller bias especially for the parametric Bayesian estimator. On average from all simulation scenario, the Bayesian empirical likelihood method has the smallest MSE, followed by the parametric Bayesian method. A similar observation can be made for the coverage probability. In all three simulations with different sample sizes, the design-based estimator does not maintain the nominal coverage probability. When the allele frequency is large (p=0.5), all three methods cannot maintain the nominal coverage probability. In fact, the genotype frequencies are similar in this situation for all inbreeding coefficient values as show in Table 3. When the allele frequency is small (p=0.1) and the inbreeding coefficient is not near the border of the parameter space, the Bayesian pseudo-empirical likelihood estimator has much better MSE and coverage than both the parametric Bayesian estimator and design-based estimator. This is expected as the design-based method is not stable when the data is skewed so the asymptotic assumption may not hold well. Similarly as shown in Chapter 2, the design-based estimator may produce an estimator outside of the parameter space, especially when the allele frequency is small (p=0.1) and the inbreeding coefficient is also small (f=-0.1, 0). For example, when sample size is 40 per stratum and the true inbreeding coefficient is 0, 14.1% of the simulated design-based estimators have the values lower 60 than -0.1111, which is the lower boundary of the parameter space for f under p=0.1 (Table 9, Figure 7 Panel B). 61 Table 8. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling，n=20 per Strata) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length               p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0045  0.0042  85.9 0.1874 45.5 0.1604 0.0328 76.3 0.5693 0.2286 0.0588 48.2 0.6095 0  ‐0.0244  0.027  43.8 0.4758 25.4 0.107 0.0274 94.7 0.5978 0.1565 0.034 99.3 0.6522 0.05  ‐0.0204  0.0414  50.5 0.6315 17.4 0.0936 0.0311 94.6 0.6175 0.1323 0.0304 98.8 0.6765 0.1  ‐0.0184  0.054  56.3 0.6956 12.9 0.0817 0.0353 94 0.6274 0.11 0.028 98.9 0.6945 0.2  ‐0.028  0.0679  69.8 0.8677 4.8 0.0438 0.0351 95.2 0.6525 0.0541 0.0228 98.5 0.7311 0.3  ‐0.1  ‐0.0161  0.0317  86.3 0.5913 3.1 0.0277 0.0275 89.2 0.5306 0.0225 0.0216 96.1 0.5901 0  ‐0.015  0.0336  88.8  0.6332  1.2  0.0192  0.0282  89.7  0.5495  0.0043  0.0239  95.3  0.6197  0.05  ‐0.0242  0.0359  88.9 0.6442 0.5 0.0055 0.0287 90.2 0.5518 ‐0.0093 0.0252 96.1 0.6263 0.1  ‐0.0157  0.0368  88.5 0.6549 0 0.0078 0.0291 89.9 0.5639 ‐0.0096 0.0269 95.3 0.6362 0.2  ‐0.0153  0.0384  88.9 0.6653 0 ‐0.0025 0.0306 89.2 0.5663 ‐0.0222 0.0302 93.9 0.6425 0.5  ‐0.1  ‐0.0112  0.0339  90.4 0.631 0 0.0116 0.0296 90 0.5606 ‐0.0054 0.0303 90.9 0.5894 0  ‐0.013  0.0341  91 0.6365 0 0.0027 0.0296 90.2 0.5662 ‐0.0121 0.0307 91.8 0.595 0.05  ‐0.0264  0.038  88.1 0.6366 0 ‐0.0134 0.0326 87.6 0.5654 ‐0.0277 0.0345 88.6 0.5948 0.1  ‐0.0157  0.0353  90.8 0.6355 0 ‐0.0069 0.0307 90.9 0.5686 ‐0.0197 0.032 91.3 0.5939 0.2  ‐0.0134  0.0379  88.6 0.6275 0 ‐0.0108 0.0334 87.8 0.563 ‐0.0224 0.0347 89.6 0.5875 62 Table 9. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling，n=40 per Strata) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0028  0.0027  89.2 0.144 43.8 0.093 0.0121 80.9 0.3747 0.1678 0.0312 51.4 0.4696 0  ‐0.0097  0.0175  62.4 0.4074 14.1 0.0605 0.0169 92.3 0.4285 0.1036 0.0166 99.4 0.5266 0.05  ‐0.015  0.0223  75 0.4965 7.5 0.0464 0.0187 93.8 0.4506 0.076 0.0133 99.8 0.5516 0.1  ‐0.0163  0.0285  77.4 0.5412 5.1 0.039 0.0216 93.7 0.4742 0.0557 0.0132 98.8 0.5731 0.2  ‐0.0246  0.0388  85.2 0.6286 1.6 0.015 0.0269 92.7 0.506 0.0102 0.0163 98.9 0.6112 0.3  ‐0.1  ‐0.0094  0.0168  88.4 0.419 0.3 0.0111 0.0153 89 0.3911 0.0014 0.0133 96.8 0.4665 0  ‐0.0076  0.0176  89  0.4422  0  0.0089  0.016  89.3  0.4076  ‐0.0053  0.0149  95.9  0.484  0.05  ‐0.0076  0.0181  89.5 0.4505 0 0.0063 0.0162 90.5 0.4133 ‐0.0086 0.0158 95.8 0.4896 0.1  ‐0.0071  0.0185  89.7 0.4567 0 0.0042 0.0164 90.5 0.4203 ‐0.0113 0.0167 95.2 0.4906 0.2  ‐0.005  0.0191  90.3 0.4616 0 0.0002 0.017 89.9 0.4244 ‐0.0137 0.0179 93.8 0.4845 0.5  ‐0.1  ‐0.0029  0.0172  90.6 0.4415 0 0.0078 0.0161 90 0.4131 ‐0.0003 0.0163 90.6 0.4265 0  ‐0.0047  0.0169  91.5 0.4442 0 0.0031 0.0158 91.7 0.4182 ‐0.0043 0.0161 91.9 0.4293 0.05  ‐0.0042  0.0172  90.4 0.444 0 0.0015 0.0159 90.3 0.419 ‐0.0052 0.0164 90.8 0.4292 0.1  ‐0.005  0.0178  89.3 0.443 0 ‐0.001 0.0165 89.7 0.4187 ‐0.0071 0.017 89.8 0.428 0.2  ‐0.0039  0.0176  88.4 0.4372 0 ‐0.004 0.0164 88.3 0.4142 ‐0.0085 0.0168 89.1 0.4225 63 Table 10 Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling，n=80 per Strata) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0003  0.0014  90.5 0.1111 39.7 0.0529 0.0044 84.7 0.2333 0.1222 0.0161 61.2 0.3531 0  ‐0.0062  0.0089  82.2 0.3121 5.2 0.0347 0.0089 93 0.3078 0.0606 0.0069 99.9 0.4175 0.05  ‐0.003  0.0121  87.5 0.3755 1.9 0.0309 0.0114 91.3 0.3371 0.0404 0.0068 99.6 0.4481 0.1  ‐0.0086  0.0143  87.4 0.4053 0.9 0.0227 0.0125 90.9 0.361 0.0219 0.0072 98.8 0.4696 0.2  ‐0.0077  0.0178  90.1 0.4574 0.1 0.0123 0.0148 90.2 0.3951 ‐0.0057 0.0113 98.4 0.5081 0.3  ‐0.1  ‐0.0057  0.0074  91 0.2957 0 0.0054 0.0071 91.6 0.2859 ‐0.005 0.0069 96.7 0.3567 0  ‐0.0007  0.0086  90.5  0.3112  0  0.0079  0.0082  89.8  0.2991  ‐0.0025  0.0082  95.3  0.3609  0.05  ‐0.0058  0.0086  91.1 0.3163 0 0.0017 0.0081 90.3 0.302 ‐0.0083 0.0083 95.7 0.3613 0.1  ‐0.006  0.0092  89.9 0.3205 0 ‐0.0005 0.0086 90.8 0.3065 ‐0.0094 0.009 94.6 0.3575 0.2  ‐0.0054  0.01  89.3 0.3237 0 ‐0.0031 0.0093 89.2 0.3092 ‐0.0102 0.0098 91.7 0.3481 0.5  ‐0.1  0.0002  0.0094  88.7 0.3101 0 0.0055 0.0091 88.2 0.2993 0.0016 0.0092 89.1 0.3047 0  ‐0.0045  0.008  91.2 0.312 0 ‐0.0008 0.0078 91.9 0.3022 ‐0.0043 0.0078 91.4 0.3067 0.05  ‐0.0071  0.0091  90.4 0.3117 0 ‐0.0045 0.0088 89.7 0.3022 ‐0.0075 0.0089 90.6 0.3065 0.1  ‐0.0068  0.0082  90.9 0.3112 0 ‐0.0051 0.0079 90.3 0.3028 ‐0.0078 0.008 91.1 0.306 0.2  ‐0.0033  0.0082  90.7 0.3071 0 ‐0.0035 0.008 90.7 0.2992 ‐0.0056 0.008 91.2 0.302 64 A. n=20 B. n=40 C. n=80 Figure  7.  Density  Plot  of  Simulated  Estimators  under  Different  Settings  (Stratified  Simple  Random  Sampling).  BPELE=Bayesian  Pseudo‐Empirical  Likelihood  Estimator,  DBE=Design‐Based  Estimator,  PBE=Parametric Bayesian Estimator. Parameter p is the allele frequency and f is the inbreeding coefficient  in the simulation. Panel A, B and C are for stratified sample size 20, 40 and 80 respectively. Red dashed  vertical line is the true parameter f; red dotted vertical line is the lower bound of the parameter space for  f conditional on allele frequency p.  65 3.4.2 Proportional to Population Size Sampling For our second simulation design, we used proportional to size (PPS) sampling. To generate a size variable X for each individual in the population of size N=4,800, the number of the minor allele A count plus a log normal distributed random error (mean 0, standard deviation 0.2) is used. A total of 20, 40 or 80 samples are randomly selected from the population using the PPS sampling. Therefore, the sample selection probability is the sample size times the relative proportion of the size measure. The sampling weight for analysis is the inverse of the sample selection probability, ݐݓ௜ ൌ 1/ሺ݊ ൈ ௫೔∑ ௫೔೔ಿసభ ሻ, where i is the index of each individual in the population and n is the sample size. The weights from this PPS sampling design is correlated with the genotype and, therefore, is informative. Table 11, Table 12, and Table 13 report the simulation results in comparing the design-based estimator and two Bayesian estimators for the bias, MSE, coverage probability (CP), out of range probability (ORP), and average length (AL). For the design-based estimator, the PPS sampling generates higher coverage probability than the stratified simple random sampling under large allele frequency (p=0.5); while the trend is opposite for smaller allele frequency (p=0.1, 0.3). This is likely due to the sample selection probability is correlated with the genetic data. Individuals with genotype AA are more likely to be selected in the PPS sampling than individuals with genotype Aa. Individuals with genotype aa have the lowest probability to be selected. Both Bayesian estimators maintain the nominal coverage probability; while the design-based estimator has lower coverage in all simulations. The design-based estimator also shows higher average length of the 95% confidence interval. Comparing with the simple random sampling, the design-based estimator from the PPS sampling is more likely to produce result outside of the parameter space. For example, when the sample size is 40 and the inbreeding coefficient is f=0, 25.9% of the simulated design-based estimators have the values lower than -0.1111, which is the lower boundary of the parameter space for f under p=0.1 (Table 12, Figure 8 Panel B). It is about ten percent more than the simulation result from the simple random sampling. 66 Overall, the comparison result among three methods has similar pattern as that under the stratified simple random sampling design. The Bayesian method is recommended since it maintains nominal coverage probability and it has smaller MSE and average length of the 95% credible interval. 67 Table 11. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Proportional to Size Sampling, n=20) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0092  0.0068  81.7 0.2106 40 0.2719 0.0894  83.2 0.8494 0.2807 0.0896 39.3 0.718 0  ‐0.0452  0.0424  11.5 0.4112 27.6 0.2071 0.0603  98.1 0.8812 0.2029 0.0554 94.5 0.7383 0.05  ‐0.0727  0.0616  13 0.4653 25.9 0.1731 0.0503  98.4 0.8862 0.1658 0.044 94.6 0.7485 0.1  ‐0.0929  0.0767  15.7 0.525 22.3 0.142 0.0401  98 0.8965 0.1292 0.0339 93.9 0.7512 0.2  ‐0.1319  0.113  23 0.6843 14.1 0.0694 0.0272  99.5 0.9238 0.0406 0.0235 91 0.7512 0.3  ‐0.1  ‐0.0239  0.0591  64.2 0.7286 7.9 0.0648 0.0412  94.7 0.7411 0.0741 0.0323 96.9 0.7296 0  ‐0.0516  0.0666  62.8  0.7735  5  0.0278  0.0406  94.9  0.7555  0.0323  0.0309  98.2  0.7474  0.05  ‐0.0522  0.0703  66.8 0.8204 3.3 0.0181 0.0421  94.6 0.7662 0.0169 0.0315 97.6 0.7618 0.1  ‐0.0433  0.0737  70.2 0.863 2 0.0171 0.0431  95.3 0.7777 0.0108 0.0341 97.1 0.7744 0.2  ‐0.0446  0.0843  77 0.9328 0.8 ‐0.0065 0.0481  94.4 0.7876 ‐0.0168 0.041 96.3 0.7871 0.5  ‐0.1  ‐0.0248  0.056  90.8 0.8715 0 0.0268 0.0411  93 0.7608 ‐0.0058 0.0417 95.3 0.7824 0  ‐0.0371  0.055  91.1 0.8879 0 0.0023 0.0391  94.7 0.7701 ‐0.0266 0.0412 95.6 0.7922 0.05  ‐0.0423  0.0584  91.2 0.8953 0 ‐0.0096 0.0412  94.2 0.7723 ‐0.0381 0.0445 94.6 0.7963 0.1  ‐0.032  0.0565  91.5 0.9029 0 ‐0.0065 0.0403  95.4 0.7742 ‐0.0342 0.0441 95.1 0.7967 0.2  ‐0.0254  0.0554  92.6 0.9051 0 ‐0.0161 0.0402  95.5 0.7692 ‐0.0382 0.0434 95.4 0.7938 68 Table 12. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Proportional to Size Sampling, n=40) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0037  0.0051  84.7 0.1533 43.3 0.1914 0.0466 78.2 0.6373 0.2282 0.058 46.9 0.6059 0  ‐0.0326  0.0315  20.5 0.3509 25.9 0.1375 0.034 96.1 0.6776 0.1577 0.0342 98.6 0.6468 0.05  ‐0.042  0.0452  27.6 0.4477 20.2 0.1167 0.0322 97 0.6951 0.1291 0.0286 98.9 0.666 0.1  ‐0.0502  0.0549  34.6 0.5464 15.1 0.0961 0.0297 98.4 0.7095 0.1024 0.024 98.7 0.6829 0.2  ‐0.072  0.0804  45.7 0.7269 6.9 0.046 0.0281 98.2 0.7372 0.042 0.0212 97.8 0.7103 0.3  ‐0.1  ‐0.0201  0.0292  79.7 0.5429 1.8 0.0307 0.0232 93.6 0.56 0.0225 0.0175 98 0.5861 0  ‐0.0232  0.034  83.6  0.5912  0.8  0.019  0.0257  92.7  0.5841  0.0037  0.0209  96.6  0.6108  0.05  ‐0.023  0.0357  84.8 0.6165 0.2 0.0135 0.0264 92.9 0.5957 ‐0.0039 0.0228 96.8 0.6215 0.1  ‐0.0248  0.0368  86.9 0.6397 0 0.0048 0.0265 93 0.6058 ‐0.013 0.0243 96.7 0.6286 0.2  ‐0.0267  0.0388  87.8 0.6824 0 ‐0.0078 0.0277 93.2 0.6209 ‐0.0286 0.0284 95 0.6367 0.5  ‐0.1  ‐0.0119  0.0269  92.3 0.6136 0 0.0129 0.0233 94.7 0.58 ‐0.0057 0.0238 94.3 0.5893 0  ‐0.0161  0.0283  92.5 0.6236 0 0.001 0.0241 94 0.5862 ‐0.0146 0.0252 94.3 0.5941 0.05  ‐0.0146  0.0281  93 0.628 0 ‐0.0017 0.0238 94.4 0.5892 ‐0.016 0.0252 94 0.5948 0.1  ‐0.0151  0.0285  93.2 0.6299 0 ‐0.0048 0.0246 94.5 0.5892 ‐0.0188 0.0258 93.7 0.5938 0.2  ‐0.0177  0.027  94.8 0.6299 0 ‐0.0147 0.023 95.6 0.5862 ‐0.0261 0.0247 95.1 0.5884 69 Table 13. