University of Iowa Iowa Research Online Theses and Dissertations Summer 2011 Penalized methods in genome-wide association studies Jin Liu University of Iowa Copyright 2011 Jin Liu This dissertation is available at Iowa Research Online: http://ir.uiowa.edu/etd/1242 Recommended Citation Liu, Jin. "Penalized methods in genome-wide association studies." PhD (Doctor of Philosophy) thesis, University of Iowa, 2011. http://ir.uiowa.edu/etd/1242. Follow this and additional works at: http://ir.uiowa.edu/etd Part of the Statistics and Probability Commons
PENALIZED METHODS IN GENOME-WIDE ASSOCIATION STUDIES by Jin Liu An Abstract Of a thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Statistics in the Graduate College of The University of Iowa July 2011 Thesis Supervisors: Professor Jian Huang Associate Professor Kai Wang
1 ABSTRACT Penalized regression methods are becoming increasingly popular in genomewide association studies (GWAS) for identifying genetic markers associated with disease. However, standard penalized methods such as the LASSO do not take into account the possible linkage disequilibrium between adjacent markers. We propose a novel penalized approach for GWAS using a dense set of single nucleotide polymorphisms (SNPs). The proposed method uses the minimax concave penalty (MCP) for marker selection and incorporates linkage disequilibrium (LD) information by penalizing the difference of the genetic effects at adjacent SNPs with high correlation. A coordinate descent algorithm is derived to implement the proposed method. This algorithm is efficient and stable in dealing with a large number of SNPs. A multi-split method is used to calculate the p-values of the selected SNPs for assessing their significance. We refer to the proposed penalty function as the smoothed minimax concave penalty (SMCP) and the proposed approach as the SMCP method. Performance of the proposed SMCP method and its comparison with a LASSO approach are evaluated through simulation studies, which demonstrate that the proposed method is more accurate in selecting associated SNPs. Its applicability to real data is illustrated using data from a GWAS on rheumatoid arthritis. Based on the idea of SMCP, we propose a new penalized method for group variable selection in GWAS with respect to the correlation between adjacent groups. The proposed method uses the group LASSO for encouraging group sparsity and a
2 quadratic difference for adjacent group smoothing. We call it smoothed group LASSO, or SGL for short. Canonical correlations between two adjacent groups of SNPS are used as the weights in the quadratic difference penalty. Principal components are used to reduced dimensionality locally within groups. We derive a group coordinate descent algorithm for computing the solution path of the SGL. Simulation studies are used to evaluate the finite sample performance of the SGL and group LASSO. We also demonstrate its applicability on rheumatoid arthritis data. Abstract Approved: Thesis Supervisor Title and Department Date Thesis Supervisor Title and Department Date
PENALIZED METHODS IN GENOME-WIDE ASSOCIATION STUDIES by Jin Liu A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Statistics in the Graduate College of The University of Iowa July 2011 Thesis Supervisors: Professor Jian Huang Associate Professor Kai Wang
Copyright by JIN LIU 2011 All Rights Reserved
Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Jin Liu has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Statistics at the July 2011 graduation. Thesis Committee: Jian Huang, Thesis Supervisor Kai Wang, Thesis Supervisor Kung-Sik Chan Aixin Tan Dale Zimmermann
ACKNOWLEDGEMENTS I am greatly grateful to all those people for their advice, help and support. It is a really good opportunity for me to learn many important things with their exemplification both in academia and in every day life. Without them, this work would be impossible. I would like to gratefully acknowledge the invaluable supervision of Dr. Jian Huang during this work. His supervision is like the torchlight in the darkness which gives me courage and confidence to walk through. He also exemplifies himself by his attitude toward research. His insights and patience are vital to inspire my interest in research and form my future career path. I am also deeply indebted to my co-advisor, Dr. Kai Wang for his patience in correcting my manuscripts and adding comments on them. His knowledge in genetics plays a vital role for this work. I would like to express my warm and sincere thanks to all committee members, Dr. Kung-Sik Chan, Dr. Aixin Tan, and Dr. Dale Zimmermann. I would also like to thank my mom and dad, for their great support in my life. ii
ABSTRACT Penalized regression methods are becoming increasingly popular in genomewide association studies (GWAS) for identifying genetic markers associated with disease. However, standard penalized methods such as the LASSO do not take into account the possible linkage disequilibrium between adjacent markers. We propose a novel penalized approach for GWAS using a dense set of single nucleotide polymorphisms (SNPs). The proposed method uses the minimax concave penalty (MCP) for marker selection and incorporates linkage disequilibrium (LD) information by penalizing the difference of the genetic effects at adjacent SNPs with high correlation. A coordinate descent algorithm is derived to implement the proposed method. This algorithm is efficient and stable in dealing with a large number of SNPs. A multi-split method is used to calculate the p-values of the selected SNPs for assessing their significance. We refer to the proposed penalty function as the smoothed minimax concave penalty (SMCP) and the proposed approach as the SMCP method. Performance of the proposed SMCP method and its comparison with a LASSO approach are evaluated through simulation studies, which demonstrate that the proposed method is more accurate in selecting associated SNPs. Its applicability to real data is illustrated using data from a GWAS on rheumatoid arthritis. Based on the idea of SMCP, we propose a new penalized method for group variable selection in GWAS with respect to the correlation between adjacent groups. The proposed method uses the group LASSO for encouraging group sparsity and a iii