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Proportional to Size Sampling, n=80) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0025  0.0026  88.8 0.111 43.9 0.1153 0.0174  79.5 0.4331 0.1707 0.0317 50.4 0.4705 0  ‐0.0145  0.0199  37.7 0.3138 18.7 0.0798 0.0177  95 0.497 0.1063 0.0167 99.5 0.5258 0.05  ‐0.0112  0.028  50.6 0.4155 12.3 0.0714 0.0209  94.6 0.5266 0.0847 0.0156 98.6 0.553 0.1  ‐0.0191  0.0351  56.5 0.4857 8 0.057 0.0224  95.6 0.5488 0.0635 0.0155 98 0.5717 0.2  ‐0.0388  0.0458  70.1 0.6119 3.7 0.0171 0.0244  95.7 0.5832 0.0099 0.0166 98.2 0.604 0.3  ‐0.1  ‐0.0087  0.0151  86.3 0.393 0.2 0.0168 0.0133  92.4 0.4168 0.0033 0.0113 97 0.4658 0  ‐0.0145  0.0166  87.9  0.424  0  0.0054  0.0142  92.6  0.437  ‐0.0105  0.0134  96.1  0.4809  0.05  ‐0.0109  0.0163  90.2 0.4409 0 0.0063 0.0138  94.6 0.4489 ‐0.0102 0.0135 96.8 0.487 0.1  ‐0.0144  0.0173  90 0.4567 0 ‐0.0004 0.0149  94.6 0.4575 ‐0.0176 0.0155 96 0.489 0.2  ‐0.0251  0.0188  90.8 0.4782 0 ‐0.0175 0.016  93.5 0.4685 ‐0.0319 0.0173 93.8 0.4846 0.5  ‐0.1  ‐0.0049  0.0134  93.3 0.4324 0 0.0067 0.0124  95 0.4289 ‐0.0022 0.0127 94.9 0.4265 0  ‐0.0024  0.0128  94.5 0.4395 0 0.0054 0.0118  96 0.433 ‐0.0021 0.0122 95.3 0.4295 0.05  ‐0.007  0.0132  94.2 0.4412 0 ‐0.001 0.0122  94.9 0.4327 ‐0.008 0.0126 94.8 0.4293 0.1  ‐0.0081  0.0132  93.8 0.4421 0 ‐0.004 0.0121  95.4 0.4334 ‐0.0102 0.0126 94.3 0.4283 0.2  ‐0.015  0.0138  94.1 0.4412 0 ‐0.014 0.0128  95.4 0.4295 ‐0.0193 0.0133 94.3 0.4235 70 A. n=20 B. n=40 C. n=80 Figure 8. Density Plot of Simulated Estimators under Different Settings (Proportional to Population Size  Sampling).  BPELE=Bayesian  Pseudo‐Empirical  Likelihood  Estimator,  DBE=Design‐Based  Estimator,  PBE=Parametric Bayesian Estimator. Parameter p is the allele frequency and f is the inbreeding coefficient  in the simulation. Panel A, B and C are for stratum sample size 20, 40 and 80 respectively. Red dashed  vertical line is the true parameter f; red dotted vertical line is the lower bound of the parameter space for  f conditional on allele frequency p.  71 3.4.3 Model Misspecification In our above simulation studies, population genotype records were generated independently from the same multinomial model with parameters p and f. In the real world, this data simulation mechanism may be modified by factors such as population stratification or other natural selection process. To investigate the consequence of model misspecification, we simulated a population with two stratums, the genotype of individual within each stratum is simulated through different multinomial models: one with the allele frequency p=0.1 and the other with p=0.3. We studied same range of the inbreeding coefficient as -0.1, 0, 0.5, 0.1 or 0.2. At each one of 1000 iterations in our first simulation study, a sample size of twenty (20) was randomly selected from each stratum. In the second simulation study, similarly as in Section 3.4.2, a sample size of twenty (20) was randomly selected from the population using the PPS sampling. Table 14 shows the results under the stratified simple random sampling. Although the design- based estimator shows smallest bias, it fails in all other criteria in terms of the MSE, coverage and average length. On the other hand, only BPELE maintains nominal coverage probability. It also has the smallest MSE and relative smaller average length than the design-based estimator. Table 15 shows a similar pattern when comparing estimators under the PPS sampling. Under the PPS sampling, the coverage probability for the design-based estimator gets improved than the simple random sampling. However, the MSE and average interval length are still less optimal when comparing to the Bayesian estimators. Two observations worth mention from the two simulation studies. First, when there is population subdivision that results in different allele frequencies among subpopulations, ignoring the population subdivision may increase the variance thus a larger confidence interval for the design- based estimator. Such problem is more serious in the PPS sampling design when the weights are correlated with the genetic data. Second, the Bayesian methods show more resistance to the model misspecification under both sampling designs, especially for the Bayesian pseudo-empirical likelihood estimator. Ignoring the population subdivision while using the average allele frequency for the whole population does not have a large negative impact on the Bayesian estimators in terms of the MSE, coverage probability and average interval length. 72  Table 14. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Non-Homogenous Population, Stratified Simple Random Sampling，n=20 per Strata) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length Table 15. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Non-Homogenous Population, Proportional to Size Sampling，n=20) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   f  Design‐Based Estimator Parametric Bayesian Estimator   Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL Bias MSE CP AL  Bias MSE CP AL ‐0.1  0.0063  0.0342  74.6 0.5611 0.0612 0.0299 90.8 0.5632 0.0599 0.0229 96.1 0.5874 0  0.0179  0.0416  82.5 0.6568 0.0567 0.0348 89.2 0.587 0.0445 0.0276 94.5 0.6226 0.05  0.0041  0.0421  85.6 0.6809 0.0374 0.0336 90.1 0.5913 0.0233 0.0276 94.5 0.6311 0.1  0.0042  0.0432  85.8 0.7007 0.0314 0.0339 90.4 0.6001 0.0155 0.0288 94.2 0.6395 0.2  0.0004  0.0458  87.6 0.7368 0.0146 0.0349 89.5 0.6061 ‐0.0033 0.0324 92.8 0.6489   f  Design‐Based Estimator Parametric Bayesian Estimator   Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL Bias MSE CP AL  Bias MSE CP AL ‐0.1  0.022  0.0363  92.9 0.8428 0.1031 0.0342 94.4 0.7144 0.0975 0.0292 99.5 0.7552 0  0.033  0.0425  89.6 0.8912 0.0855 0.0357 94.6 0.7201 0.0711 0.0273 99.2 0.7982 0.05  0.0204  0.0418  91.8 0.903 0.0648 0.0329 96.6 0.7228 0.0466 0.0247 99 0.8108 0.1  0.0214  0.0429  93.3 0.9144 0.0573 0.0326 96.7 0.7224 0.0344 0.0249 98.9 0.8218 0.2  0.0185  0.0425  94 0.9139 0.0367 0.0305 97.1 0.7222 0.0075 0.0247 99.2 0.8374 73 Chapter 4: Estimation of the Inbreeding Coefficient with Family Level Correlation National level surveys such as the NHANES sometimes collect family level genetic samples. Among these surveys, the genetic correlation is induced by offspring’s genetic inheritance from their biological parents within the household. Besides the differential population weights introduced by the sampling design, within household correlations further inflate the variance of estimates for the genetic traits. Methods developed by She (2009), Li (2011), and Li (2013) were the first frequentist approaches to test HWE considering within household genetic correlations among family members. For this chapter, we are trying to extend the Bayesian method to estimate the inbreeding coefficient when the survey collects family level genetic data. Inspired by the NHANES survey design, let’s consider a case with genetic data collected from families with both parents and one child (2P1O). As shown in Table 17, for a diallelic locus (i.e., A and a) of autosomal genes, there are a total of ten (G=10) possible genotype combinations for a sampled family. The mother and father’s genotype frequencies are independent following the general inbreeding model. For genes at normal chromosome, child will get one copy of gene from each parent. Therefore for genes located at the autosomal chromosome, switching mother and father’s genotypes generally will not affect the gene flow to the next generation. In another words, two parents’ genotypes: genotype AA for the mother and aa for the father, or, genotype aa for the mother and AA for the father, have their child carrying the same genotype Aa. For a diallelic locus, there are six different types of genotype distribution considering the mother and father together: AA-AA， AA-Aa, AA-aa, Aa-Aa, Aa-aa, or aa-aa. Conditional on the parents’ genotypes, the probability of child’s genotype follows the Mendel’s law (Table 16). Let ࢗ ൌ ሺݍଵ, ݍଶ, ݍଷ, ݍସ, ݍହ, ݍ଺, ݍ଻, ଼ݍ, ݍଽ, ݍଵ଴ሻ be the joint probability vector of each parents-child genotypes combination. It can be derived as the product of the parent genotype probability and the child genotype probability conditional on his/her parents. Table 17 also lists the number of the allele A (numAg) and the number of the heterozygotes Aa (numHg) for the g-th family for g=1,…,10. 74 Table 16. Child Genotype Probability Conditional on the Parents’ Genotypes (Mendel’s Law)   Child Genotype Parents’ Genotypes AA Aa aa AA‐AA  1 ‐ ‐ AA‐Aa  1/2 1/2 ‐ AA‐aa  ‐ 1 ‐ Aa‐Aa  1/4 1/2 1/4 Aa‐aa  ‐ 1/2 1/2 aa‐aa  ‐ ‐ 1 Table 17. Family Genotype Distribution for Both Parents One Child Triads (2P1O) Parent  Genotype  Child  Genotype  Parent  Genotype  Probability  Conditional  Child Genotype  Probability  (Mendel’s Law)  Joint  Probability  (qg)  #   Allele A  (numAg)  #  Heterozygotes  (numHg)  AA ‐ AA  AA  ݌ଵଵଶ   1 ݍଵ ൌ ݌ଵଵଶ 6  0 AA ‐ Aa  AA  2݌ଵଵ݌ଵଶ  ½ ݍଶ ൌ ݌ଵଵ݌ଵଶ 5  1 AA ‐ Aa  Aa  2݌ଵଵ݌ଵଶ  ½ ݍଷ ൌ ݌ଵଵ݌ଵଶ 4  2 AA ‐ aa  Aa  2݌ଵଵ݌ଶଶ  1 ݍସ ൌ 2݌ଵଵ݌ଶଶ 3  1 Aa ‐ Aa  AA  ݌ଵଶ݌ଵଶ  ¼ ݍହ ൌ ݌ଵଶଶ /4 4  2Aa ‐ Aa  Aa  ݌ଵଶ݌ଵଶ  ½ ݍ଺ ൌ ݌ଵଶଶ /2 3  3Aa ‐ Aa  aa  ݌ଵଶ݌ଵଶ  ¼ ݍ଻ ൌ ݌ଵଶଶ /4 2  2 Aa ‐ aa  Aa  2݌ଵଶ݌ଶଶ  ½ ଼ݍ ൌ ݌ଵଶ݌ଶଶ 2  2 Aa ‐ aa  aa  2݌ଵଶ݌ଶଶ  ½ ݍଽ ൌ ݌ଵଶ݌ଶଶ 1  1 aa ‐ aa  aa  ݌ଶଶଶ   1 ݍଵ଴ ൌ ݌ଶଶଶ 0  0 Similar as the individual genotype counts that follow a multinomial distribution, the family genotype counts also follow a multinomial distribution with probability vector ࢗ ൌ ሺݍଵ, ݍଶ, ݍଷ, ݍସ, ݍହ, ݍ଺, ݍ଻, ଼ݍ, ݍଽ, ݍଵ଴ሻ. It is easy to verify that the total joint probability of the family genotype is ෍ ݍ௚ ଵ଴ ௚ୀଵ ൌ ݌ଵଵଶ ൅ 2݌ଵଵ݌ଵଶ ൅ 2݌ଵଵ݌ଶଶ ൅ 2݌ଵଶ݌ଶଶ ൅ ݌ଵଶଶ ൅ ݌ଶଶଶ ൌ ሺ݌ଵଵ൅݌ଵଶ ൅ ݌ଶଶሻଶ ൌ 1. Furthermore, the expected number of the allele A count and the expected number of the heterozygote Aa count for the family are, respectively, 75 ܧሺ#݈݈݈݁݁ܣ ܣሻ ൌ෍ ݍ௚݊ܣ݉ݑ௚ ଵ଴ ௚ୀଵ ൌ 6݌ଵଵଶ ൅ 9݌ଵଵ݌ଵଶ ൅ 6݌ଵଵ݌ଶଶ ൅ 3݌ଵଶ݌ଶଶ ൅ 3݌ଵଶଶ ൌ 6݌ଵଵሺ݌ଵଵ൅݌ଵଶ ൅ ݌ଶଶሻ ൅ 3݌ଵଶሺ݌ଵଵ൅݌ଵଶ ൅ ݌ଶଶሻ ൌ 6݌ଶ ൅ 6݌ሺ1 െ ݌ሻ݂ ൅ 6ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻ ൌ 6݌, and ܧሺ# Heterozygotesሻ ൌ෍ ݍ௚݊ܪ݉ݑ௚ ଵ଴ ௚ୀଵ ൌ 3݌ଵଵ݌ଵଶ ൅ 2݌ଵଵ݌ଶଶ ൅ 3݌ଵଶ݌ଶଶ ൅ 52݌ଵଶ ଶ ൌ 6݌ሺ1 െ ݌ሻ െ 4݂݌ሺ1 െ ݌ሻ. The expected number of the allele A for a family using the familywise 2P1O genotype data is 6p. It is three times of 2p, the expected number of the allele A for an individual using the individual genotype data. Treating the 2P1O as if they are independent individuals and ignoring the within family correlation, provides an unbiased estimator of the allele frequency. However, the expected number of the heterozygotes for a 2P1O family is 6݌ሺ1 െ ݌ሻ െ 4݂݌ሺ1 െ ݌ሻ ൌ 2ሺ3 െ 2݂ሻ݌ሺ1 െ ݌ሻ, which is different from the expected number of the heterozygotes, 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻ among individuals without genetic correlations. Ignoring the within family genetic correlation will produce a biased estimator of the inbreeding coefficient. Under a simple random sampling setting, suppose there are 2P1O families of size of n. Let totalH be the total number of the heterozygotes, totalA be the total number of the allele A from these n families (3n individuals). Let ̂݌ be the estimator of the allele frequency. Based on the above calculation, the estimator of the inbreeding coefficient with respect to the random sampling of families ( መ݂୤ୟ୫୧୪୷) as well as the estimator ignoring the within family correlations ( መ݂୧୬ୢ୧୴୧ୢ୳ୟ୪) are መ݂୤ୟ୫୧୪୷ ൌ 32 െ ܪ݈ܽݐ݋ݐ ݊ 2 ൈ 2̂݌ሺ1 െ ̂݌ሻ ൌ 1 2 ൤3 െ ܪ݈ܽݐ݋ݐ ݊ ൈ 2̂݌ሺ1 െ ̂݌ሻ൨ , 76 መ݂୧୬ୢ୧୴୧ୢ୳ୟ୪ ൌ 1 െ 2ܪ݈ܽݐ݋ݐ ൈ ሺ3݊ሻ ൈ ̂݌ሺ1 െ ̂݌ሻ ൌ 1 3 ൤3 െ ܪ݈ܽݐ݋ݐ ݊ ൈ 2̂݌ሺ1 െ ̂݌ሻ൨ , where ̂݌ ൌ ௧௢௧௔௟஺଺ൈ௡ ൌ ௧௢௧௔௟஺ ଶൈଷ௡ ൌ ௧௢௧௔௟஺ ଺௡ . From above calculation, ignoring the within-family sampling design results in a 1/3 decrease of the measure of f. As we discussed in Chapter 1, the inbreeding coefficient is a correlation measure between two homologous genes in uniting gametes. Therefore, considering the within family correlation inflates the inbreeding coefficient (correlation) estimate. Although the final estimator could be corrected through this factor under our simplest scenario with only 2P1O families in the population, the justification for the standard error calculation is not trivial even under the simple random sampling. With the introduction of different sample selection probability as well as within family genotype correlation, the discussion in this chapter fills the gap in estimating the inbreeding coefficient from complex survey which collects family level genotype data. 4.1 Direct Design-Based Estimator Assume a finite population consists of N families, let Gi (i=1, 2, …, N) be the 2P1O family genotypes, numAi be the number of the allele A and numHi be the number of the heterozygotes for the ith family in the population. If this finite population is assumed to be selected from a superpopulation with family genotype frequencies described in Table 17, the finite population mean of the number of the allele A is therefore ݊ܣ݉ݑതതതതതതതത ൌ ଵே∑ ݊ܣ݉ݑ௜ே௜ୀଵ ൌ 6݌. Similarly, the finite population mean of the number of the heterozygotes is ݊ܪ݉ݑതതതതതതതത ൌ ଵே∑ ݊ܪ݉ݑ௜ே௜ୀଵ ൌ 6݌ሺ1 െ ݌ሻ െ 4݂݌ሺ1 െ ݌ሻ. The finite population parameter f can be viewed as a function of the finite population averages of the familywise allele A count (݊ܣ݉ݑതതതതതതതത) and heterozygotes count (݊ܪ݉ݑതതതതതതതത): ݂ ൌ 32 െ 9 ∗ ݊ܪ݉ݑതതതതതതതത ݊ܣ݉ݑതതതതതതതത ∗ ሺ3 െ ݊ܣ݉ݑതതതതതതതതሻ . Under unequal probability sampling, each sampled families of size n has a design weight ݀௜, which is the inverse of the family unit sampling inclusion probability. From survey like the NHANES, the family level weights can be constructed by taking the average of the remaining weights (equal 77 to the final sample weight divided by the within-family weight) of 2 parents and 1 child in each of the sampled families (Korn & Graubard, 2003). The design-based direct survey estimator for the finite population inbreeding coefficient can be expressed as መ݂ ൌ 32 െ 9 ∗ ݊ܪ݉ݑ෣ ݊ܣ݉ݑ෣ ∗ ൫3 െ ݊ܣ݉ݑ෣ ൯ , where ݊ܪ݉ݑ෣ and ݊ܣ݉ݑ෣ are Horvitz–Thompson type estimators of population means. The standard error of the mean can be estimated from jackknife method as described earlier. 4.2 Parametric Bayesian Estimator with Survey Weight Again, let ݀ప෩ ൌ ௗ೔∑ ௗ೔೔∈ೞ be the normalized family sampling weight and is assumed to be independent of the observation of family genotype. We incorporate the complex sampling design through the effective sample size and effective family genotype counts in the Bayesian multinomial model: ܲݎሺࢗ|∗࢔ሻ ൌ ܲݎ൫ሺ݊ଵ∗, ݊ଶ∗, … , ݊ଵ଴∗ሻหሺݍଵ, ݍଶ, … , ݍଵ଴ሻ൯ ൌ ݊ ∗! ∏ ݊௚∗ଵ଴௚ୀଵ ෑ ݍ௚ ௡೒∗ ଵ଴ ௚ୀଵ , where ݊௚∗ ൌ ݊∗ ∑ ሚ݀௜௡௜ୀଵ ܫሾீ೔ୀ௚ሿ . The design-based variance is calculated as ܸܽୢݎ ୣୱ୧୥୬ሺ݊ܣ݉ݑതതതതതതതത෣ ሻ. Under simple random sampling without replacement, the variance is ܸܽݎୱ୰ୱ୵୭୰ሺ݊ܣ݉ݑതതതതതതതത෣ ሻ . The design effect is Deff ൌ ܸܽୢݎ ୣୱ୧୥୬ሺ݊ܣ݉ݑതതതതതതതത෣ ሻ/ܸܽݎୱ୰ୱ୵୭୰ሺ݊ܣ݉ݑതതതതതതതത෣ ሻ and therefore the effective sample size is ݊∗ ൌ ݊/Deff. Same as in Chapter 3, we use a non-informative proper uniform prior for the allele frequency p. Conditional on p, we apply a uniform prior on its parameter space for the inbreeding coefficient f. The OPENBUGS Bayesian model is listed below: Level 1: ݈ܽ݅݉݋݊݅ݐ݈ݑܯ~∗࢔ሺ݊∗, ࢗሻ, ࢔∗ ൌ ሺ݊1∗, ݊2∗, … , ݊10∗ሻ, ࢗ ൌ ሺݍଵ, ݍଶ, … , ݍଵ଴ሻ, Reparametrize q as a function of ࢖෥ ൌ ሺ݌ଵଵ, ݌ଵଶ, ݌ଶଶሻ shown in Table 17; ቐ ݌ଵଵ ൌ ݌ଶ ൅ ݌ሺ1 െ ݌ሻ݂, ݌ଵଶ ൌ 2݌ሺ1 െ ݌ሻሺ1 െ ݂ሻ, ݌ଶଶ ൌ ሺ1 െ ݌ሻଶ ൅ ݌ሺ1 െ ݌ሻ݂. Level 2: ݉ݎ݋݂ܷ݅݊~݌ሾ0, 1ሿ, 78 ݂|݉ݎ݋݂ܷ݅݊~݌ ቂି୫୧୬ሺ௣,ଵି௣ሻଵି୫୧୬ሺ௣,ଵି௣ሻ , 1ቃ. model=function(){ q11 <- (1 - f) * p * p + f * p q12 <- 2 * (1 - f) * p * (1 - p) q22 <- (1 - f) * (1 - p) * (1 - p) + f * (1 - p) q[1] <- q11 * q11 q[2] <- q11 * q12 q[3] <- q11 * q12 q[4] <- 2 * q11 * q22 q[5] <- q12 * q12/4 q[6] <- q12 * q12/2 q[7] <- q12 * q12/4 q[8] <- q12 * q22 q[9] <- q12 * q22 q[10] <- q22 * q22 y[ ] ~ dmulti(q[ ], n) p ~ dunif(0, 1) f <- w * (1 - f.min) + f.min f.min <- max(-p/(1 - p), -(1 - p)/p) w ~ dunif(0, 1) n <- sum(y[ ]) } Alternatively by substituting the joint probability ݍ௚ (g=1,…,10) with ሺ݌ଵଵ, ݌ଵଶ, ݌ଶଶሻ as shown in Table 17, the likelihood is then simplified as ܲݎሺ࢖|∗࢔ሻ ൌ ݊ ∗! ∏ ݊௚∗ଵ଴௚ୀଵ ෑ ݍ௚ ௡೒∗ ଵ଴ ௚ୀଵ ∝ ݌ଵଵଶ௡భ∗ା௡మ∗ା௡య∗ା௡ర∗݌ଵଶ௡మ∗ା௡య∗ାଶሺ௡ఱ∗ା௡ల∗ା௡ళ∗ሻା௡ఴ∗ା௡వ∗݌ଶଶଶ௡భబ∗ା௡వ∗ା௡ఴ∗ା௡ర∗. Therefore, the data can be modelled through the multinomial with parameter ࢖ ൌ ሺ݌ଵଵ, ݌ଵଶ, ݌ଶଶሻ and the consolidated effective number of genotype count. The total count of ݊ଶ∗ ൅ ݊ଷ∗ ൅ 2ሺ݊ହ∗ ൅ ݊଺∗ ൅ ݊଻∗ሻ ൅ ଼݊∗ ൅ ݊ଽ∗ is the number of parents with heterozygotes Aa; the total count of 2݊ଵ∗ ൅ ݊ଶ∗ ൅ ݊ଷ∗ ൅ ݊ସ∗ is the number of parents with genotype AA; and the total count of 2݊ଵ଴∗ ൅ ݊ଽ∗ ൅ ଼݊∗ ൅ ݊ସ∗ is the number of parents with genotype aa. In another words, the child genotype is not included in the above simplified multinomial likelihood. This is consistent with 79 the biological fact that the child genotype is produced under the Mendel’s law conditioning on the parents’ genotypes. Therefore, no additional parameters need be included in the hierarchical Bayesian model. Accordingly, we expect that analyzing the whole family genotype data using the method discussed in this chapter will produce a similar result as analyzing a subset of the data that only includes parents’ genotypes. 4.3 Bayesian Pseudo-Empirical Likelihood Estimator Let ݈௉ா௅ ൌ ݊∑ ݀ప෩ log ሺߨ௜ሻ௜∈௦ be the pseudo empirical log-likelihood for the above design, where S is a randomly selected families. In order to maximize the likelihood ݈௉ா௅ with the following constraints on the total probability and the unbiased estimating function: ߨ௜ ൐ 0,෍ ߨ௜௜∈௦ ൌ 1,෍ ߨ௜ሾ݊ܪ݉ݑ௜௜∈௦ െ ሺ6݌ሺ1 െ ݌ሻ െ 4݂݌ሺ1 െ ݌ሻሻሿ ൌ 0. Lagrange method can therefore be used. Again, a survey design-based unbiased estimator ̂݌ of the allele frequency is used to replace p. BPELE is derived by maximizing the profiled pseudo- empirical log-likelihood with modified constraints: ߨ௜ ൐ 0,෍ ߨ௜௜∈௦ ൌ 1,෍ ߨ௜ሾ݊ܪ݉ݑ௜௜∈௦ െ ሺ6̂݌ሺ1 െ ̂݌ሻ െ 4݂̂݌ሺ1 െ ̂݌ሻሻሿ ൌ 0. Similarly as in Chapter 3, define ݑ௜ ൌ ݊ܪ݉ݑ௜ െ ሺ6̂݌ሺ1 െ ̂݌ሻ െ 4݂̂݌ሺ1 െ ̂݌ሻሻ and thus ∑ ߨ௜ݑ௜௜∈௦ ൌ 0. Lagrange multipliers and grid search method are used to derive the BPELE from the posterior mean. 4.4 Simulation Study A simulation study for a single biallelic locus with alleles A and a is conducted to evaluate the performance of the above three estimators. Let a finite population of size N= 4,800 families of each two parents and one child (2P1O) consisting of two stratum with subpopulation size ratio of 5:1. For each family, the genotypes for the mother and the father are generated independently from the multinomial model with frequencies of ܲሺܽܽሻ ൌ ሺ1 െ ݌ሻଶ ൅ ݌ሺ1 െ ݌ሻ݂ , ܲሺܽܣሻ ൌ 2ሺ1 െ ݂ሻ݌ሺ1 െ ݌ሻ , and ܲሺܣܣሻ ൌ ݌ଶ ൅ ݌ሺ1 െ ݌ሻ݂ . Conditional on the parents’ genotypes within a randomly generated family, the child’s genotype is generated under the Mendel’s law. 80 The parameters for the simulation are varied with the allele A frequency p=0.1, 0.3, or 0.5 and the inbreeding coefficient f=-0.1, 0, 0.05, 0.1, or 0.2. For the realized data at each simulation, estimators of the inbreeding coefficient are calculated by the methods described above. The bias, variance, mean square error (MSE) of the estimators are calculated similarly as in Chapter 2, except that the true population parameter in computing the bias and the MSE is based on the finite population of size N=4,800 families. When the number of sampled families is 20, there is small chance that all sampled families have the same family genotypes, which is replaced by extra simulation runs. 4.4.1 Stratified Simple Random Sampling At each of 1000 iterations, a same sample size of twenty (20), forty (40) or eighty (80) families are randomly selected from each stratum. Therefore, the weights, computed as the inverse of selection probabilities, are different across the two strata. Table 18, Table 19, Table 20, and Figure 9 report the simulation results to compare the design- based estimator and two Bayesian estimators for the bias, MSE, coverage probability (CP), out of range probability (ORP), and average interval length (AL). The design-based estimator generally has a negative bias. Both Bayesian estimators tend to have negative bias expect when the allele frequency is small (p=0.1). As in Chapter 3, when the allele frequency is small (p=0.1), the Bayesian methods have larger bias especially when the inbreeding coefficient is small (f=-0.1, 0). In general for all simulation scenario, the parametric Bayesian estimator has the smallest MSE when the sample size is small. When sample size is larger, all methods have relative similar MSE. For coverage, the BPELE maintains the best coverage but with larger average interval length in all three simulations. When the allele frequency is p=0.5, the BPELE does not maintain the nominal coverage probability. As expected, the design-based estimator is likely to lay outside of the parameter space, especially when the allele frequency is small (p=0.1). Increasing the sample size slightly reduces the out of range possibility. The situation gets even worse when the actual inbreeding coefficient is small (f=-0.1, 0). 81 Table 18. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling of Families，n=20 Families per Strata) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0071  0.0107  83 0.2778 63.6 0.1025 0.0139 74.4 0.3792 0.2578 0.0723 40.8 0.6836 0  ‐0.0102  0.0275  65.8 0.4761 31.7 0.0716 0.0184 91.7 0.4391 0.1883 0.0435 99.3 0.7384 0.05  ‐0.0252  0.0351  78.3 0.5975 17.3 0.037 0.0211 93.9 0.462 0.1283 0.0258 99.6 0.7654 0.1  ‐0.0222  0.0338  76.4 0.5878 18.2 0.0495 0.0214 93.3 0.4713 0.1377 0.0278 100 0.7663 0.2  ‐0.0284  0.0467  84.8 0.7281 6.5 0.0028 0.024 92 0.5081 0.0665 0.0183 99.4 0.8129 0.3  ‐0.1  ‐0.0109  0.0246  88.4 0.5158 2.8 0.0275 0.0152 89.5 0.3921 0.0281 0.0166 97.8 0.5749 0  ‐0.0162  0.0237  91.3  0.5378  0.6  0.0109  0.0155  89.5  0.4063  0.0004  0.0177  97.3  0.6147  0.05  ‐0.0163  0.0293  88 0.5518 0.3 ‐0.0095 0.0177 89 0.418 ‐0.008 0.0235 95.9 0.6305 0.1  ‐0.0126  0.0271  89.2 0.5496 0 0.0051 0.0171 88.8 0.4162 ‐0.0076 0.0224 95.6 0.6313 0.2  ‐0.0108  0.0296  88.9 0.5813 0 ‐0.0069 0.0182 88.7 0.4178 ‐0.0146 0.0261 95.2 0.6479 0.5  ‐0.1  ‐0.0199  0.0276  89.4 0.5506 0 0.0035 0.0178 87.1 0.4096 ‐0.008 0.0289 89.7 0.5459 0  ‐0.0226  0.0268  90.5 0.5568 0 ‐0.0076 0.0174 89 0.4189 ‐0.0188 0.0261 90.6 0.5482 0.05  ‐0.0147  0.0288  89.4 0.5504 0 ‐0.0069 0.0174 89.6 0.4175 ‐0.0193 0.0284 90 0.5468 0.1  ‐0.0174  0.0276  89.4 0.5426 0 ‐0.0002 0.0159 90.9 0.4189 ‐0.0239 0.0277 90.6 0.5474 0.2  ‐0.0096  0.0258  91 0.5415 0 ‐0.0023 0.017 88.8 0.4083 ‐0.0351 0.0318 91.2 0.5604 82 Table 19. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling of Families，n=40 Families per Strata) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0018  0.0056  79.7 0.2175 56.5 0.0573 0.0047 80.4 0.2334 0.1887 0.0378 44.4 0.5137 0  ‐0.0152  0.0122  81.2 0.3485 20.3 0.0369 0.009 93.5 0.3079 0.1214 0.0181 100 0.5775 0.05  ‐0.0113  0.0174  86.4 0.4459 5 0.0142 0.0115 91.5 0.3435 0.0748 0.0114 99.9 0.6216 0.1  ‐0.0107  0.0162  87.2 0.4377 5.3 0.026 0.012 91.1 0.3496 0.079 0.0114 99.9 0.6179 0.2  ‐0.0187  0.0245  87.9 0.5153 0.7 ‐0.0122 0.0154 88.8 0.3886 0.0232 0.0113 99.4 0.6756 0.3  ‐0.1  0.0008  0.0116  89 0.3626 0 0.021 0.0073 90.2 0.2862 0.0113 0.0094 98.5 0.4532 0  ‐0.0089  0.0132  89.7  0.3778  0  0.0087  0.0086  87.9  0.296  ‐0.007  0.012  96.7  0.4699  0.05  ‐0.008  0.0133  89.9 0.388 0 ‐0.0067 0.0088 89.5 0.3064 ‐0.009 0.0125 96.5 0.4724 0.1  ‐0.0057  0.0139  89.2 0.3857 0 0.0088 0.0087 89.2 0.3051 ‐0.0074 0.0133 94.9 0.4694 0.2  ‐0.0133  0.0149  89.5 0.4065 0 ‐0.003 0.0095 88.7 0.3042 ‐0.017 0.0146 95.1 0.468 0.5  ‐0.1  ‐0.0103  0.013  91 0.3875 0 0.0083 0.0085 89.5 0.2973 ‐0.009 0.0127 90.1 0.3791 0  ‐0.0054  0.0136  90 0.3928 0 ‐0.0025 0.0086 89.4 0.3045 ‐0.0055 0.0133 89.9 0.3854 0.05  ‐0.0116  0.0137  90.1 0.3891 0 ‐0.01 0.0093 88.4 0.3023 ‐0.012 0.0135 89.7 0.3862 0.1  ‐0.0093  0.0126  90.8 0.3799 0 0.0038 0.0085 89.9 0.303 ‐0.0099 0.0124 91.5 0.3776 0.2  ‐0.0156  0.0132  89.2 0.38 0 ‐0.0068 0.0089 88.9 0.2946 ‐0.0171 0.013 89.5 0.372 83 Table 20. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Stratified Simple Random Sampling of Families，n=80 Families per Strata) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0001  0.0029  82.1 0.1599 51.2 0.0322 0.0017 81.7 0.1373 0.1379 0.02 44.1 0.3793 0  ‐0.0046  0.0066  86.3 0.2571 7.8 0.0188 0.0044 91.9 0.2203 0.0738 0.0074 100 0.449 0.05  ‐0.0027  0.0096  88.2 0.3229 0.4 0.0017 0.0067 87.2 0.2517 0.0369 0.0057 100 0.4988 0.1  0.0008  0.0085  90 0.3171 0.7 0.0186 0.0062 91.1 0.2606 0.0413 0.0054 99.8 0.4967 0.2  ‐0.0059  0.012  91.5 0.3705 0.1 ‐0.014 0.0085 89.1 0.294 0.0016 0.0078 99.4 0.5556 0.3  ‐0.1  ‐0.0037  0.0059  89.7 0.2568 0 0.0128 0.004 89.3 0.2041 ‐0.0021 0.0054 97.7 0.3418 0  ‐0.0038  0.0068  87.8  0.2654  0  0.0089  0.0046  88.7  0.2129  ‐0.0044  0.0067  96.7  0.3395  0.05  ‐0.006  0.0065  89.3 0.271 0 ‐0.0099 0.0043 89.7 0.2206 ‐0.0073 0.0065 96 0.3382 0.1  ‐0.0015  0.0069  89.5 0.2715 0 0.0083 0.0047 89.1 0.2198 ‐0.0027 0.0068 95.5 0.3332 0.2  ‐0.0084  0.0076  88.8 0.2885 0 ‐0.0038 0.005 87.1 0.2184 ‐0.0104 0.0075 93.8 0.3338 0.5  ‐0.1  ‐0.0038  0.0064  91.1 0.2729 0 0.0066 0.0046 89.1 0.2128 ‐0.0032 0.0063 90.6 0.2677 0  ‐0.0058  0.0071  89.8 0.2773 0 ‐0.0047 0.0045 89.2 0.2179 ‐0.0059 0.0071 90 0.2736 0.05  ‐0.0033  0.0068  89.7 0.2727 0 ‐0.0071 0.0044 89.3 0.2168 ‐0.0035 0.0068 90.1 0.272 0.1  0.0004  0.0064  90.4 0.2685 0 0.0111 0.0045 89 0.2174 0.0001 0.0063 90.8 0.2682 0.2  ‐0.0013  0.0063  90.5 0.2673 0 0.0013 0.0042 89.5 0.2109 ‐0.0022 0.0062 90.1 0.2638 84 A. n=20 families B. n=40 families C. n=80 families Figure 9. Density Plot of Simulated Estimators under Different Settings (Stratified Simple Random  Sampling). BPELE=Bayesian Pseudo‐Empirical Likelihood Estimator, DBE=Design‐Based Estimator,  PBE=Parametric Bayesian Estimator. Parameter p is the allele frequency and f is the inbreeding  coefficient in the simulation. Panel A, B and C are for stratum family sample size 20, 40 and 80  families respectively. Red dashed vertical line is the true parameter f; red dotted vertical line is  the lower bound of the parameter space for f conditional on allele frequency p.  85 4.4.2 Proportional to Population Size Sampling For the second simulation study, we used proportional to size (PPS) sampling. For each family, the size variable X is generated by the number of the allele A count plus a log normal distributed random error (mean 0, standard deviation 0.2). A total of 20, 40 or 80 families are randomly selected from this population using the PPS sampling. Therefore, the sample selection probability is the sample size times the relative proportion of the size measure. The sampling weight for analysis is the inverse of the sample selection probability, ݐݓ௜ ൌ 1/ሺ݊ ൈ ௫೔∑ ௫೔೔ಿసభ ሻ, where i is the index of each family in the population and n is the sample size. The sampling weight from this PPS sampling design depends on the number of the allele A, therefore, is informative. Table 21, Table 22, Table 23 and Figure 10 report the simulation results to compare design-based estimator and two Bayesian estimators for the bias, MSE, coverage probability (CP), out of range probability (ORP), and average interval length (AL). The design-based estimator generally has a negative bias, while both Bayesian methods tend to have negative bias when the allele frequency is medium or large (p=0.3, 0.5). When the allele frequency is small (p=0.1), the Bayesian methods have larger bias especially when the inbreeding coefficient is small (f=-0.1, 0). On average from all simulation scenario, the parametric Bayesian estimator has the smallest MSE. For coverage, the design-based estimator does not maintain the nominal level. The BPELE maintains the coverage closest to the nominal level, but with largest average interval length in all three simulations. The design-based estimator is likely to lay outside of the parameter space when the allele frequency is small (p=0.1, 0.3). The situation gets worse when the actual inbreeding coefficient is small (f=-0.1, 0). Increasing the sample size slightly reduces the out of range possibility. 