quadratic difference for adjacent group smoothing. We call it smoothed group LASSO, or SGL for short. Canonical correlations between two adjacent groups of SNPS are used as the weights in the quadratic difference penalty. Principal components are used to reduced dimensionality locally within groups. We derive a group coordinate descent algorithm for computing the solution path of the SGL. Simulation studies are used to evaluate the finite sample performance of the SGL and group LASSO. We also demonstrate its applicability on rheumatoid arthritis data. iv
TABLE OF CONTENTS LIST OF TABLES................................. vii LIST OF FIGURES................................ viii CHAPTER 1 INTRODUCTION............................. 1 1.1 Literature Review.......................... 2 1.2 Proposed Penalized Methods.................... 8 1.3 Overview of this thesis........................ 11 2 SMOOTHED MINIMAX CONCAVE PENALIZATION........ 13 2.1 The SMCP method......................... 13 2.1.1 Comparison with fused LASSO............... 16 2.2 Genome-wide screening incorporating LD............. 17 2.2.1 Computation......................... 19 2.3 Joint SMCP Model in GWAS.................... 23 2.3.1 Joint Loss Function..................... 23 2.3.2 Coordinate Descent Algorithm............... 24 2.4 Methods............................... 28 2.4.1 Tuning parameter selection................. 28 2.4.2 P -values for the selected SNPs............... 29 2.5 Simulation and Empirical Studies................. 30 2.5.1 Simulation Studies and Results............... 30 2.5.2 Application to GAW 16 RA Data............. 33 2.5.3 Application to GAW 17 Data................ 38 3 SMOOTHED GROUP LASSO...................... 49 3.1 The SGL method.......................... 49 3.2 Genome-wide group screening.................... 51 3.2.1 A GCD Algorithm in Marginal Model........... 53 3.2.2 Properties of the Second Penalty.............. 55 3.3 Joint SGL Model in GWAS..................... 56 3.4 Methods............................... 59 3.4.1 Principal Components.................... 60 3.4.2 Canonical Correlation.................... 61 3.4.3 Selection of the Tuning Parameters............. 61 v
3.4.4 P -values for the Selected SNPs............... 63 3.5 Simulation and Empirical Studies................. 64 3.5.1 Simulation Studies and Results............... 64 3.5.2 Application to GAW 16 RA Data............. 67 4 REGULARIZED LOGISTIC REGRESSION.............. 72 4.1 Unpenalized Logistic Regression.................. 73 4.2 Logistic Regression Model with SMCP............... 74 4.2.1 Marginalized Logistic Regression with SMCP....... 74 4.2.2 Joint Logistic Regression with SMCP........... 77 4.3 MM algorithm in Logistic Regression................ 78 4.4 Logistic Regression with SGL penalty............... 79 4.5 Simulation and Empirical Studies................. 80 4.5.1 Simulation Studies and Results............... 80 4.5.2 Rheumatoid Arthritis Data Example............ 81 5 DISCUSSION AND FUTURE WORK.................. 87 REFERENCES.................................. 92 APPENDIX..................................... 96 vi
LIST OF TABLES 2.1 Simulation results for marginalized linear model with SMCP....... 32 2.2 Simulation results for joint linear model with SMCP............ 32 2.3 List of 50 SNPs selected by the marginalized SMCP method and marginalized LASSO method for a simulated data set................ 34 2.4 SNPs found significant other than chromosome 6............. 37 2.5 SNPs selected by the SMCP and LASSO for trait Q1 in replicate 1... 40 2.6 Mean and standard error of true positives and false positives for selected SNPs over 200 replicates for trait Q1.................... 40 3.1 True positive, false discovery rate (FDR) and false negative rate (FNR) for simulated data from example 1..................... 66 3.2 True positive, false discovery rate (FDR) and false negative rate (FNR) for simulated data from example 2..................... 67 3.3 True positive, false discovery rate (FDR) and false negative rate (FNR) for simulated data from example 3 under joint model........... 68 3.4 True positive, false discovery rate (FDR) and false negative rate (FNR) for simulated data from example 3 under marginal model......... 68 3.5 Multi-split p-values for a simulated data in example 3........... 69 4.1 True positive, false discovery rate (FDR) and false negative rate (FNR) for simulated data under marginalized logistic regression model with SMCP 82 4.2 True positive, false discovery rate (FDR) and false negative rate (FNR) for simulated data under joint logistic regression model with SMCP... 82 4.3 True positive(number of groups), false discovery rate (FDR) and false negative rate (FNR) for simulated data under logistic regression model with SGL................................... 83 vii
LIST OF FIGURES 1.1 The l 1 penalty for the LASSO along with the MCP and SCAD, λ=1, γ=3 6 2.1 Correlation plots in Chromosome 6 from Genetic Analysis Workshop 16 Rheumatoid Arthritis dataset........................ 15 2.2 Graph of Function R under different a j,b j and c j............. 41 2.3 Plot of log 10 (p-value) for SNPs on chromosome 6 selected by (a) the SMCP method and (b) the LASSO method for the rheumatoid arthritis data. These p-values are generated using the multi-split method. The horizontal line corresponds to significance level 0.05............. 42 2.4 Genome-wide plot of β estimates for the SMCP and the MCP methods on binary trait................................ 43 2.5 Genome-wide plot of β estimates for the LASSO and regular single-snp linear regression on binary trait....................... 44 2.6 Genome-wide plot of β estimates for the SMCP and MCP methods on quantitative trait............................... 45 2.7 Genome-wide plot of β estimates for the LASSO and regular single-snp linear regression on quantitative trait.................... 46 2.8 Correlation plots along the genome from Genetic Analysis Workshop 17 simulated dataset............................... 47 2.9 Genome-wide plot of β estimates for regular single-snp linear regression. 48 2.10 Boxplots for comparison of residuals on genotypes of selected SNPs. Y axis is for residuals and x axis for genotypes................ 48 3.1 Solution path for a simulated data for (a) group LASSO, and SGL with (b) η = 0.5, (c) η = 0.2 and (d) η = 0.1. Black lines are paths of non-zero groups and grey lines are paths of irrelevant groups............ 57 3.2 Genome-wide plot of β for SGL and group LASSO and β for simple linear regression................................ 71 viii
4.1 β estimates on chromosome 6 for marginalized logistic regression with the SMCP and the LASSO, and the regular single-snp linear regression. 84 4.2 β estimates on chromosome 6 for joint logistic regression with the SMCP and the LASSO................................ 85 4.3 β estimates on chromosome 6 for logistic regression with the SGL and the group LASSO............................... 86 ix
1 CHAPTER 1 INTRODUCTION Variable selection is an important topic in statistics. In practice, a large number of predictors are measured to catch all the possible factors related to the outcome. Usually, only a small fraction of predictors are important. As model gets more complicated, it will decrease bias but increase variance. Hence one needs to balance between bias and variance. Traditionally, researchers use forward, backward and subset methods to do model selection. The computation for these methods grows exponentially as the number of predictors increases. In genome-wide association studies (GWAS), there are usually hundreds of thousands of single nucleotide polymorphisms (SNPs) for only hundreds of subjects. The high-dimensionality of SNPs data makes these traditional methods computational infeasible. Penalized methods are effective tools for variable selection which has been shown to be applicable both theoretically and empirically for dealing with high-dimensional data. This chapter first introduces the background of GWAS and the development of penalized methods. Several widely used penalized methods are briefly presented. Then the proposed penalized methods are briefly introduced. More specific details will be presented in the relevant chapters.
2 1.1 Literature Review A genome-wide association study (GWAS) involves rapidly scanning genetic markers across genome of individuals of a particular species to find genetic variations associated with a particular disease. GWAS are particularly useful to find genetic variations that contribute to common or complex diseases, such as rheumatoid arthritis, glaucoma, diabetes and so on. Currently, GWAS rely on the single-marker analysis testing individual associations between each SNP and the trait either binary or quantitative since individual identification is simple and readily applicable (Cho et al., 2010). While single-marker analysis can be used when a single genetic variation is responsible for a particular trait, it may not be appropriate for investigating a complex polygenic trait due to following reasons. Due to the large number of genetic markers, certain multiple testing is required to reduce the increased probability of false positive findings. Second, single-marker analysis does not take into account the correlation structure linkage disequilibrium (LD) in the SNPs data. Therefore, it is important to develop a method capable of considering all genetic factors. In another perspective, this kind of problem can also be viewed as variable selection in high-dimensional data. Penalized methods seem to be suitable to this purpose. Model selection is a classical topic in statistics. It is well studied for the traditional models. Much work has been done in this field. Penalized regression methods arise in the last decade. Those approaches include LASSO (Tibshirani, 1996), bridge (Frank and Friedman, 1993), SCAD (Fan and Li, 2001), nonnegative garotte (Breiman, 1995), the elastic net (Zou and Hastie, 2005), adaptive LASSO
3 (Zou, 2006), fused LASSO (Tibshirani et al., 2005), and minimax concave penalty (MCP) (Zhang, 2010). Much work has been done to understand the properties of these penalties under the situation of p < n and p n for various models. The LASSO (least absolute shrinkage and selection operator) is a penalized method imposing an l 1 -penalty on the regression coefficients (Tibshirani, 1996). It was shown in Tibshirani (1996) that the LASSO can shrink regression coefficients to 0. Knight and Fu (2000) studied the asymptotical properties for LASSO-type estimators and showed that the estimates have positive probability of being 0. Therefore, it can be used for estimation and automatic variable selection in regression analysis. Consider a linear regression model with p variables: y i = β 0 + x iβ + ɛ i (1.1) where y i is the ith response variable, ɛ i is the ith error term, x i is a p 1 covariate vector, and β is the regression coefficient vector. Without loss of generality, I can assume that the features have been standardized such that n i=1 x ij = 0 and 1 n n i=1 x2 ij = 1. Then the objective function Q for a generalized loss function L n (β) with LASSO penalty can be written as: Q(β) = 1 n L n(β) + λ j β j where λ 0 is the penalty parameter. The LASSO provides a computationally feasible way for variable selection in high-dimensional settings (Tibshirani, 1996). Recently, this approach has been applied to GWAS for selecting associated SNPs (Wu et al., 2009). It has been shown