86 Table 21. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Proportional to Size Sampling, n=20 Families) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0179  0.018  84.3 0.2789 68.8 0.1945 0.0456  75.6 0.6294 0.3402 0.1238 37 0.8548 0  ‐0.0514  0.0428  25.9 0.4569 49.7 0.1376 0.0339  96.5 0.665 0.2579 0.0763 98.5 0.8826 0.05  ‐0.0677  0.0594  38.2 0.5993 37.1 0.0952 0.0268  97.7 0.689 0.196 0.0487 98.8 0.9029 0.1  ‐0.0505  0.0631  39.1 0.6204 37.9 0.1069 0.0325  97 0.6871 0.2042 0.0529 99.2 0.9011 0.2  ‐0.0919  0.0912  49.9 0.7968 24.7 0.0301 0.0257  97.9 0.7147 0.1122 0.0269 98.8 0.919 0.3  ‐0.1  ‐0.0394  0.0464  84 0.6649 9.6 0.0394 0.0241  94.2 0.5634 0.0659 0.0239 97.8 0.7266 0  ‐0.0265  0.0522  86.2  0.7144  2.2  0.0276  0.0284  91.8  0.5893  0.0357  0.0262  97  0.7807  0.05  ‐0.0428  0.06  86.1 0.7407 2.3 0.002 0.0294  90.3 0.6015 0.0057 0.0275 97.9 0.8006 0.1  ‐0.0426  0.0616  85.5 0.7577 0.8 0.008 0.0289  93.1 0.6067 ‐0.0028 0.0315 96.9 0.807 0.2  ‐0.0604  0.0709  83.7 0.7828 0.5 ‐0.0203 0.0325  91.7 0.617 ‐0.0438 0.0383 96.6 0.8425 0.5  ‐0.1  ‐0.0395  0.0518  87.5 0.725 0 0.0105 0.0268  93.4 0.586 0.056 0.0599 97 0.8912 0  ‐0.035  0.0557  87.5 0.7402 0 ‐0.0128 0.0273  93.3 0.5971 ‐0.0355 0.049 94.2 0.8253 0.05  ‐0.0309  0.0547  86.9 0.746 0 ‐0.0171 0.0284  92.8 0.5987 ‐0.0426 0.0494 94.8 0.8263 0.1  ‐0.0418  0.0562  87.8 0.7305 0 ‐0.0045 0.0268  93.3 0.5975 ‐0.0565 0.0512 95 0.8452 0.2  ‐0.039  0.055  87 0.7182 0 ‐0.0225 0.0276  93.2 0.5932 ‐0.113 0.063 94.2 0.8908 87 Table 22. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Proportional to Size Sampling, n=40 Families) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0115  0.0116  76.3 0.2279 69.1 0.122 0.0185  76.9 0.4344 0.2583 0.0715 39.6 0.6818 0  ‐0.0271  0.0287  46.3 0.406 39.6 0.0824 0.0175  95.8 0.495 0.1867 0.042 98.7 0.7321 0.05  ‐0.038  0.0374  61.2 0.519 23.5 0.0472 0.0166  97.4 0.5294 0.1317 0.0258 99.4 0.7641 0.1  ‐0.0309  0.0365  62.4 0.5292 25.1 0.0629 0.0203  96.4 0.5307 0.1386 0.0278 99.2 0.7642 0.2  ‐0.0489  0.0573  72.4 0.6679 12.7 0.0072 0.021  97.8 0.5747 0.068 0.0196 98.2 0.8044 0.3  ‐0.1  ‐0.0151  0.0243  86.3 0.475 2.2 0.0256 0.0144  92.8 0.4214 0.0298 0.015 97.5 0.5697 0  ‐0.0313  0.0307  84.5  0.5029  0.6  0.0062  0.0162  91.8  0.4428  ‐0.0053  0.0204  96.8  0.6044  0.05  ‐0.0249  0.0342  83.1 0.5242 0.2 ‐0.0111 0.0179  91.6 0.4562 ‐0.0097 0.0248 94.9 0.6166 0.1  ‐0.0433  0.0322  85.6 0.5233 0.2 ‐0.0056 0.0158  93.2 0.4603 ‐0.0313 0.0236 96 0.6212 0.2  ‐0.0347  0.0379  84.8 0.5565 0 ‐0.0147 0.0177  93.4 0.4741 ‐0.0343 0.0317 94.2 0.6453 0.5  ‐0.1  ‐0.0246  0.0285  87.4 0.5131 0 0.006 0.015  92.5 0.4384 ‐0.0108 0.0311 90.8 0.5501 0  ‐0.0219  0.0305  85.4 0.522 0 ‐0.0035 0.0164  92.3 0.4485 ‐0.0228 0.0294 89.1 0.5506 0.05  ‐0.0202  0.0287  87.9 0.5273 0 ‐0.017 0.016  92.6 0.4485 ‐0.0251 0.0284 91.1 0.5501 0.1  ‐0.0377  0.0298  86.4 0.5158 0 ‐0.0114 0.0156  93.6 0.4483 ‐0.0462 0.0305 89.5 0.5459 0.2  ‐0.0343  0.0274  86.5 0.5124 0 ‐0.0183 0.0143  93.5 0.4443 ‐0.0562 0.0323 89.3 0.5626 88 Table 23. Comparison of Design-Based, Parametric Bayesian and Bayesian Empirical Likelihood Estimators (Proportional to Size Sampling, n=80 Families) * MSE=mean square error, CP=coverage probability, ORP=out of range probability, AL=average length   p    f  Design‐Based Estimator Parametric Bayesian Estimator  Bayesian Pseudo‐Empirical Likelihood  Estimator  Bias  MSE  CP AL ORP Bias MSE CP  AL Bias MSE CP AL 0.1    ‐0.1  ‐0.0097  0.0055  66.5 0.178 65 0.067 0.0059  83.8 0.2694 0.1867 0.0366 42.6 0.5094 0  ‐0.0085  0.0162  72.4 0.3361 22.4 0.0461 0.01  94.8 0.3522 0.126 0.0203 99.4 0.5781 0.05  ‐0.0192  0.0218  80 0.4117 10.7 0.0184 0.0115  95.8 0.3911 0.0742 0.0125 98.7 0.6127 0.1  ‐0.0154  0.0195  81.5 0.4087 10.1 0.0326 0.0121  95.2 0.3944 0.0802 0.012 99.6 0.6134 0.2  ‐0.0191  0.0293  84.4 0.5172 3 0.0023 0.0151  92.9 0.4493 0.0282 0.0137 98.3 0.6725 0.3  ‐0.1  ‐0.0016  0.0142  85.1 0.3402 0.3 0.023 0.008  92.4 0.3116 0.0134 0.0108 96.4 0.4454 0  ‐0.017  0.0148  85.4  0.3568  0  0.0033  0.0084  93  0.3278  ‐0.0128  0.0129  95.7  0.4639  0.05  ‐0.0176  0.0168  84.1 0.3703 0 ‐0.0093 0.0096  92.4 0.3393 ‐0.0164 0.0152 94.7 0.4683 0.1  ‐0.0159  0.0184  82.4 0.3732 0 0.0033 0.0102  92 0.342 ‐0.0157 0.0171 92.6 0.4624 0.2  ‐0.0247  0.0215  80.2 0.3896 0 ‐0.0094 0.011  91.8 0.3514 ‐0.0273 0.0208 91 0.4683 0.5  ‐0.1  ‐0.0088  0.0129  88.4 0.3631 0 0.0123 0.0079  92.2 0.3214 ‐0.0078 0.0127 90.5 0.3789 0  ‐0.007  0.0142  87.3 0.3718 0 ‐0.0028 0.0079  93.5 0.3279 ‐0.0074 0.014 89 0.3861 0.05  ‐0.0126  0.014  87.6 0.3699 0 ‐0.0065 0.0082  93.5 0.3259 ‐0.0132 0.0138 90 0.3845 0.1  ‐0.0162  0.015  86.6 0.3643 0 0.0016 0.008  93.4 0.3278 ‐0.017 0.0148 87.7 0.3785 0.2  ‐0.0114  0.0151  84 0.3614 0 ‐0.0021 0.0079  92.7 0.3232 ‐0.0146 0.0153 85.4 0.3746 89 A. n=20 families B. n=40 families C. n=80 families Figure  10.  Density  Plot  of  Simulated  Estimators  under  Different  Settings  (Proportional  to  Population Size Sampling). BPELE=Bayesian Pseudo‐Empirical Likelihood Estimator, DBE=Design‐ Based Estimator, PBE=Parametric Bayesian Estimator. Parameter p is the allele frequency and f  is the inbreeding coefficient in the simulation. Panel A, B and C are for stratum family sample size  20, 40 and 80 families respectively. Red dashed vertical line is the true parameter f; red dotted  vertical line is the lower bound of the parameter space for f conditional on allele frequency p.  90 Chapter 5: Estimation of the Inbreeding Coefficient Incorporating Between Subject Correlation In Chapter 4, we incorporated within family level correlation in estimating the inbreeding coefficient. Inspired by the NHANES survey design, we now discuss the feasibility in incorporating the between family correlation to estimate the inbreeding coefficient. Because of the similarity in analysis between surveyed individual genotypes and surveyed family genotypes as noted earlier, it is reasonable to concentrate on the between subject correlation only. Most complex surveys usually have a clustered sampling design. Within cluster, it is expected to have between subject correlations. Similarly, between family correlations can happen when families are sampled from clusters. The hierarchical Bayesian model discussed in the following can be generalized to the family level genotype data. 5.1 Population Subdivision Population subdivision means that the study population is not panmictic, e.g., individuals do not mate at random regardless of any environmental, genetic, or social preference. This departure from panmixia introduces amount of inbreeding at various levels of subdivision. Therefore, the genetic difference among individuals is a result of difference among members of the subpopulation, among members of different subpopulations in the same geographical region, and among members of subpopulations in the same geographical region (Holsinger K. , 1999). In another words, individuals who are from the same cluster, therefore, tend to be more genetically similar than those who are not. Failure to incorporate population cluster or stratification can severely affect statistical test of the genetic association (Pritchard, Stephens, & Donnelly, 2000). There are two types of variability to consider: variability due to sampling and that due to the population substructure. Even if two subpopulations are maintained under the same evolutionary conditions, they may still have different allele frequencies because of the stochastic nature (Weir & Hill, 2002). When the sample size is large and the population substructure is non-negligible, simulation studies show that the estimates from the subpopulations are closer to the true values than the pooled estimates. The opposite is also true for small sample size from more homogeneous subpopulations (Lockwood, Roeder, & Devlin, 2001). 91 The fixation index F is a measure of correlation between homologous genes relative to a pair of genes randomly selected from a population. In a simple two-level sampling hierarchy where individuals are within subpopulations and subpopulations are within total population, Wright classified the F statistics into three categories: FIT, the correlation between genes within individual relative to the genes from population; FIS, the correlation between genes within individual relative to the genes from subpopulation; FST, the correlation between genes within subpopulation relative to the genes from total population (Wright S. , 1965). By partitioning the genetic diversity and using the modeling approach, we can study whether any difference among populations are statistically important. Let HI be the actual heterozygosity for individuals within subpopulation, HS be expected heterozygosity within subpopulation assuming HWE within subpopulations, and HT be expected heterozygosity in the total population assuming HWE for the total population. Based on the definition of F statistics, it maintains the following relationship: ܨூ் ൌ 1 െ ܪூ்ܪ , ܨ തூௌ ൌ 1 െ ܪூܪௌ , ܨௌ் ൌ 1 െ ܪௌ ்ܪ, 1 െ ܨூ் ൌ ሺ1 െ ܨതூௌሻሺ1 െ ܨௌ்ሻ, where ܨതூௌ is unweighted average of FIS values across subpopulations with proportional to the subpopulation size. A weighted average calculation which includes relative deme effect sizes and subpopulation allele frequencies can be found in Wright (1969). Genetic subdivision can be estimated based on the above observed and expected heterozygosity. The genetic diversity due to the allele frequency difference among subpopulation is, therefore, GST = (HT – HS) / HT (Nei, 1973). From its hierarchical population structure, under the framework of the analysis of variance (ANOVA), Cockerham decomposed the total variance of gene frequencies into variance components associated with various levels of subpopulation to estimate the correlation of genes (Cockerham, 1973). For a simple case of two alleles from a single locus, FST is equal to the variance among populations divided by the maximum possible variance given the observed mean allele frequencies, e.g., ௩௔௥ೄሺ௣ሻ௉തሺଵି௣̅ሻ (Weir & Cockerham, 1984). Thus, it is a measure of proportion of genetic diversity in the whole population that is due to the allele frequency difference among subpopulations. Small FST means that the allele frequencies within each population are similar. When the number of populations sampled is relatively large, both 92 estimators of FST – Nei’s GST and Wei & Cockerham’s θ, are typically very similar (Holsinger K. , 1999). 