4 that the LASSO is selection consistency if the predictors meet the irrepresentable condition and this condition is almost necessary (Zhao and Yu, 2006). This condition is stringent and there is no known mechanism to verify it in GWAS. Zhang and Huang (2008) studied the sparsity and the bias of the LASSO in high-dimensional linear regression models. They showed that under reasonable conditions, the LASSO selects a model of the correct order of dimensionality. However, the LASSO tends to overselect unimportant variables. Therefore, direct application of the LASSO to GWAS tends to generate findings with high false positive rates. Another limitation of the LASSO is that, if there is a group of variables among which the pairwise correlations are high, then the LASSO tends to select only one variable from the group and does not care which one is selected (Zou and Hastie, 2005). As shown by Fan and Li (2001), a good penalty function should have three properties: unbiasedness, sparsity and continuity. The LASSO lacks the property in terms of unbiasedness. The LASSO estimates are always biased, since its penalization rate is constant no matter how large the parameters are. Hence, the LASSO can do variable selection but always produces biased estimates. Several methods attempting to improve the performance of the LASSO have been proposed. The adaptive LASSO (Zou, 2006) uses data-dependent weights for penalizing coefficients in the l 1 penalty. Zou (2006) chose the inverse of ordinary least-square estimates for the weights. It enjoys the oracle properties under some mild regularity conditions; in other words, it performs as well as if the underlying model was known beforehand. In the case that the number of predictors is much larger
5 than sample size, adaptive weights cannot be initiated easily. Elastic net method (Zou and Hastie, 2005) can effectively deal with certain correlation structures in the predictors by using a combination of ridge and LASSO penalties. Fan and Li (2001) introduced a smoothly clipped absolute deviation (SCAD) method. Zhang (2010) proposed a flexible minmax concave penalty (MCP) which attenuates the effect of shrinkage leading to bias. Both the SCAD and the MCP belong to the same family of quadratic spline penalties and both lead to oracle selection results (Zhang, 2010). The penalty is illustrated in Fig. 1.1. The MCP has a simpler form and requires weaker conditions for the oracle property. We refer to Zhang (2010) and Mazumder et al. (2009) for detailed discussion. For SNPs data, some of the correlations among them are very high due to LD. In the context of GWAS, none of the methods mentioned above consider the natural ordering along the genome with respect to their physical positions and possible linkage disequilibrium (LD) between adjacent SNPs. Thus this inadequacy prevents them from being applied in GWAS. Fused LASSO considers the ordering of coefficients but it is inappropriate, since the effect of association for a SNP (as measured by its regression coefficient) is only identifiable up to its absolute value a homozygous genotype can be equivalently coded as either 0 or 2 depending on the choice of the reference allele. A new penalized method incorporating linkage disequilibrium suitable to GWAS is briefly introduced in next section and more details can be found in Chapter 2. To accommodate the situation of existing grouping structures among variables,
6 (a) Penalty (b) Penalization Rate Figure 1.1: The l 1 penalty for the LASSO along with the MCP and SCAD, λ=1, γ=3 the group LASSO (Bakin, 1999) was introduced using an adequate extension of the LASSO penalty over the group norm. In group LASSO, an l 2 norm of the coefficients associated with a group of variables is used in the penalty function. Yuan and Lin (2006) considered the group selection methods by imposing the LASSO, the LARS and the nonnegative garrote and proposed algorithms to these types of group selection. Kim et al. (2006) considered group selection in generalized linear models. They proposed an efficient algorithm and a blockwise standardization method. Meier et al. (2008) extended the group LASSO to logistic regression models. They showed that the group LASSO for logistic regression is consistent under certain conditions and developed a block coordinate descent algorithm that can be used to high-dimensional data. Huang et al. (2009) proposed a group bridge method than can carry out feature selection at the group and within-group individual variable levels simultaneously. Wei and Huang (2008) studied the selection and estimation properties of the group
7 LASSO in high-dimensional settings when the number of groups exceeds the sample size. Huang et al. (2010) later extended group LASSO to group MCP (gmcp). They showed that gmcp enjoys an oracle property meaning that with high probability, gmcp estimator is equal to the oracle estimator assuming the true model was known. They proposed an efficient and stable group coordinate descent algorithm for computing the solution path of the gmcp. Another important aspect of the penalized methods is to solve the penalized objective function efficiently. Closed form solutions are not available in high dimensional setting for penalized methods. Therefore, optimization algorithms must be developed. Least angle regression (LARS) (Efron et al., 2004) is an algorithm that can be easily modified to produce the solutions for the LASSO. LARS produces a full piece-wise linear solution path, which is very useful in cross-validation. For the SCAD or the MCP, optimization is complicated. Fan and Li (2001) proposed a local quadratic approximation (LQA) to the penalty function. Then Newton-Raphson algorithm can be modified to optimize the penalized objective function. Zou and Li (2008) proposed a local linear approximation (LLA) algorithm to make a linear approximation to the penalty function. Then LARS algorithm can be implemented to compute the solution. The LLA algorithm is inefficient in that it cannot produce solution path at one time. Coordinate descent algorithm is found to be efficient in penalized methods. Much work has been done in this field (Breheny and Huang, 2011, Friedman et al., 2007, 2010, Mazumder et al., 2009, Wu and Lange, 2007).