5.2 Bayesian Hierarchical Model Based on the similarities between Bayesian modeling and hierarchical approach to partition the genetic diversity, Holsinger developed a Bayesian simple multi-locus bi-allelic beta-binomial model to estimate the fixation index (1999). The allele frequency p in each subpopulation has a beta distribution Beta(a, b) with ܽ ൌ ݔሺ1 െ ߠሻ/ߠ and ܾ ൌ ሺ1 െ ݔሻሺ1 െ ߠሻ/ߠ , where θ corresponds to FST that can be calculated from the allele frequencies across all subpopulations and x corresponds to the empirical estimate of the mean allele frequency (Rorder, Escobar, Kadane, & Balazs, 1998). With appropriate priors for x and θ, the allele frequencies for each subpopulation and θ can be estimated from the posterior distribution. Then GST can be further estimated from a random effects model. From Wright’s classification, FIS is the inbreeding coefficient f for the subpopulation and is often used as a measure of the genotype frequencies departure from HWE within subpopulations. Within each subpopulation, the genotype counts follow a multinomial distribution with subpopulation specific inbreeding coefficient fi and allele frequency pi similar as previously discussed. If the genotype frequencies in the total population differs from the HWE, expectations can be modeled based on the average allele frequency. For the bi-allelic case, they are given by: ቐ ݌ଵଵ ൌ ݌ଶ ൅ ݌ሺ1 െ ݌ሻܨ, ݌ଵଶ ൌ 2݌ሺ1 െ ݌ሻሺ1 െ ܨሻ, ݌ଶଶ ൌ ሺ1 െ ݌ሻଶ ൅ ݌ሺ1 െ ݌ሻܨ, where p is the average allele frequency and F is the inbreeding coefficient in the total population FIT. If there is no population subdivision, the above model is reduced to the one we used in previous chapters, therefore F=FIS=f. For the subdivided population, let f be the average within- subpopulation departure from the HWE. Then, we haveሺ1 െ ܨሻ ൌ ሺ1 െ ݂ሻሺ1 െ ߠሻ. We will use the following notations: s: index for the subpopulation (s = 1, 2, …, S); 93 k: index for the allele at a single locus (k = 1, 2, …, K); m: index for the genotype (m = 1, 2, …, M), M=K(K+1)/2; Aij: index for the genotype with the alleles Ai and Aj (i =1, 2, …, K; j = i, i+1, …, K); ݊௦௜௝: index for the genotype count of Aij for the subpopulation s; ࢙࢔ : a vector of size M for the genotype counts ሺ݊௦ଵଵ, ݊௦ଵଶ, … , ݊௦ଵ௄, ݊௦ଶଶ, … , ݊௦௄௄ሻ for the subpopulation s; ࢖෥࢙: a vector of size M for the genotype frequencies ሺ݌௦ଵଵ, ݌௦ଵଶ, … , ݌௦ଵ௄, ݌௦ଶଶ, … , ݌௦௄௄ሻ for the subpopulation s; ࢙࢖: a vector of size K for the allele frequencies ሺ݌௦ଵ, ݌௦ଶ, … , ݌௦௄ሻ for the subpopulation s. In the first level of hierarchy, the genotype counts of each subpopulation are modeled as ݈ܽ݅݉݋݊݅ݐ݈ݑܯሺ࢙࢔, ࢖෥࢙ሻ, e.g., ܲሺ࢖|࢙࢔෥࢙ሻ ൌ ݊௦!∏ ∏ ݊௦௜௝!௄௝ୀ௜௄௜ୀଵ ෑ ෑ ݌௦௜௝ ௡ೞ೔ೕ ௄ ௝ୀ௜ ௄ ௜ୀଵ . The genotype frequencies can be re-parameterized using the inbreeding model as ݌௦௜௜ ൌ ሺ1 െ ݂ሻ݌௦௜ଶ ൅ ݂݌௦௜ ൌ ݌௦௜ଶ ൅ ݌௦௜ሺ1 െ ݌௦௜ሻ݂, ݅ ൌ 1,… , ݇; ݌௦௜௝ ൌ 2ሺ1 െ ݂ሻ݌௦௜݌௦௝, ݅ ൌ 1,… , ݇; ݆ ൌ ݅ ൅ 1,… , ݇. We assume a common f, which is the same as the average within-population inbreeding coefficient FIS. A subpopulation specific inbreeding coefficient fs can also be modelled when local inbreeding is of interest. In the second level, the allele frequency ࢙࢖ is modeled by ݐ݈݄݁ܿ݅ݎ݅ܦሺࢇሻ; ࢇ ൌ ሺܽଵ, ܽଶ, … , ܽ௄ሻ (Beta when K=2) (Lange, 1995): ܲሺࢇ|࢙࢖ሻ ൌ 1ܤሺࢇሻෑ ݌௦௞ ௔ೖିଵ ௄ ௞ୀଵ , where ܽ ൌ ∑ ܽ௞௄௞ୀଵ is the dispersion parameter with large value implying smaller dispersion among subpopulations. Therefore ak/a is the expected value of the allele Ak frequency (Navarroa, Griffithsb, Steyversc, & Lee, 2006). Since the dispersion parameter a represents the amount of 94 variation that we expect to see in a finite sample, it can be approximated by the Wright’s FST (or Wei & Cockerham’s θ) as a=(1- θ)/ θ (Lockwood, Roeder, & Devlin, 2001). Therefore, our first re-parameterization of a is as follows: ܽ௞ ൌ 1 െ ߠߠ ݌௞, ݇ ൌ 1,… , ܭ, where ࢖ ൌ ሺ݌ଵ, … , ݌௄ሻ corresponds to the mean of the allele frequencies distribution across subpopulations. From the property of Dirichlet distribution, the mean and variance of the allele frequency estimator is given below (Lockwood, Roeder, & Devlin, 2001): ܧሺ̂݌௦௞ሻ ൌ ܽ௞ܽ ൌ 1 െ ߠ ߠ ݌௞ ∑ 1 െ ߠߠ ݌௞௄௞ୀଵ ൌ ݌௞; ܸܽݎሺ̂݌௦௞ሻ ൌ ܽ௞ሺܽ െ ܽ௞ሻሺܽ ൅ 1ሻܽଶ ൌ 1 െ ߠ ߠ ݌௞ሺ 1 െ ߠ ߠ െ 1 െ ߠ ߠ ݌௞ሻ ሺ1 െ ߠߠ ൅ 1ሻሺ 1 െ ߠ ߠ ሻଶ ൌ ݌௞ሺ1 െ ݌௞ሻߠ. Let ߮ሺ݂ሻ, ߮ሺߠሻ, ߮ሺ࢖ሻ be the prior distribution for f, θ, and p, respectively. The joint posterior distribution for p, θ, and f is given by ߮ሺ࢖, ݂, ࢙࢔|ߠሻ ∝ ൜ෑ ߮ሺ࢙࢖|࢙࢔ሻ ௌ ௦ୀଵ ߮ሺࢇ|࢙࢖ሻൠ߮ሺ࢖ሻ߮ሺߠሻ߮ሺ݂ሻ. If X has a Dirichlet distribution, ࢄ ൌ ሺ ଵܺ, … , ܺ௄ሻ~ݎ݅ܦሺܽሻ, the vector X is neutral in the sense that XK is independent of ܺሺି௄ሻ ൌ ሺ ௑భଵି௑಼ , ௑మ ଵି௑಼ , … , ௑಼షభ ଵି௑಼ሻ . Any permutation of X is also neutral (Connor & Mosimann, 1969). Therefore, for k = 1, 2, …, K-1, it is possible to take log- transformation of the Dirichlet distribution parameters. Let ߛ௞ ൌ ݈݃݋ ሺ௔ೖ௔಼ሻ and ߚ ൌ ݈݃݋ ሺܽሻ. Under this re-parameterization, the prior distribution of ࢽ ൌ ሺߛଵ, … , ߛ௄ିଵሻ, ߮ሺࢽሻ, can be assumed to have a multivariate normal distribution with mean and variance-covariance matrix derived from the observed data. The prior distribution of β, ߮ሺߚሻ, can also be assumed to have a normal distribution with parameters estimated form the data. The posterior distribution therefor is given by ߮ሺ࢖, ݂, ࢙࢔|ߠሻ ∝ ൜ෑ ߮ሺ࢙࢖|࢙࢔ሻ ௌ ௦ୀଵ ߮ሺࢇ|࢙࢖ሻൠ߮ሺࢽሻ߮ሺߚሻ߮ሺ݂ሻ. 95 One of the advantages of this Multinomial-Dirichlet hierarchical model is that the estimated minor allele frequency will not be forced to be zero if there is no observed allele count in a particular subpopulation. We will use this model parameterization in our future research. The above model can be generalized to samples with unequal sample selection probabilities. In such case, we use the effective sample size ࢙࢔∗as discussed in Chapters 3 and 4. When K=2, our first OPENBUGS hierarchical Bayesian model for the cluster sampling is as follows: Level 1: ݈ܽ݅݉݋݊݅ݐ݈ݑܯ~∗࢙࢔ሺ݊௦∗, ࢙࢖ሻ, ࢙࢔∗ ൌ ሺ݊௦଴∗, ݊௦ଵ∗, ݊௦ଶ∗ሻ, ࢙࢖ ൌ ሺ݌௦଴, ݌௦ଵ, ݌௦ଶሻ, s=1, 2, …, S ݊௦∗ ൌ ݊௦଴∗ ൅ ݊௦ଵ∗ ൅ ݊௦ଶ∗, ቐ ݌௦଴ ൌ ݌௦ଶ ൅ ݌௦ሺ1 െ ݌௦ሻ݂, ݌௦ଵ ൌ 2݌௦ሺ1 െ ݌௦ሻሺ1 െ ݂ሻ, ݌௦ଶ ൌ ሺ1 െ ݌௦ሻଶ ൅ ݌௦ሺ1 െ ݌௦ሻ݂. Level 2: ݌௦~ܽݐ݁ܤሾሺଵିఏሻ௣ఏ ， ሺଵିఏሻሺଵି௣ሻ ఏ ሿ. Level 3: p ~ Uniform [0, 1], ݂|p ~ Uniform [-min(p, 1-p)/(1-min(p,1-p)) ,1], ߠ ~ Uniform [0, 1]. model=function() { for (i in 1:nsub){ q[i,1]<-(1-f)*p[i]*p[i] + f*p[i] q[i,2]<-2*(1-f)*p[i]*(1-p[i]) q[i,3]<-(1-f)*(1-p[i])*(1-p[i]) + f*(1-p[i]) y[i,1:3] ~ dmulti(q[i,],n[i]) n[i]<-sum(y[i,]) p[i] ~ dbeta(alpha, beta) } alpha<-(1-theta)*pai/theta beta<-(1-theta)*(1-pai)/theta pai~dunif(0,1) f <- w*(1 - f.min) + f.min f.min <- max(-pai/(1-pai), -(1-pai)/pai) 96 w ~ dunif(0, 1) theta~dunif(0,1) } Instead of using a common inbreeding coefficient at level one, cluster specific inbreeding models can be applied. The population average of the inbreeding coefficient, therefore, can be estimated from the average of the posterior cluster specific fs, s=1, …, S (Cavalli-Sforza & Bodmer, 1971). The following is our second OPENBUGS hierarchical Bayesian model for the cluster sampling: Level 1: ݈ܽ݅݉݋݊݅ݐ݈ݑܯ~∗࢙࢔ሺ݊௦∗, ࢙࢖ሻ, ࢙࢔ ൌ ሺ݊௦଴∗, ݊௦ଵ∗, ݊௦ଶ∗ሻ, ࢙࢖ ൌ ሺ݌௦଴, ݌௦ଵ, ݌௦ଶሻ, s=1, 2, …, S; ݊௦∗ ൌ ݊௦଴∗ ൅ ݊௦ଵ∗ ൅ ݊௦ଶ∗, ቐ ݌௦଴ ൌ ሺ1 െ ݌௦ሻଶ ൅ ݌௦ሺ1 െ ݌௦ሻ ௦݂, ݌௦ଵ ൌ 2݌௦ሺ1 െ ݌௦ሻሺ1 െ ௦݂ሻ, ݌௦ଶ ൌ ݌௦ଶ ൅ ݌௦ሺ1 െ ݌௦ሻ ௦݂. Level 2: ݌௦~ܽݐ݁ܤ ቂሺଵିఏሻ௣ఏ ， ሺଵିఏሻሺଵି௣ሻ ఏ ቃ, ௦݂|݌௦~ܷ݂݊݅݉ݎ݋ ቂെ ୫୧୬ሺ௣ೞ,ଵି௣ೞሻଵି୫୧୬ሺ௣ೞ,ଵି௣ೞሻ , 1ቃ. Level 3: p ~ Uniform [0, 1], ߠ ~ Uniform [0, 1]. model=function() { for (i in 1:nsub){ q[i,1]<-(1-f[i])*p[i]*p[i] + f[i]*p[i] q[i,2]<-2*(1-f[i])*p[i]*(1-p[i]) q[i,3]<-(1-f[i])*(1-p[i])*(1-p[i]) + f[i]*(1-p[i]) y[i,1:3] ~ dmulti(q[i,],n[i]) p[i] ~ dbeta(alpha,beta) f[i]<-w[i]*(1 - f.min[i]) + f.min[i] f.min[i]<- -min(p[i],1-p[i])/(1-min(p[i],1-p[i])) w[i]~dunif(0,1) n[i]<-sum(y[i,]) } alpha<-(1-theta)*pai/theta beta<-(1-theta)*(1-pai)/theta pai~dunif(0,1) 97 theta~dunif(0,1) pf<-mean(f[]) } 5.3 Simulation Study We compare the parametric Bayesian estimator with the direct sample estimator using a Monte Carlo simulation study. First, a direct sample estimate f(s) is calculated based on simulated total sample genotypic data. This is compared with the Bayesian estimator with a common inbreeding coefficient model. Secondly, ܨതூௌ, an average of FIS values across subpopulations with proportional to the subpopulation size is derived with respect to the clustered data. This is compared with the Bayesian estimator of the population average of cluster specific inbreeding coefficients. The following parameters are used in the simulation: (i) H=23, the number of sample strata; (ii) L=2, the number of sample clusters per sample stratum; (iii) m=60, the number of individuals sampled from each cluster; (iv) ߩ ൌ 0.017, the intra-class correlation coefficient measuring the degree of correlation due to the clustering of the sample; This correspondent to an inflation of variance of 1 ൅ ߩሺ݉ ൅ 1ሻ ൌ 2 (Brier, 1980) for proportion compared to a simple random sample with the same sample size. Therefore, the design effect at this setting is Deff=2. (v) w, the non-informative sample weight, which is randomly assigned to one third each of the samples with values 1, 3 or 5. (vi) G=5, the number of analyzing groups; (vii) ࡳ࢖ ൌ ሺ݌ଵ, ݌ଶ, ݌ଷ, ݌ସ, ݌ହሻ ൌ ሺ0.2, 0.25, 0.3, 0.35, 0.4ሻ , the allele frequencies for G groups; We use this parameter setting to represent slightly different allele frequencies among subpopulations. (viii) f=-0.1, 0, 0.05, 0.1 or 0.2. These simulation settings are in line with the NHANES III study design (NCHS, 1994) and are modified from Li (2009) when the performance of testing HWE under complex sampling is studied. For the g-th group in each cluster, genotype frequencies are first generated from 98 independent Dirichlet-Multinomial distribution, then individual genotypes are simulated from multinomial with those generated genotype frequencies. The group size is generated by multinomial with cluster sample size and equal probabilities among groups. For the direct sample estimator, the variance is estimated by the jackknife method. For the Bayesian approach, effective sample size and effective number of genotype counts are used in the multinomial model. Table 24 and Table 25 display the simulated (with 100 replications) bias, MSE, relative root mean square error (RRMSE), coverage probability and average interval length for the direct sample estimator and hierarchical Bayesian estimator. For the model with common f, the direct sample estimator is calculated based on the overall sample; while for the cluster specific fs model, it is calculated as the mean of the subpopulation inbreeding coefficients weighted by the subpopulation size. In all our simulations, only average direct sample estimator using cluster specific inbreeding coefficients can maintain the nominal coverage probability. The Bayesian estimator has less coverage than the design-based estimator. However, it has smaller simulated bias, MSE and average interval length. For example, when the true inbreeding coefficient f=0, the direct sample estimator has a relative large positive bias of 0.0228 with MSE of 0.001 when treating the total population without subdivision; while the Bayesian estimator has a small negative bias of -0.0012 with MSE of 0.0005. The RRMSE between the Bayesian and design-based estimator is 71%. The large positive bias observed for the direct sample estimator in Table 24 is likely due to the Wahlund effect caused by subpopulation structure. As we discussed in Chapter 1, the inbreeding coefficient is a measure of correlation between two homologous genes in uniting gametes. With the introduction of the intra-class correlation coefficient due to the clustering of the sample, it inflates the overall correlation between genes. By estimating the average subpopulation inbreeding coefficients, the bias of the direct sample estimator is reduced to -0.002. The bias for the Bayesian method is also reduced slightly to -0.0003. To compare the design-based estimator with the Bayesian estimator under clustered survey sampling, a second simulation study is conducted. Instead of generating a random sample as above in each simulation iteration, we first create a finite population with 1000 PSUs with other simulation parameters remaining the same. The finite population inbreeding coefficient is 99 calculated as the average of the inbreeding coefficients across clusters. Next at each one of 1000 simulation iteration, we randomly select 50 PSUs from the finite population. All units within the selected PSUs are included in the final analysis sample. Based on the result from the first simulation study, we focus on the sample direct estimator ܨതூௌ and the Bayesian estimator from a cluster specific inbreeding coefficient model. The effective sample size and genotype counts are used in the Bayesian model. Table 26 indicates that both estimators have similar bias and MSE.  Table 24. Comparison of Direct Sample Estimator and Parametric Bayesian Estimators (Single Common f Model)   f  Direct Sample Estimator Parametric Bayesian Estimator  Bias  MSE  CP AL Bias MSE CP AL  RRMSE ‐0.1  0.0222  0.001  88 0.0979 ‐0.0044 0.0005 77 0.0561  0.71 0  0.0228  0.001  90 0.1029 ‐0.0012 0.0005 85 0.0606  0.71 0.05  0.0244  0.0012  86 0.1041 0.0011 0.0005 82 0.062  0.65 0.1  0.0203  0.001  93 0.1061 ‐0.0015 0.0006 81 0.0628  0.77 0.2  0.0202  0.0009  90 0.1059 0.0000 0.0005 87 0.0635  0.75 *MSE=mean square error, CP=coverage probability, AL=average length, RRMSE=relative root mean square error (vs. Direct Sample Estimator) Table 25. Comparison of Direct Sample Estimator and Parametric Bayesian Estimators (Cluster Specific f Model)   f  Direct Sample Estimator Parametric Bayesian Estimator  Bias  MSE  CP AL Bias MSE CP AL  RRMSE ‐0.1  ‐0.0056  0.0006  95 0.0942 ‐0.0033 0.0005 74 0.0566  0.91 0  ‐0.002  0.0005  97 0.1028 ‐0.0003 0.0005 83 0.0606  1.0 0.05  0.0000  0.0005  96 0.1042 0.0015 0.0005 81 0.0618  1.0 0.1  ‐0.0019  0.0006  95 0.1063 ‐0.0008 0.0006 80 0.0628  1.0 0.2  ‐0.0004  0.0005  98 0.1066 0.0000 0.0005 87 0.0635  1.0 *MSE=mean square error, CP=coverage probability, AL=average length, RRMSE=relative root mean square error (vs. Direct Sample Estimator) Table 26. Comparison of Design-Based Estimator and Parametric Bayesian Estimator (Cluster Specific f Model)   f  Design‐Based Estimator Parametric Bayesian Estimator  Bias  MSE Bias MSE  ‐0.1  0.0002  0.0003 0.0005 0.0003  0  ‐0.001  0.0004 ‐0.0008 0.0004  0.05  ‐0.0007  0.0004 ‐0.0005 0.0004  0.1  ‐0.0009  0.0004 ‐0.0008 0.0004  0.2  ‐0.0009  0.0004 ‐0.0008 0.0004  100 Chapter 6: Application to Health and Retirement Study To illustrate our proposed Bayesian methods to estimate the inbreeding coefficient, we analyzed the HRS genetic data that consist of SNPs from two widely studied candidate genes. HRS employs a multi-stage area probability sampling design. The details of HRS study design can be found in Chapter 1. The main HRS survey data is public and can be download once registered at its website https://ssl.isr.umich.edu/hrs/start.php. To obtain the access to HRS restricted genetic data, we first applied to dbGaP (https://dbgap.ncbi.nlm.nih.gov/aa/wga.cgi?page=login) genetic data repository. Once access to dbGaP was granted, we further applied to HRS restrict data access (http://hrsonline.isr.umich.edu/index.php?p=xxgen0) directly for the access to HRS-dbGaP cross- reference file as well as a well-selected subset of candidate genes and SNPs. The cross-reference file provides the ID crosswalk (HHID and PN) between the HRS genetic data files and the HRS public data files. After all datasets were downloaded from HRS data repository, we merged all data files by those unique identifiers. A total of 12,507 subjects were included in our merged data file. Among them, 6658 subjected were genotyped in 2006 and the remaining 5849 subjects were genotyped in 2008. From released 2006 genetic data, sample weights are provided for the subjects whose biomarkers were selected for genotyping. Respondents with at least one valid biomarker result were assigned biomarker weights. The initial weights were calculated by dividing the HRS 2006 sample weights by the predicted probabilities of responses to each component. These interim weights were further trimmed at the 5th and 95th percentiles and were finally post-stratified back to the entire 2006 HRS sample by age, sex, and race/ethnicity. A similar process was conducted for the 2008 released genetic data. However, for this chapter we only focus on the 2006 genetic data for our application. For the genetic data download, the value provided for each SNP is actually the dosage (number of coded alleles) for a person. Since the genotypes are further imputed using the external 1000 Genomes data (1000 Genomes, 2015), the dosage ranges from 0 to 2 and may not be exactly the integer values for each person. The data file also includes an INFO metric, which is the observed statistical information associated with the allele frequency estimate. It is considered as a measure 101 of SNP imputation quality (1000 Genomes Project Reference Panel, 2012). INFO score greater than 0.8 is considered as a conservative filter, while greater than 0.3 is a liberal one. As recommended in the Quality Control document, we used the following algorithm to define final genotype: genotype=0 if dosage value ≤0.5; genotype=1 if 0.51.5. As documented in the Quality Control Report for Genotypic Data (2012), the inbreeding coefficient is given by 1 - (number of observed heterozygotes) / (number of expected heterozygotes), same as we discussed in Chapter 1 and Chapter 2 under the simple random sampling. The report shows an approximately symmetric distribution of the inbreeding coefficient estimates for all autosomal SNPs, with mean of 0.0023 for European and mean of 0.0019 for African (Figure 11). The results do not suggest an excess of positive inbreeding coefficient, or, equivalently, an excess of homozygotes. As a benchmark for the comparison, we computed the estimate of the inbreeding coefficient same as the one described in the Quality Report. Moreover, we computed the survey design-based estimator, parametric Bayesian estimator, and Bayesian pseudo-empirical likelihood estimator as discussed in Chapter 3. Among many genes recorded in the HRS genetic data file, we selected two genes to demonstrate our application. First, we selected SNPs on APOC1 gene that encodes Apolipoprotein C1, a component of lipoproteins. This protein is responsible for the activation of esterified lecithin cholesterol with an important role in the exchange of esterified cholesterol between lipoproteins and in removal of cholesterol from tissues (Tata, et al., 1985). We also studied SNPs on BDNF gene that encodes brain-derived neurotrophic factor (BDNF), a protein mainly expressed in the central nervous system. Neurotrophic factors promote the survival of neurons by preventing associated signals that initiate programmed cell death. Study suggests that this gene is associated with depression (Binder & Scharfman, 2004; Zuccato, 2009). We did not apply the recommended composite quality filter documented in the QC report. Such filter is typically used as initial screen tool for the downstream gene-disease association studies to exclude those potential questionable SNPs. The QC document reports that the percentage of all SNPs with minor allele frequency (MAF) less than 1% is 24.5% for autosomes and 17.2% for the X chromosome. To reduce computation time, we only studied a subset of the total reported SNPs 102 for both genes. First, we selected SNPs with MAF greater than 10%. For APOC1, there are a total of 105 SNPs in the HRS data repository. After we applied the MAF filter, 27 SNPs remained in the final analysis. For BDNF, there are a total of 357 SNPs. Because of the limitation of R memory size for our computer, we only selected first two hundred SNPs in the file. After the application of the MAF filter, there are 49 SNPs included in the comparison. In order to show results when MAF is just over the limit for SNP, our second study selected 38 SNPs from BDNF gene with MAF restricted from 1% to 5%. Figure 11. Distribution of Estimated Inbreeding Coefficients for all Autosomal SNPs (“Figure 26”  in Quality Control Report for Genotypic Data, 2012)  103 Table 27, Table 28 and Figure 12 Panel A & B list four estimates of inbreeding coefficient for the selected SNPs with MAF>10%: naïve estimator for a simple random sampling, design-based survey sample estimator, parametric Bayesian estimator under a multinomial likelihood, and Bayesian pseudo-empirical likelihood estimator. When the inbreeding coefficient is small, ignoring the sampling design is more likely to generate a statistical significant estimator under a nominal 95% confidence level, e.g., the 95% confidence interval does not cover the zero. This observation is consistent with the QC report, which recommends to use a stricter filter threshold of p=0.0001 to exclude SNPs from further downstream analysis. Furthermore, for autosomal SNPs, the QQ plot of all p-values for the testing of HWE suggests deviation of observed from expected p-values is between 0.001 and 0.01. Such filter results in some potential good SNPs being screened out from further downstream analysis. Our analysis suggests that most of these estimates with small values are not considered statistically significant at 0.05 level any more when taking the sampling design into account. For these SNPs with all four estimates are consistently significant, we also notice that the size of the inbreeding coefficient is generally smaller when the sampling design feature is considered in analysis. Furthermore, the Bayesian methods generally have smaller credible intervals and, therefore, a more precise estimation of the inbreeding coefficient. The Bayesian pseudo-empirical likelihood estimator is the most conservative estimator among all four estimators we considered. Table 29 and Figure 12 Panel C lists same four estimates for the selected SNPs with MAF greater than 1% but less than 5%. For all inbreeding coefficient estimate greater than 0.1, all four methods detect such large deviation from HWE. For smaller coefficient less than 0.1, Bayesian pseudo- empirical likelihood estimate suggests that all those are not worth mention at the 0.05 confidence level, while the results are sporadic for all other three methods. 104 Table 27. Estimated Inbreeding Coefficients for Selected SNPs from APOC1 Gene (MAF>10%). SNP  Unweighted  Estimator  Design‐Based Estimator  Parametric Bayesian  Estimator  Bayesian Pseudo‐ Empirical Likelihood  Estimator  kgp658335  0.0136 (‐0.0107, 0.0378)  ‐0.0004 (‐0.0473, 0.0477)  ‐0.0004 (‐0.0238, 0.0234)  ‐0.0004 (‐0.0258, 0.0262)  kgp7807118  0.0074 (‐0.017, 0.0319)  ‐0.0065 (‐0.0576, 0.0446)  ‐0.0067 (‐0.0296, 0.0175)  ‐0.0067 (‐0.0437, 0.0293)  kgp8411531  0.0068 (‐0.018, 0.0315)  ‐0.024 (‐0.0627, 0.0146)  ‐0.0227 (‐0.044, ‐0.004)  ‐0.0245 (‐0.074, 0.024)  rs10414043  0.0103 (‐0.0145, 0.0352)  0.0156 (‐0.0276, 0.0587)  0.017 (‐0.0081, 0.0435)  0.0152 (‐0.0308, 0.0602)  rs111789331  0.0142 (‐0.0109, 0.0393)  0.0107 (‐0.0343, 0.0556)  0.0118 (‐0.0124, 0.037)  0.0103 (‐0.0361, 0.0559)  rs12721046  0.0168 (‐0.0085, 0.042)  0.0132 (‐0.0288, 0.0551)  0.014 (‐0.0099, 0.042)  0.0128 (‐0.033, 0.058)  rs12721051  0.0125 (‐0.0123, 0.0373)  0.0056 (‐0.0386, 0.0497)  0.0066 (‐0.0189, 0.0326)  0.0053 (‐0.0356, 0.0454)  rs12721056  0.006 (‐0.0183, 0.0302)  ‐0.005 (‐0.0505, 0.0405)  ‐0.004 (‐0.0284, 0.0236)  ‐0.005 (‐0.0315, 0.0215)  rs157594  ‐0.0102 (‐0.0344, 0.014)  ‐0.0262 (‐0.0759, 0.0235)  ‐0.0259 (‐0.05, ‐0.0015)  ‐0.0262 (‐0.05, ‐0.002)  rs157595  0.0404 (0.0161, 0.0647)  0.018 (‐0.0294, 0.0655)  0.0178 (‐0.0067, 0.0425)  0.018 (‐0.0075, 0.0435)  rs3826688  0.041 (0.0166, 0.0653)  0.0223 (‐0.0282, 0.0729)  0.0217 (‐8e‐04, 0.0449)  0.0223 (‐0.0034, 0.0486)  rs390082  0.0298 (0.0039, 0.0557)  ‐0.0081 (‐0.0495, 0.0332)  ‐0.0078 (‐0.0296, 0.0146)  ‐0.0085 (‐0.0552, 0.0368)  rs3925681  0.0126 (‐0.0116, 0.0368)  0.0219 (‐0.0153, 0.0592)  0.0222 (‐0.0015, 0.0467)  0.0219 (‐0.0034, 0.0466)  rs438811  0.0415 (0.0167, 0.0663)  0.0158 (‐0.028, 0.0595)  0.0159 (‐0.0079, 0.0399)  0.0156 (‐0.0152, 0.0468)  rs439401  0.0363 (0.0119, 0.0607)  0.0191 (‐0.031, 0.0693)  0.019 (‐0.0053, 0.0442)  0.0191 (‐0.0072, 0.0458)  rs4420638  0.0177 (‐0.0071, 0.0426)  0.0125 (‐0.0283, 0.0533)  0.0138 (‐0.0111, 0.0378)  0.0123 (‐0.0272, 0.0508)  rs445925  0.0273 (0.0015, 0.053)  ‐0.011 (‐0.0543, 0.0323)  ‐0.011 (‐0.0341, 0.0138)  ‐0.0114 (‐0.0584, 0.0346)  rs483082  0.0299 (0.0053, 0.0546)  0.0078 (‐0.0365, 0.0522)  0.0083 (‐0.0168, 0.0332)  0.0077 (‐0.0239, 0.0391)  rs484195  0.0412 (0.0169, 0.0656)  0.0229 (‐0.0231, 0.0689)  0.0229 (‐8e‐04, 0.0489)  0.0229 (‐0.0028, 0.0482)  rs5117  0.0134 (‐0.0111, 0.0379)  ‐0.0112 (‐0.0613, 0.0388)  ‐0.0102 (‐0.0338, 0.0154)  ‐0.0114 (‐0.0446, 0.0224)  rs56131196  0.0179 (‐0.007, 0.0428)  0.0126 (‐0.0282, 0.0534)  0.0132 (‐0.011, 0.0386)  0.0123 (‐0.0272, 0.0508)  rs584007  0.0332 (0.0089, 0.0576)  0.0158 (‐0.0343, 0.0659)  0.0153 (‐0.0091, 0.0391)  0.0158 (‐0.0107, 0.0423)  rs59325138  0.0114 (‐0.0129, 0.0356)  ‐3e‐04 (‐0.0447, 0.044)  ‐1e‐04 (‐0.025, 0.0229)  ‐4e‐04 (‐0.0261, 0.0249)  rs7256200  0.0109 (‐0.014, 0.0357)  0.016 (‐0.0272, 0.0593)  0.0171 (‐0.0078, 0.0435)  0.0156 (‐0.0306, 0.0614)  rs73052335  0.023 (‐0.003, 0.049)  0.0152 (‐0.0261, 0.0565)  0.0161 (‐0.009, 0.0426)  0.0147 (‐0.0347, 0.0633)  rs78959900  ‐0.001 (‐0.0252, 0.0232)  ‐0.0079 (‐0.0544, 0.0387)  ‐0.0073 (‐0.0318, 0.0174)  ‐0.0079 (‐0.0347, 0.0183)  rs814573*  0.0363 (0.011, 0.0615)  0.0347 (‐0.0093, 0.0786)  0.0351 (0.0116, 0.0593)  0.0345 (‐0.0031, 0.0709)  * The INFO score is 0.786. 