8 1.2 Proposed Penalized Methods Before presenting the proposed penalized methods, the background of genetics related to our study is introduced first. DNA is a nucleic acid that contains instructions in the development and functioning of all known living organisms except for some viruses. To some extend, DNA is like blueprint, since it contains the instructions needed to construct other components of cells. The DNA segments carrying this genetic information are called gene. DNA consists of two chains of subunits twisted around one another to form a double strand helix. The subunits of each strand are nucleotides, each of which contains four chemical constituents called bases. These are adenine (A), thymine(t), guanine(g) and cytosine(c). Bases are all paired. Adenine pairs with thymine while guanine pairs with cytosine. Inside the nucleus of a living cell, genes are arranged in linear order along thread-like objects. These threads are called chromosomes. Normally, there are 23 pairs of chromosomes in human beings. In genetics, SNPs data is obtained from some chips testing DNAs on a dense set of predetermined genetic markers. In most cases, those genetic markers are singlenucleotide polymorphisms (SNP). SNPs are basic elements which DNAs are composed of. There are four bases in DNA, i.e. A,T,G and C. In most of time, chromosomes are paired. Hence, one specific marker can be homozygous and heterozygous. Assume that A and T are alleles for one locus. The genotypes for this locus can be AA,AT and TT. AA and TT are homozygous, but AT is heterozygous. Since researchers arbitrary choose A and T for the reference allele, homozygous AA and TT are not distinguishable. Usually, homozygous genotypes are coded as 0 or 2 depending on
9 the choice reference allele, while heterozygous genotypes are coded as 1. Most often, the alleles at two linked loci are associated through linkage disequilibrium (LD). LD can occur for a number of reasons, including genetic linkage, selection, the rate of recombination, the rate of mutation, genetic drift, non-random mating, and population structure. It is rare that disease markers are selected for assay. But the LD between disease markers and physically close SNPs makes the detection of association between SNP markers and phenotypic trait possible. LD is vulnerable to decay exponentially due to the recombinant. This is the reason why scientists can easily find newly developed mutation other than mutation occurred hundreds of generations ago. But if the physical loci of disease marker and SNP marker are close enough, it is safe to assume that recombination rate is zero, i.e. no recombinant between disease marker and SNP marker. Therefore, the LD between disease marker and SNP marker will remain almost the same as it was at the time the mutation occurred. SNPs are naturally ordered along the genome with respect to their physical positions. In the presence of linkage disequilibrium (LD), adjacent SNPs are expected to show similar strength of association. Making use of LD information from adjacent SNPs is highly desirable as it should help to better delineate association signals while reducing randomness seen in single SNP analysis. Fused LASSO (Tibshirani et al., 2005) is not appropriate for this purpose, since the effect of association for a SNP (as measured by its regression coefficient) is only identifiable up to its absolute value a homozygous genotype can be equivalently coded as either 0 or 2 depending on the
10 choice of the reference allele. Further discussion on this point is provided in Chapter 2. A new penalized regression method is proposed for identifying associated SNPs in genome-wide association studies. The proposed method uses a novel penalty, which I shall refer to as the smoothed minimax concave penalty, or SMCP, for sparsity and smoothness in absolute values. The SMCP is a combination of the MCP and a penalty consisting of the squared differences of the absolute effects of adjacent markers. The MCP promotes sparsity in the model and does automatic selection of associated SNPs. The penalty for squared differences of the absolute effects takes into account the natural ordering of SNPs and adaptively incorporates possible LD information between adjacent SNPs. It explicitly uses correlation between adjacent markers and penalizes the differences of the genetic effects at adjacent SNPs with high correlation. I derive a coordinate descent algorithm for implementing the SMCP. Furthermore, I use a resampling method for computing p-values of the selected SNPs in order to assess their significance. The proposed SMCP model is similar to the fused LASSO which uses l 1 norm over the difference of adjacent parameters. The SMCP model, however, is different from the fused LASSO in two aspects. First, it uses the MCP penalty in place of the LASSO penalty. It has been demonstrated that the MCP penalty is superior to the LASSO in terms of estimation bias and selection accuracy. Second, it imposes penalty on the difference in adjacent β s instead of the difference in adjacent βs. Therefore, the penalty would not be affected by which allele is used as the reference
11 to score genotypes. Although β would be of the opposite sign, the value β remains the same. In the same manner, the proposed SMCP model is extended to the model with grouping structure. SNPs are correlated through LD. Hence, there are existing grouping structure in the SNPs or sequencing data. I propose a new group selection approach using a combination of the group LASSO and the quadratic difference of the norm of adjacent groups. I call this approach the smoothed group LASSO, or simply SGL model. In the SGL model, I use canonical correlation to measure the correlation between two groups of variables. Principal components are used to reduce the dimensions within groups. It helps eliminate the presence of the highly correlated SNPs which results in small eigenvalues in Cholesky decomposition that makes the transformation unstable. 1.3 Overview of this thesis In Chapter 2, I introduce the proposed SMCP method and present genomewide screening incorporating the proposed SMCP method. I also describe a coordinate descent algorithm for estimating model parameters and discuss selection of the values of the tuning parameters and p-value calculation. In Chapter 3, I present the model framework and setting of the smoothed group LASSO. A group coordinate descent algorithm for fitting the proposed model is described. I also show the miscellaneous methods used in the model, e.g. canonical correlation, principal components, selection of the tuning parameters and evaluation
12 of the significance of selected groups. In Chapter 4, I implement the SMCP method in logistic regression model. Both the marginalized and the joint logistic regression models with SMCP are studied. The coordinate descent algorithms for them are scratched. Moreover, MM algorithm is used to convert penalized weighted least-squares into penalized least-squares. I also implement SGL in logistic regression model in this section. Within these chapters, the empirical properties of the methods are investigated under simulation studies as well as applied to real data sets from genetic analysis workshop (GAW) 16 rheumatoid arthritis (RA) data and GAW 17 simulated data.