105 Table 28. Estimated Inbreeding Coefficients for Selected SNPs from BDNF Gene (MAF>10%). SNP  Unweighted Estimator  Design‐Based Estimator  Parametric Bayesian  Estimator  Bayesian Pseudo‐ Empirical Likelihood  Estimator  kgp10709149  0.0199 (‐0.0057, 0.0455)  ‐0.0027 (‐0.0495, 0.0442)  ‐0.0012 (‐0.0238, 0.0229)  ‐0.0031 (‐0.0527, 0.0453)  kgp2818969  0.0049 (‐0.0195, 0.0292)  0.0045 (‐0.0401, 0.0492)  0.0048 (‐0.0189, 0.0316)  0.0043 (‐0.0317, 0.0403)  kgp3196024  0.0049 (‐0.0195, 0.0292)  0.0045 (‐0.0401, 0.0492)  0.0048 (‐0.0189, 0.0316)  0.0043 (‐0.0317, 0.0403)  rs10767658  0.0069 (‐0.0174, 0.0312)  0.0081 (‐0.0344, 0.0506)  0.0088 (‐0.0178, 0.034)  0.008 (‐0.0205, 0.0365)  rs10767659  0.1108 (0.0866, 0.1351)  0.0805 (0.0375, 0.1234)  0.0803 (0.0567, 0.1035)  0.0804 (0.0551, 0.1061)  rs10767662  0.0214 (‐0.0028, 0.0456)  0.017 (‐0.0275, 0.0615)  0.0176 (‐0.0061, 0.0414)  0.017 (‐0.0075, 0.0415)  rs10835210  0.0578 (0.0335, 0.0821)  0.0452 (0.0079, 0.0825)  0.0457 (0.0212, 0.0694)  0.0452 (0.0205, 0.0705)  rs10835211  0.0221 (‐0.0026, 0.0468)  0.013 (‐0.0214, 0.0475)  0.0134 (‐0.0098, 0.0379)  0.0129 (‐0.0199, 0.0451)  rs10835213*  ‐0.0381 (‐0.0619,‐0.0144)  ‐0.0318 (‐0.0717, 0.0081)  ‐0.0319 (‐0.0546, ‐0.0082)  ‐0.0319 (‐0.0616, ‐0.0026)  rs11030101  0.0561 (0.0318, 0.0803)  0.0424 (6e‐04, 0.0842)  0.0426 (0.0185, 0.0668)  0.0424 (0.0182, 0.0672)  rs11030102  0.0261 (0.0014, 0.0509)  0.0168 (‐0.0198, 0.0533)  0.017 (‐0.0066, 0.0418)  0.0167 (‐0.0162, 0.0488)  rs11030104  0.0084 (‐0.016, 0.0329)  0.0109 (‐0.0323, 0.0541)  0.0112 (‐0.0121, 0.0358)  0.0107 (‐0.0255, 0.0465)  rs11030107  0.0263 (0.0016, 0.0511)  0.017 (‐0.0196, 0.0535)  0.0174 (‐0.0065, 0.0427)  0.0168 (‐0.0161, 0.0489)  rs11030108  0.0109 (‐0.0134, 0.0352)  0.0082 (‐0.0359, 0.0523)  0.008 (‐0.0163, 0.0344)  0.0081 (‐0.021, 0.037)  rs11030112  0.0137 (‐0.0107, 0.038)  0.0091 (‐0.0342, 0.0524)  0.0092 (‐0.0146, 0.0342)  0.0091 (‐0.0199, 0.0381)  rs114907865  0.1072 (0.083, 0.1315)  0.0811 (0.0349, 0.1273)  0.0809 (0.0579, 0.1045)  0.081 (0.0559, 0.1069)  rs11500197  0.0233 (‐0.0014, 0.048)  0.0139 (‐0.0216, 0.0494)  0.0138 (‐0.0107, 0.0392)  0.0138 (‐0.0183, 0.0457)  rs117567901  0.0052 (‐0.0192, 0.0295)  0.0035 (‐0.0364, 0.0433)  0.0035 (‐0.0211, 0.0286)  0.0033 (‐0.0322, 0.0388)  rs12575096  0.0126 (‐0.012, 0.0372)  0.0136 (‐0.0302, 0.0574)  0.0147 (‐0.0106, 0.0384)  0.0134 (‐0.0233, 0.0497)  rs12790234  0.0052 (‐0.0192, 0.0295)  0.0035 (‐0.0364, 0.0433)  0.0035 (‐0.0211, 0.0286)  0.0033 (‐0.0322, 0.0388)  rs138385919  0.0263 (0.0016, 0.0511)  0.017 (‐0.0196, 0.0535)  0.0174 (‐0.0065, 0.0427)  0.0168 (‐0.0161, 0.0489)  rs1401635  0.0106 (‐0.0137, 0.0349)  0.0066 (‐0.0381, 0.0513)  0.0071 (‐0.0174, 0.0333)  0.0065 (‐0.0218, 0.0352)  rs143956188  0.0052 (‐0.0192, 0.0295)  0.0035 (‐0.0364, 0.0433)  0.0035 (‐0.0211, 0.0286)  0.0033 (‐0.0322, 0.0388)  rs1519480  0.1086 (0.0843, 0.1329)  0.079 (0.0374, 0.1206)  0.0789 (0.0549, 0.1018)  0.0789 (0.0531, 0.1051)  rs16917237  0.0106 (‐0.0139, 0.0351)  0.0133 (‐0.0307, 0.0573)  0.0134 (‐0.0108, 0.0375)  0.0131 (‐0.0228, 0.0492)  rs1808124  0.0333 (0.0089, 0.0577)  0.0283 (‐0.0085, 0.0651)  0.0285 (0.0031, 0.0548)  0.0283 (0.001, 0.055)  rs189740576  ‐0.0033 (‐0.0273, 0.0206)  ‐0.0212 (‐0.0642, 0.0218)  ‐0.0194 (‐0.0403, 0.0043)  ‐0.0217 (‐0.0722, 0.0288)  rs1949513  0.0652 (0.0409, 0.0896)  0.0512 (0.0125, 0.09)  0.0515 (0.0274, 0.0754)  0.0512 (0.0257, 0.0767)  rs2049045  0.0084 (‐0.0161, 0.033)  0.0119 (‐0.0334, 0.0573)  0.012 (‐0.0134, 0.0372)  0.0117 (‐0.0283, 0.0507)  rs34379767  0.0052 (‐0.0192, 0.0295)  0.0035 (‐0.0364, 0.0433)  0.0035 (‐0.0211, 0.0286)  0.0033 (‐0.0322, 0.0388)  rs35038967  0.0124 (‐0.0122, 0.037)  0.0133 (‐0.0304, 0.057)  0.0141 (‐0.0107, 0.0399)  0.0131 (‐0.0235, 0.0485)  rs4378341  0.0248 (6e‐04, 0.049)  0.0208 (‐0.0242, 0.0657)  0.021 (‐0.0031, 0.0452)  0.0208 (‐0.0036, 0.0454)  rs4385847  0.0244 (2e‐04, 0.0486)  0.0207 (‐0.0238, 0.0653)  0.0207 (‐0.0035, 0.045)  0.0207 (‐0.0037, 0.0453)  rs4517468  0.1007 (0.0764, 0.1249)  0.0689 (0.0255, 0.1122)  0.0688 (0.0462, 0.0933)  0.0688 (0.0433, 0.0943)  rs4542361  0.0251 (9e‐04, 0.0493)  0.022 (‐0.0229, 0.0668)  0.0218 (‐0.003, 0.0462)  0.022 (‐0.0024, 0.0466)  rs4633417  0.0207 (‐0.0035, 0.045)  0.0148 (‐0.0321, 0.0617)  0.0144 (‐0.0088, 0.0382)  0.0148 (‐0.01, 0.039)  rs4923463  0.0045 (‐0.0198, 0.0289)  0.0032 (‐0.0411, 0.0475)  0.0042 (‐0.0208, 0.0297)  0.003 (‐0.0336, 0.0394)  rs4923464  0.0091 (‐0.0154, 0.0336)  0.0109 (‐0.0325, 0.0543)  0.0115 (‐0.0134, 0.0363)  0.0107 (‐0.0252, 0.0468)  rs4923466  0.0052 (‐0.0192, 0.0295)  0.0035 (‐0.0364, 0.0433)  0.0035 (‐0.0211, 0.0286)  0.0033 (‐0.0322, 0.0388)  rs61888762  0.0126 (‐0.0118, 0.0369)  0.0087 (‐0.0356, 0.0531)  0.0089 (‐0.0144, 0.033)  0.0087 (‐0.0204, 0.0376)  rs6265  0.0124 (‐0.0123, 0.037)  0.012 (‐0.0345, 0.0585)  0.0122 (‐0.011, 0.0365)  0.0118 (‐0.0264, 0.0496)  rs6484320  0.0037 (‐0.0206, 0.028)  0.0062 (‐0.0327, 0.0452)  0.0067 (‐0.0184, 0.032)  0.006 (‐0.0292, 0.0408)  rs6484321  0.0052 (‐0.0191, 0.0296)  0.0065 (‐0.0333, 0.0464)  0.007 (‐0.0171, 0.0329)  0.0064 (‐0.0287, 0.0413)  rs7103411  0.0037 (‐0.0206, 0.028)  0.0061 (‐0.0327, 0.0449)  0.0064 (‐0.017, 0.0308)  0.0059 (‐0.0293, 0.0407)  rs7103873  0.024 (‐2e‐04, 0.0482)  0.0208 (‐0.0226, 0.0642)  0.0205 (‐0.0035, 0.045)  0.0208 (‐0.0039, 0.0451)  rs7104207  0.0237 (‐5e‐04, 0.048)  0.0207 (‐0.0227, 0.0641)  0.0208 (‐0.0038, 0.0452)  0.0207 (‐0.004, 0.045)  rs7124442  0.0347 (0.0103, 0.0591)  0.032 (‐0.0062, 0.0702)  0.0319 (0.0069, 0.0572)  0.032 (0.0042, 0.0592)  rs7482752  0.0251 (9e‐04, 0.0493)  0.0199 (‐0.0253, 0.0651)  0.0209 (‐0.0031, 0.0458)  0.0199 (‐0.0042, 0.0438)  rs7926362  0.0037 (‐0.0206, 0.028)  0.0063 (‐0.0327, 0.0453)  0.0073 (‐0.0165, 0.0318)  0.0061 (‐0.0292, 0.0408)  * The INFO score is 0.726. 106 Table 29. Estimated Inbreeding Coefficients for Selected SNPs from BDNF Gene (<1%MAF<5%). SNP  Unweighted Estimator  Design‐Based Estimator  Parametric Bayesian  Estimator  Bayesian Pseudo‐ Empirical Likelihood  Estimator  kgp12568216  0.0081 (‐0.0192, 0.0355)  ‐0.0063 (‐0.0362, 0.0236)  6e‐04 (‐0.0186, 0.0292)  0.0214 (‐0.0252, 0.0998)  kgp12594720  0.038 (‐0.0063, 0.0822)  0.0649 (‐0.0329, 0.1628)  0.0728 (0.0267, 0.1297)  0.0844 (‐0.0049, 0.2111)  kgp12631228  0.0998 (0.0534, 0.1462)  0.095 (0.0209, 0.1691)  0.0986 (0.0555, 0.1501)  0.0946 (‐0.0019, 0.1951)  kgp12991896  0.0475 (0.004, 0.091)  0.0434 (‐0.0232, 0.1099)  0.0487 (0.0108, 0.0996)  0.0638 (‐0.011, 0.173)  kgp1796534  ‐0.002 (‐0.0244, 0.0205)  ‐0.0043 (‐0.03, 0.0213)  0.0084 (‐0.0112, 0.045)  0.0468 (‐0.0107, 0.1493)  kgp370455  0.0092 (‐0.0218, 0.0403)  ‐0.0045 (‐0.0295, 0.0205)  0.0078 (‐0.0113, 0.0457)  0.0472 (‐0.0105, 0.1505)  kgp782640  0.0485 (0.0118, 0.0853)  0.0638 (‐0.0108, 0.1384)  0.0667 (0.0294, 0.1087)  0.0639 (‐0.0171, 0.1489)  rs112989286  0.0325 (‐0.0109, 0.0758)  0.0545 (‐0.0434, 0.1523)  0.0697 (0.0246, 0.1356)  0.0822 (‐0.0045, 0.2125)  rs114813081  0.0541 (0.0028, 0.1055)  0.0761 (‐0.0481, 0.2002)  0.0807 (0.0306, 0.1475)  0.0965 (‐0.0027, 0.2353)  rs116590817  0.0581 (0.0112, 0.105)  0.0462 (‐0.0219, 0.1142)  0.05 (0.0142, 0.1012)  0.0664 (‐0.0104, 0.1776)  rs117314519  9e‐04 (‐0.0237, 0.0255)  ‐0.0113 (‐0.0363, 0.0136)  ‐0.0065 (‐0.0225, 0.0194)  0.0191 (‐0.0253, 0.0957)  rs117392835*  1e‐04 (‐0.0242, 0.0245)  0.0239 (‐0.0573, 0.1051)  0.0332 (0.0014, 0.0779)  0.0486 (‐0.0148, 0.1472)  rs11819808  0.2048 (0.1521, 0.2574)  0.1902 (0.1013, 0.2792)  0.1944 (0.1441, 0.2466)  0.187 (0.0819, 0.2859)  rs11826087  0.0733 (0.0193, 0.1274)  0.0789 (‐0.0356, 0.1933)  0.0868 (0.039, 0.1509)  0.0953 (‐0.0029, 0.2291)  rs12284158  0.2036 (0.1511, 0.2561)  0.1885 (0.0996, 0.2775)  0.1899 (0.1428, 0.242)  0.1854 (0.0807, 0.2837)  rs12284724  0.2036 (0.1511, 0.2561)  0.1885 (0.0996, 0.2775)  0.1899 (0.1428, 0.242)  0.1854 (0.0807, 0.2837)  rs140500062  0.0585 (0.0114, 0.1056)  0.0465 (‐0.0217, 0.1148)  0.0519 (0.0159, 0.0984)  0.0668 (‐0.0104, 0.1786)  rs140893479*  ‐0.0106 (‐0.0124, ‐0.0089)  ‐0.011 (‐0.0141, ‐0.0078)  0.0029 (‐0.0111, 0.0403)  0.0524 (‐0.008, 0.163)  rs142615576  ‐0.0015 (‐0.0254, 0.0224)  0.0081 (‐0.0372, 0.0534)  0.0119 (‐0.0112, 0.0396)  0.0132 (‐0.0433, 0.0837)  rs145216856  0.0585 (0.0114, 0.1056)  0.0465 (‐0.0217, 0.1148)  0.0519 (0.0159, 0.0984)  0.0668 (‐0.0104, 0.1786)  rs145596917  0.0195 (‐0.0139, 0.0529)  ‐0.0026 (‐0.0283, 0.0231)  0.0012 (‐0.0165, 0.035)  0.0319 (‐0.0186, 0.1184)  rs146296986  0.1825 (0.1258, 0.2392)  0.1886 (0.0711, 0.3062)  0.1919 (0.1373, 0.2541)  0.1855 (0.0808, 0.2838)  rs183649896*  ‐0.0117 (‐0.0136, ‐0.0099)  ‐0.0111 (‐0.0144, ‐0.0077)  0.0017 (‐0.0111, 0.037)  0.0521 (‐0.0081, 0.1619)  rs190640500*  ‐0.0121 (‐0.014, ‐0.0102)  ‐0.0106 (‐0.0132, ‐0.0079)  0.0026 (‐0.0104, 0.0375)  0.0539 (‐0.0076, 0.1664)  rs34043390*  ‐0.0079 (‐0.0259, 0.01)  ‐0.0165 (‐0.02, ‐0.013)  ‐0.0076 (‐0.017, 0.0165)  0.0359 (‐0.0135, 0.1275)  rs4528281  0.0321 (‐0.011, 0.0753)  0.0538 (‐0.0432, 0.1508)  0.0686 (0.0231, 0.128)  0.0816 (‐0.0046, 0.2114)  rs55699391  0.1181 (0.0702, 0.1659)  0.1359 (0.0538, 0.2181)  0.1384 (0.0923, 0.1885)  0.1333 (0.0314, 0.2304)  rs56274324  0.1072 (0.0602, 0.1542)  0.1086 (0.0225, 0.1947)  0.112 (0.0691, 0.1637)  0.107 (0.0064, 0.2064)  rs57083135  0.0574 (0.0108, 0.1039)  0.0456 (‐0.0228, 0.114)  0.0501 (0.0129, 0.1016)  0.0659 (‐0.0105, 0.1765)  rs58597651  0.0574 (0.0108, 0.1039)  0.0456 (‐0.0228, 0.114)  0.0501 (0.0129, 0.1016)  0.0659 (‐0.0105, 0.1765)  rs7124665  0.0966 (0.0401, 0.1532)  0.051 (‐0.0121, 0.114)  0.0563 (0.0176, 0.1043)  0.0797 (‐0.0057, 0.2083)  rs72878164  0.1181 (0.0702, 0.1659)  0.1359 (0.0538, 0.2181)  0.1384 (0.0923, 0.1885)  0.1333 (0.0314, 0.2304)  rs72878175  0.0574 (0.0108, 0.1039)  0.0456 (‐0.0228, 0.114)  0.0501 (0.0129, 0.1016)  0.0659 (‐0.0105, 0.1765)  rs72878179  0.0574 (0.0108, 0.1039)  0.0456 (‐0.0228, 0.114)  0.0501 (0.0129, 0.1016)  0.0659 (‐0.0105, 0.1765)  rs72878181  0.0574 (0.0108, 0.1039)  0.044 (‐0.0221, 0.1101)  0.0494 (0.0133, 0.0966)  0.0644 (‐0.0109, 0.1741)  rs72878196  0.0958 (0.0504, 0.1411)  0.0908 (0.0186, 0.163)  0.0944 (0.0546, 0.14)  0.0906 (‐0.0047, 0.1893)  rs78531552  ‐0.0052 (‐0.0263, 0.0158)  ‐0.0151 (‐0.0327, 0.0025)  ‐0.0063 (‐0.0195, 0.0201)  0.028 (‐0.019, 0.112)  rs79292978  0.0178 (‐0.0177, 0.0534)  0.0055 (‐0.0337, 0.0448)  0.0158 (‐0.008, 0.0576)  0.0483 (‐0.0117, 0.1513)  * The INFO scores are less than 0.8. 107 A B C Figure  12.  Comparison  of  Inbreeding  Coefficient  Estimators  from  HRS  Genetic  Data.  BPELE=Bayesian  Pseudo‐Empirical  Likelihood  Estimator,  DBE=Design‐Based  Estimator,  PBE=Parametric  Bayesian Estimator, SRS=Simple Random Sampling Estimator. Panel A, B and C are for APOC1 SNPs with  allele  frequency  >10%,  BDNF  SNPs  with  allele  frequency  >10%,  and  BDNF  SNPs  with  allele  frequency  between 1% and 5%, respectively. Solid triangles represent estimators with 95% CI do not cover zero (a  significant deviation from HWE at 5% level).     108 Chapter 7: Summary and Future Research In this dissertation, both parametric Bayesian method with multinomial likelihood and non- parametric Bayesian method with empirical likelihood are proposed for genetic data collected from a complex survey sampling design. The proposed methods are evaluated through simulations under various sampling designs, including stratified simple random sampling, proportional to size sampling and clustered sampling. Using real genetic data from HRS study, we have demonstrated the utility of the proposed Bayesian methods for SNPs screening in GWAS based on national level complex survey. Design-based estimator is popular with survey practitioners since it automatically takes the survey design into account and provides a reliable estimate for large samples, which is often the case for large scale national survey. For non-standard statistics such as the inbreeding coefficient from genetic data, the standard error of the computed statistic is often complicated. However, it can be estimated by the Taylor linearization or widely used resampling techniques. For estimation of the inbreeding coefficient induced from general biological inbreeding model, design-based estimator may be outside of the parameter space limit. The use of such estimator needs be carefully reviewed when the minor allele frequency (MAF) is small, especially if the estimated inbreeding coefficient is also small. Bayesian method provides an alternative approach for the estimation of the inbreeding coefficient. This dissertation fills the gap in the Bayesian application to GWAS where subjects may have different sample selection probabilities. Our proposed hierarchical Bayesian multinomial model coincides with the biological inbreeding model for the genotype distribution. Unlike the design- based estimator, our parametric Bayesian estimator is naturally restricted to the parameter space of the inbreeding coefficient induced by the general biological inbreeding model. Without assuming a multinomial likelihood for the Bayesian model, our proposed non-parametric Bayesian estimator extended the widely used method proposed by Owen, Sitter, Rao & Wu to the nominal genetic data. Although the computation to maximize the empirical likelihood under the constraint of estimating equation is non-trivial, with a carefully selected algorithm and 109 approximate grid search method, it is reasonably efficient to produce a non-parametric Bayesian estimator. Our simulation studies suggests that the design-based estimator and two Bayesian estimators have similar performance when the allele frequency is not small or the sample size is large enough. However, when allele frequency is small, the design-based method often produces estimator outside of the parameter space, especially when the sample size is also small. Such limitation can be easily addressed through Bayesian prior distribution for the parameter. Furthermore, when the data is very much skewed, the Bayesian method tends to produce a more reliable estimator based on its data augmentation nature. Our simulation studies also remind us that the computation of empirical likelihood on the boundary of the parameter space is not adequately stable. Therefore, a special attention needs be paid when the observed inbreeding coefficient is closer to the lower bound of the parameter space induced by the inbreeding model. On the other hand, Sforza & Bodmer (1971) and HRS study (Weir D. R., 2012) reported that inbreeding coefficient is not likely close to the lower boundary of