13 CHAPTER 2 SMOOTHED MINIMAX CONCAVE PENALIZATION This chapter is organized as follows. Section 2.1 introduces the proposed SMCP method. Section 2.2 presents a genome-wide screening incorporating the proposed SMCP method. Section 2.2.1 describes a coordinate descent algorithm for estimating model parameters. Section 2.3 presents the joint linear model with SMCP and describes a coordinate descent algorithm for estimating parameters. Section 2.4 discusses selection of the values of the tuning parameters and p-value calculation. Section 2.5.1 evaluates the proposed method and a LASSO method using simulated data. Section 2.5.2 applies the proposed method to a GWAS on rheumatoid arthritis. 2.1 The SMCP method Let p be the number of SNPs included in the study, and let β j denote the effect of the jth SNP in a working model that describes the relationship between phenotype and markers. Here we assume that the SNPs are ordered according to their physical locations on the chromosomes. Adjacent SNPs in high LD are expected to have similar strength of association with the phenotype. To adaptively incorporate LD information, we propose the following penalty that encourages smoothness in β s at neighboring SNPs: λ 2 2 p 1 ζ j ( β j β j+1 ) 2, (2.1) j=1 where the weight ζ j is a measure of LD between SNP j and SNP (j+1). This penalty encourages β j and β j+1 to be similar to an extent inversely proportional to the
14 LD strength between the two corresponding SNPs. Adjacent SNPs in weak LD are allowed to have larger difference in their β s than if they are in stronger LD. The effect of this penalty is to encourage smoothness in β s for SNPs in strong LD. By using this penalty, we expect a better delineation of the association pattern in LD blocks that harbor disease variants while reducing randomness in β s in LD blocks that do not. We note that there is no monotone relationship between ζ and the physical distance between two SNPs. While it is possible to use other LD measures, we choose ζ j to be the absolute value of Pearson s correlation coefficient between the genotype scores of SNP j and SNP (j+1). The values of ζ j for rheumatoid arthritis data used by Genetic Analysis Workshop 16, the data set to be used in our simulation study and empirical study, are plotted for chromosome 6 (Fig. 2.1(a)). The proportion that ζ j > 0.5 over non-overlapping 100-SNP windows is also plotted (Fig. 2.1(b)). For the purpose of SNP selection, we use the MCP, which is defined as t ρ(t; λ 1, γ) = λ 1 (1 x/(γλ 1 )) + dx, 0 where λ 1 is a penalty parameter and γ is a regularization parameter that controls the concavity of ρ. Here x + is the nonnegative part of x, i.e., x + = x1 x 0. The MCP can be easily understood by considering its derivative, which is ρ(t; λ 1, γ) = λ 1 ( 1 t /(γλ1 ) ) + sgn(t), where sgn(t) = 1, 0, or 1 if t < 0, = 0, or > 0. As t increases from 0, the MCP begins by applying the same rate of penalization as the LASSO, but continuously relaxes that penalization until t > γλ 1, the condition under which the rate of penal-
15 (a) Absolute lag-one autocorrelation (b) Proportion of absolute lag-one autocorrelation coefficients > 0.5 for 100 SNPs per segment Figure 2.1: Correlation plots in Chromosome 6 from Genetic Analysis Workshop 16 Rheumatoid Arthritis dataset
16 ization drops to 0. It provides a continuum of penalties where the LASSO penalty corresponds to γ = and the hard-thresholding penalty corresponds to γ 1+. We note that other penalties, such as the LASSO penalty or the SCAD penalty, can also be used to replace the MCP. We choose MCP because it possesses all the basic desired properties of a penalty function and is computationally simple (Mazumder et al., 2009, Zhang, 2010). Given the parameter vector β = (β 1,..., β p ) and a loss function g(β) based on a working model for the relationship between the phenotype and markers, the SMCP in a working model can be expressed as minimizing the criterion L n (β) = g(β) + p j=1 ρ( β j ; λ 1, γ) + λ 2 2 p 1 ζ j ( β j β j+1 ) 2. (2.2) We minimize this objective function with respect to β, while using a bisection method to determine the regularization parameters (λ 1, λ 2 ). SNPs corresponding to ˆβ j 0 are selected as being potentially associated with disease. These selected SNPs will be subject to further analysis using a multi-split sampling method to determine their statistical significance, as described later. j=1 2.1.1 Comparison with fused LASSO Fused LASSO was designed to incorporate the feature that the order of the parameters is meaningful. Its objective function with loss function g(β) can be written as follows: p 1 p g(β) + λ 1 β j + λ 2 β j+1 β j j=1 j=1