parameter space for most human SNPs. Furthermore, practitioners are more concerned with a positive inbreeding coefficient for more heterozygotes which may indicate a deviation from HWE due to population structure. Such population structure, often exists in national level complex survey, needs be carefully taken into account when estimating the inbreeding coefficient in GWAS. In terms of the choice of estimator to be used in practice, our recommendation should be based on the purpose of the inbreeding coefficient estimation. If the main purpose is to screen out the SNPs for genotyping error, it is reasonable to use the naïve estimator as stated in the QC report with a more conservative confidence level such as 0.0001. On the other hand, if the main purpose is to have a more robust estimator for the inbreeding coefficient, we recommend to incorporate the sampling design. Our demonstration using the HRS genetic data indicates that estimator ignoring the sampling design often overestimates the inbreeding coefficient, therefore, more likely to deviate from HWE. When minor allele frequency is small, we suggested to consider the Bayesian estimators to ensure it is within the parameter space. As an initial assessment of Bayesian method in the estimation of inbreeding coefficient for genetic data from complex survey, we limit our discussion on the single locus SNPs. Perhaps the next 110 important step is to extend the method to include multiple loci SNPs. These loci share the same population history and therefore the degree of subpopulation heterogeneity across loci is expected to be similar (Lockwood, Roeder, & Devlin, 2001). Using an appropriate Bayesian model that borrows strength across loci, more accurate estimates of subpopulation allele frequency and the inbreeding coefficient can be obtained. Introducing empirical likelihood in the Bayesian method may appear to be not practically useful in our preliminary study. However, without any model assumption, it is easier to include auxiliary variables into analysis. Although the focus of this dissertation is to estimate inbreeding coefficient, our immediate future research is gene-disease association study under complex survey setting. The robustness of model based Bayesian estimator is greatly dependent on its model specification. Serious misspecification of the model for gene-disease association will lead to misleading and often bad inferences. Bayesian method with the empirical likelihood does not require a parametric model. It is relatively straightforward to include candidate variables possibly related to the disease and borrow information from other sources such as Census data. For example, for most genetic inherited disease, the onset of disease is often related to aging. While it is possible to parametrically model the gene-disease association with the consideration of age and other environmental factors, the exact functional form may not be clear. On the other hand, finite population measure for age is easily accessible from Census. Using such auxiliary variable as another constraint to maximize the empirical likelihood, we can compare the genetic traits between subjects with the disease (case) and those without (control). 111 Appendices: R Programs #-------------------------- Newton method to derive lamda from equation sum(ds*(u-mu)/(1+lamda*(u-mu)))=0 # adapted from Rao & Wu Lag1<-function(u,ds,mu){ L<--1/max(u-mu) R<--1/min(u-mu) dif<-1 tol<-1e-8 if(L>=0 | R<=0){ dif=0 M=0 } while(dif>tol){ M<-(L+R)/2 glam<-sum((ds*(u-mu))/(1+M*(u-mu))) if(glam>0) L<-M if(glam<0) R<-M dif<-abs(glam) } return(M) } #--------function to calculate Bayesian posterior mean for inbreeding coefficient f and #confidence interval. # adapted from Rao & Wu #ds- normalized weight, ys- data #phat- profiled allele frequency, use design-based estimator Bay2=function(ds,ys,phat,to=0.00001,inc=0.001){ pmin=min(phat,1-phat) R= -pmin/(1-pmin) mu=sum(ds*ys) lam=Lag1(ys,ds,mu) elm=exp(-nss*sum(ds*log(1+lam*(ys-mu)))) el=NULL pf=NULL while(R<1-inc){ R=R+inc mu=2*(1-R)*phat*(1-phat) lam=Lag1(ys,ds,mu) elr=exp(-nss*sum(ds*log(1+lam*(ys-mu)))) if (elr/elm > to){ el=c(el,elr) pf=c(pf,R)} } A=sum(el)*inc el=el/A ng=length(el) a=0 k=0 alpha=0.05 while(a<=alpha/2){ k=k+1 112 a=sum(el[(ng-k):ng])*inc } Rhat=pf[ng-k] a=0 k=1 while(a<=alpha/2){ k=k+1 a=sum(el[1:k])*inc } Lhat=pf[k] bhat=sum(el*pf*inc) #plot(pf,el,type="l",ylab="Posteria Empirical Likelihood",xlab="f") return(cbind(bhat=bhat,bl=Lhat,br=Rhat,len=Rhat-Lhat)) } #--------------------------------------------Jackknife variance estimator for f-------------------------------------------------- jnse=function(df){ ns=dim(df)[1] je=rep(0,ns) for (i in 1:ns){ repdsgn = as.svrepdesign(dstrat, type="JKn", fay.rho=0.5) repwt=repdsgn$repweights[,i] repdf=data.frame(geno=df$geno,df$n1,wt=repwt$weights) jdstrat=svydesign(ids=~1,strata=NULL, weights=~wt,data=repdf) jp=svymean(~geno, jdstrat)[1] jn=svymean(~n1, jdstrat)[1] je[i]=1-jn/(2*(1-jp/2)*(jp/2)) } return(jnse=sqrt((ns-1)*sum((je-mean(je))^2)/ns)) } #---------------------------------R program to compare the MLE with MELE------------------------------------------ library(emplik) totn=50 #sample size p=0.1 f=0.05 p.alle=c(1-p,p) nsim=250 set.seed=1234 k.alle=length(p.alle) pij=c( (1-f)*p.alle[1]^2+f*p.alle[1], 2*(1-f)*p.alle[1]*p.alle[2], (1-f)*p.alle[2]^2+f*p.alle[2] ) pij.cumsum=cumsum(pij) fin=matrix(rep(0,nsim*8),nrow=nsim,ncol=8) myfun=function(y){ y1=y y2=y*(2-y) return(cbind(y1,y2)) } for (k in 1：nsim){ tmp=runif(totn) 113 geno=sapply(1:(totn), function(k) min(which(pij.cumsum>=tmp[k]))) geno=geno - 1 print(c(k,table(geno))) if (length(table(geno))==3) fin[k,11:13]=table(geno) if (length(table(geno))==2) fin[k,11:13]=c(table(geno),0) p.bar=sum(geno)/(2*totn) #sample allele freq f.bar=1-(sum(geno*(2-geno))/totn)/(2*p.bar*(1-p.bar)) f.sd=sqrt((1-f.bar)*(1-2*p.bar*(1-p.bar)*(1-f.bar) - (2*p.bar-1)^2*(1-f.bar)^2)/(2*p.bar*(1-p.bar)*totn)) fin[k,5]=f.bar fin[k,6]=f.sd nss=totn fin[k,1:4]=Bay2(ds=1/totn,ys=geno*(2-geno),phat=p.bar,to=0.00001,inc=0.001) x=geno*(2-geno) d=rep(1,totn) np=100 nf=100 mu1=mean(geno) mu2=mean(x) sp=sf=smu=logL=rep(0,(np*nf)) for (i in 1:np){ for (j in 1:nf){ sp[(i-1)*nf+j]=i/(2*np) fmin=-sp[(i-1)*nf+1]/(1-sp[(i-1)*nf+1]) sf[(i-1)*nf+j]=fmin+(j-1)*(1-fmin)/(nf-1) smu[(i-1)*nf+j]=2*(1-sf[(i-1)*nf+j])* sp[(i-1)*nf+j]*(1- sp[(i-1)*nf+j]) # temp=el.cen.EM(x, d, fun=function(t){t}, mu=smu[(i-1)*np+j]) # logL[(i-1)*np+j]=temp$loglik mu1=2*sp[(i-1)*nf+j] mu2=smu[(i-1)*nf+j] temp=el.cen.EM2(geno, d, fun=myfun, mu=c(mu1,mu2)) logL[(i-1)*nf+j]=temp$loglik } print(i) } lh=data.frame(sp=sp,sf=sf,smu=smu,logL=logL) lmax=max(lh$logL) res=subset(lh,logL==lmax) fin[k,7]=res$sf fin[k,8]=res$sp } x=round(c(MLE_Bias=mean(fin[,5]-f,na.rm=TRUE), MLE_MSE=(mean(fin[,5])-f)^2+var(fin[,5]), MLE_Cover=100*sum(fin[,5]-1.96*fin[,6]<=f & f<=fin[,5]+1.96*fin[,6],na.rm=TRUE)/nsim, MLE_Len=mean(2*1.96*fin[,6],na.rm=TRUE), MLE_Out=100*(sum(fin[,5]>1,na.rm=TRUE)+sum(fin[,5] < -min(p,1-p)/(1-min(p,1- p)),na.rm=TRUE))/nsim, EL_Bias=mean(fin[,7]-f,na.rm=TRUE), EL_MSE=(mean(fin[,7],na.rm=TRUE)-f)^2+var(fin[,7],na.rm=TRUE), EL_Out=100*(sum(fin[,7]>1,na.rm=TRUE)+sum(fin[,7] < -min(p,1-p)/(1-min(p,1- p)),na.rm=TRUE))/nsim), 3) 114 #---------------------Code to analyze the HRS genetic data for APOC gene---------------------------------------- options(stringsAsFactors=FALSE) source("C:/Research/R/allfunction.R") library(survey) library(arm) library(rms) library (BRugs) library(coda) library(dplyr) #######Attach main survey, crosslink and genetic data####### attach("R/Longevity.RData") attach("R/crosslink.RData") attach("R/track.RData") survey=merge(htrack,crosslink,by=c("HHID","PN")) survey=subset(survey,GENETICS06==1 | GENETICS08==1， select=c(STRATUM,SECU,GENETICS06,GENETICS08,KBIOWGTR,LBIOWGTR, HHID,PN,LOCAL_ID)) survey$LBIOWGTRn=as.numeric(as.character(survey$LBIOWGTR)) survey$KBIOWGTRn=as.numeric(as.character(survey$KBIOWGTR)) survey$LOCAL_ID=as.numeric(as.character(survey$LOCAL_ID)) ######################genetic data######################## varlist=names(df.APOC1.SNP) nvar=length(varlist) eval(parse(text=paste("df.APOC1.SNP$",varlist[2]))) #######assign SNP to 0/1/2 according QC report guidance####### df.APOC1.SNP=as.data.frame(cbind(df.APOC1.SNP[,1],sapply(df.APOC1.SNP[,-1], function(x) ifelse(x<=0.5,0,ifelse(x>0.5 & x<=1.5,1,2))))) sapply(df.APOC1.SNP[,-1], function(x) table(x)) tt=stack(df.APOC1.SNP,select=-V1) #######preselect SNP based on MAF####### sumct=data.frame(unclass(table(tt$ind,tt$values))) lsnp=sumct %>% mutate(pX0=100*X0/(X0+X1+X2),pX1=100*X1/(X0+X1+X2),pX2=100*X2/(X0+X1+X2), maf=100*(X1+2*X2)/(2*(X0+X1+X2)), snpname=row.names(sumct)) %>% filter(maf>10 & maf<90) lsnp lnsnp=lsnp$snpname df.APOC1.SNP$LOCAL_ID=df.APOC1.SNP$V1 #######combine survey and genetic data####### dim(survey) dim(df.APOC1.SNP) fdf=merge(survey,df.APOC1.SNP[,c("LOCAL_ID",lnsnp)],by="LOCAL_ID") fdf06=subset(fdf,GENETICS06==1 & is.na(KBIOWGTRn)==FALSE) summary(fdf06$KBIOWGTRn) ############Parametric Bayesian Model####### 115 model1=function() { q[1]<-(1-f)*p*p + f*p q[2]<-2*(1-f)*p*(1-p) q[3]<-(1-f)*(1-p)*(1-p) + f*(1-p) y[1:3] ~ dmulti(q[],n) #prior p ~ dunif(0, 1) f <- w*(1 - f.min) + f.min f.min <- max(-p/(1-p), -(1-p)/p) w ~ dunif(0, 1) n<-sum(y[]) } writeModel(model1, "model1.txt") nss=dim(fdf06)[1] myAna=function(var,data=fdf06){ geno=eval(substitute(var),data, parent.frame()) n1=geno*(2-geno) wt=fdf06$KBIOWGTRn stype=fdf06$STRATUM psu=fdf06$SECU gtype=paste("N",geno,sep="") df=data.frame(geno,psu,n1,wt,stype,gtype) dstrat=svydesign(ids=~psu,strata=~stype, weights=~wt, data=df,nest=TRUE) p.svy=svymean(~geno, dstrat) n1.svy=svymean(~n1, dstrat) f_design=1-n1.svy/(2*(1-p.svy/2)*(p.svy/2)) ns=length(table(df$stype)) lst=unique(df$stype) je=rep(0,ns*2) for (i in 1:ns){ for (j in 1:2){ #sdf=subset(df,stype!=lst[i] & psu!=j) sdf=df sdf$wt[sdf$stype==lst[i] & sdf$psu==j]=0 sdf$wt[sdf$stype==lst[i] & sdf$psu!=j]=sdf$wt[sdf$stype==lst[i] & sdf$psu!=j]*2 jdstrat=svydesign(ids=~psu,strata=~stype, weights=~wt,data=sdf,nest=TRUE) jp=svymean(~geno, jdstrat)[1] jn=svymean(~n1, jdstrat)[1] je[(i-1)*2+j]=1-jn/(2*(1-jp/2)*(jp/2)) } } jnse=sqrt((2*ns-1)*sum((je-mean(je))^2)/(2*ns)) nef=round(length(geno)/(svyvar(~geno, dstrat)/(sd(geno))^2),0) nys=data.frame(svytotal(~gtype,dstrat)) ny=c(ifelse(is.na(nys["gtypeN0","total"]),0,nys["gtypeN0","total"]), ifelse(is.na(nys["gtypeN1","total"]),0,nys["gtypeN1","total"]), ifelse(is.na(nys["gtypeN2","total"]),0,nys["gtypeN2","total"])) ny=round(nef*ny/(sum(ny)),0) 116 bdata=list(y=ny[3:1]) inits1 = list(p=0.5,w=0.5) pbl=BRugsFit(data = bdata, inits = inits1, para = c("f"), modelFile = "model1.txt", numChains = 1,nBurnin = 1000, nIter = 10000, nThin = 10)$Stats #srs p_srs=sum(geno)/(2*nss) #sample allele freq f_srs=1-mean(n1)/(2*p_srs*(1-p_srs)) f_sd=sqrt((1-f_srs)*(1-2*p_srs*(1-p_srs)*(1-f_srs) - (2*p_srs-1)^2*(1-f_srs)^2)/(2*p_srs*(1-p_srs)*nss)) return(cbind(sn0=ny[3],sn1=ny[2],sn2=ny[1],f_srs=f_srs,srsl=f_srs-1.96*f_sd,srsr=f_srs+1.96*f_sd, f_design=f_design,dl=f_design-1.96*jnse, dr=f_design+1.96*jnse, f_eb=Bay2(ds=wt/sum(wt),ys=geno*(2-geno),phat=1-p.svy/2,to=0.00001,inc=0.001), f_pbl=pbl$mean,pbll=pbl$val2.5pc,pblr=pbl$val97.5pc)) } nv=length(lnsnp) suma=matrix(rep(0,16*nv),ncol=16,nrow=nv) for (i in 1:nv){ suma[i,]=myAna(eval(parse(text=paste(lnsnp[i]))),fdf06) } fsum=as.data.frame(suma) names(fsum)=c("sn0","sn1","sn2","f_srs","srsl","srsr","f_design","dl","dr","bhat", "bl","br","len","f_pbl","pbll","pblr") dfsum=data.frame(snp=lnsnp, srs=paste(round(fsum$f_srs,4), " (",round(fsum$srsl,4), ", ", round(fsum$srsr,4), ")",sep=""), des=paste(round(fsum$f_design,4), " (",round(fsum$dl,4), ", ", round(fsum$dr,4), ")",sep=""), bel=paste(round(fsum$bhat,4), " (",round(fsum$bl,4), ", ", round(fsum$br,4), ")",sep=""), bpb=paste(round(fsum$f_pbl,4), " (",round(fsum$pbll,4), ", ", round(fsum$pblr,4), ")",sep="")) write.csv(dfsum,file="APOC1.csv") #######Plot####### library(ggplot2) srs=fsum[,c("f_srs","srsl", "srsr")] names(srs)=c("p","l","h") srs$seq=1:dim(srs)[1] srs <- srs[order(srs$p),] srs$gp=rep("SRS",dim(srs)[1]) des=fsum[,c("f_design","dl", "dr")] names(des)=c("p","l","h") des$seq=1:dim(des)[1] des <- des[order(des$p),] des$gp=rep("DBE",dim(des)[1]) pbe=fsum[,c("bhat","bl","br")] names(pbe)=c("p","l","h") pbe$seq=1:dim(pbe)[1] pbe <- pbe[order(pbe$p),] pbe$gp=rep("PBE",dim(pbe)[1]) pbl=fsum[,c("f_pbl","pbll","pblr")] names(pbl)=c("p","l","h") pbl$seq=1:dim(pbl)[1] 117 pbl <- pbl[order(pbl$p),] pbl$gp=rep("BPELE",dim(pbl)[1]) df=rbind(srs,des,pbe,pbl) df_s=subset(df,df$l>0 | df$h<0) ggplot(df, aes(x=seq, y=p, group=gp,linetype=gp))+ geom_line()+ scale_linetype_manual(values = c("solid","dotted","longdash","dotdash"))+ geom_hline(aes(yintercept=0),linetype=2) + geom_point(data=df_s,aes(x=seq,y=p),shape=17,size=3)+ xlab("SNPs Number")+ylab("Estimate of Inbreeding Coefficient") 118 Bibliography 1000 Genomes. 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