17 As one can see, the biggest difference between SMCP and fused LASSO lies in the second penalty. SMCP uses a l 2 penalty on the absolute difference which makes soft smoothing. But fused LASSO uses l 1 for the smoothing penalty. Hence, it will put adjacent parameters to be exactly the same. It is like hard smoothing. Furthermore, SMCP consider the nature of GWAS. Not only the order of SNPs matters, but also the sign of the predictor effects. At one locus, the effect is positive while at the adjacent locus, the effect may be negative. The last point is that in GWAS, the number of SNPs can be hundreds of thousands. Therefore, we need a fast algorithm to fit the model. Despite that Friedman et al. (2007) proposed a generalized algorithm that yields a solution to fused LASSO, it is not as efficient as coordinate descent algorithm for SMCP which has explicit solution. 2.2 Genome-wide screening incorporating LD A basic method for GWAS is to conduct genome-wide screening of a large number of dense SNPs individually and look for those with significant association with phenotype. Although several important considerations, such as adjustment for multiple comparisons and possible population stratification, need to be taken into account in the analysis, the essence of the existing genome-wide screening approach is single-marker based analysis without considering the structure of SNP data. In particular, the possible LD between two adjacent SNPs are not incorporated in the analysis. Our proposed SMCP method can be used for screening a dense set of SNPs
18 incorporating LD information in a natural way. To be specific, here we consider the standard case-control design for identifying SNPs that potentially associated with disease. Let the phenotype be scored as 1 for cases and 1 for controls. Let n j be the number of subjects whose genotypes are non-missing at SNP j. The standardized phenotype of the ith subject with non-missing genotype at SNP j is denoted by y ij. The genotype at SNP j is scored as 0, 1, or 2 depending on the number of copies of a reference allele in a subject. Let x ij denote the standardized genotype score satisfying i x ij = 0 and n j i=1 x2 ij = n j. Consider the penalized criterion L n (β) = 1 2 p 1 n j=1 j n j (y ij x ij β j ) 2 + i=1 p j=1 ρ( β j ; λ 1, γ)+ λ 2 2 p 1 ζ j ( β j β j+1 ) 2. (2.3) j=1 Here the loss function is g(β) = 1 2 p 1 n j=1 j n j (y ij x ij β j ) 2. (2.4) i=1 We note that switching the reference allele used for scoring the genotypes changes the sign of β j but β j remains the same. It may be counter-intuitive to use a quadratic loss in (2.4) for case-control designs. However, we now show that this is appropriate. Regardless of how the phenotype is scored, the least squares regression slope of the phenotype over the genotype score at SNP j (i.e., a regular single SNP analysis) equals n j n j y ij x ij / x 2 ij = 2(ˆp 1j ˆp 2j )/φ j (1 φ j ), i=1 i=1 where φ j is the proportion of cases out of total subjects computed from the subjects with non-missing genotype and ˆp 1j and ˆp 2j are allele frequencies of the SNP j in cases
19 and controls, respectively. This shows that the β j in the squared loss function (2.4) can be interpreted as the effect size of SNP j. In the classification literature, quadratic loss has also been used for indicator response variables (Hastie et al., 2009). An alternative loss function for binary phenotype would be the sum of negative marginal log-likelihood based on a working logistic regression model. We have found that the selection results using this loss function are in general similar to those based on (2.4). In addition, the computational implementation of the coordinate descent algorithm described in the next subsection using the loss function (2.4) is much more stable and efficient and can easily handle tens of thousands of SNPs. 2.2.1 Computation In this section, we derive a coordinate descent algorithm for computing the solution to objective function (2.3). This algorithm was originally proposed for criterions with convex penalties such as LASSO (Friedman et al., 2010, Knight and Fu, 2000, Wu and Lange, 2007). It has been proposed to calculate nonconvex penalized regression estimates (Breheny and Huang, 2011, Mazumder et al., 2009). This algorithm optimizes a target function with respect to one parameter at a time, iteratively cycling through all parameters until convergence is reached. It is particularly suitable for problems such as the current one that have a simple closed form solution in a single dimension but lack one in higher dimensions. We wish to minimize the objective function L n (β) in (2.3) with respect to β j while keeping all other β k, k j, fixed at their current estimates. Thus only the
20 terms involving β j in L n matter. That is, this problem is equivalent to minimizing R(β j ) defined as R(β j ) = 1 2n j n j (y ij x ij β j ) 2 + ρ( β j ; λ 1, γ) i=1 + 1 2 λ 2[ζ j ( β j β j+1 ) 2 + ζ j 1 ( β j 1 β j ) 2 ] = C + a j β 2 j + b j β j + c j β j, j = 2,..., p 1, (2.5) where C is a term free of β j, β j+1 and β j 1 are current estimates of β j+1 and β j 1, respectively, and a j, b j, and c j are determined as follows: For β j < γλ 1, ( a j = 1 n j ) 1 x 2 ij + λ 2 (ζ j 1 + ζ j ) 1, 2 γ n j n j i=1 b j = 1 x ij y ij, n j i=1 and c j = λ 1 λ 2 ( β j+1 ζ j + β j 1 ζ j 1 ). (2.6) For β j γλ 1, ( a j = 1 1 2 n j n j ) x 2 ij + λ 2 (ζ j 1 + ζ j ), i=1 c j = λ 2 ( β j+1 ζ j + β j 1 ζ j 1 ), (2.7) while b j remains the same as in the previous situation. Note that function R(β j ) is defined for j 1, p. It can be defined for j = 1 by setting β j 1 = 0 and for j = p by setting β j+1 = 0 in the above two situations.
21 Minimizing R(β j ) with respect to β j is equivalent to minimizing a j β 2 j + b j β j + c j β j, or equivalently, ( a j β j + b ) 2 j + c j β j. (2.8) 2a j The first term is convex in β j if a j > 0. In the case β j γλ 1, a j > 0 is trivially true. In the case β j < γλ 1, a j > 0 holds when γ > 1. Let ˆβ j denote the minimizer of R(β j ). It has the following explicit expression: ˆβ j = sign(b j ) ( b j c j ) + 2a j. (2.9) This is because if c j > 0, minimizing (2.8) becomes a regular one dimensional LASSO problem. SMCP estimator ˆβ j is the soft-threshold operator. If c j < 0, it can be shown that ˆβ j and b j are of opposite sign. If b j 0, expression (2.8) becomes ( a j β j + b ) 2 j c j β j. 2a j Hence ˆβ j = (b j c j )/2a j < 0. If b j < 0, then ˆβ j = ˆβ j and ˆβ j = (b j + c j )/2a j > 0. In summary, expression (2.9) holds in all situations. From Fig. (2.2), one can observe that the optimizer of the objective function depends on the choice of b j and c j. If c j > 0, the estimates are equal to the LASSO estimates. Else if c j <= 0, the objective function will have two modes. The minimum of R(β j ) depends on the sign of b j. Fig. (2.2) represents the equation (2.5) graphically. We note that a j and b j do not depend on any β j. They only need to be computed once for each SNP. Only c j needs to be updated after all β j s are updated. In the special case of λ 2 = 0, the SMCP method becomes the MCP method. In that
22 case, even c j no longer depends on β j 1 and β j+1 : c j = λ 1 if β j < γλ 1 and c j = 0 otherwise. Expression (2.9) gives the explicit solution for β j. Generally, an iterative algorithm is required to estimate these parameters. Let β (0) (0) = ( β 1,..., (0) β p ) be the initial value of the estimate of β. The proposed coordinate descent algorithm proceeds as follows: 1. Compute a j and b j for j = 1,..., p. 2. Set s = 0. 3. For j = 1,..., p, (a) Compute c j according to expressions (2.6) or (2.7). (b) Update β (s+1) j according to expression (2.9). 4. Update s s + 1. 5. Repeat steps (3) and (4) until the estimate of β converges. In practice, the initial values β (0) j, j = 1,..., p are set to 0. Each β j is then updated in turn using the coordinate descent algorithm described above. One iteration in completed when all β j s are updated. In our experience, convergence is typically reached after about 30 iterations for the SMCP method. For now, the property of the second penalty is discussed. We assume that currently, λ 1 and λ 2 are fixed and we want to minimize the objective function (2.2). Suppose that in the most recent step (s 1), β j 1 was updated and compare the (s) (s 1) value of estimates under adjacent steps, δ = β j 1 β j 1. We further assume that
23 at the most recent step (s 1), only now go into the step s to update β j. β (s 1) j 1 is non-zero and δ is usually positive. We If corr(x j, x j 1 ) > 0, then ζ j 1 = corr(x j, x j 1 ). We have c (s) j Note that c (s) j = c (s 1) j λ 2 δζ j 1. < c (s 1) j, since ζ j 1 > 0. From expression (2.9), we know that β (s) j will be non-zero if c j is less than b j. One can see that with stronger correlation (i.e. larger ζ j 1 ) and/or larger λ 2, c (s) j will be smaller. Consequently, β(s) j is more likely to be non-zero. Also, the sign of β j is positive if it is non-zero. It makes sense that the correlation between the (j 1)th and the jth predictors is positive. It is similar when corr(x j, x j 1 ) < 0. Thus, incorporating the second penalty increases the chance that adjacent SNPs with high correlation are selected together. 2.3 Joint SMCP Model in GWAS Here we describe the proposed penalized regression approach for GWAS under joint modeling. We first describe the loss function used in the objective function. We then describe a coordinate descent algorithm to fit the joint linear model with SMCP. 2.3.1 Joint Loss Function Let p be the number of SNPs and n the number of subjects. The centered phenotype of the ith subject is denoted by y i. For quantitative trait analysis, the phenotype is continuous. The genotype at SNP j is scored as 0, 1, or 2 depending on
24 the number of copies of a reference allele in a subject. Let x ij denote the standardized genotype score such that i x ij = 0 and 1 n n i=1 x2 ij = 1, and y the standardized phenotype such that i y i = 0. We consider the linear loss function g(β) = 1 2n n (y i i=1 p x ij β j ) 2. (2.10) Thus, for joint linear model with loss function (2.10), the objective function (2.2) is needed to minimize with respect to (β). I show the coordinate descent algorithm to the joint least squares in the following section. j=1 2.3.2 Coordinate Descent Algorithm We wish to minimize the objective function (2.2) with respect to β j while keeping other β k, k j, fixed at their current estimates. Thus only the terms involving the β j in L n matter. That is, this problem is equivalent to minimizing R(β j ) defined as R(β j ) = 1 2n n ( r i( j) x ij β j ) 2 + ρ( β j ; λ 1, γ) i=1 + 1 2 λ 2[ζ j ( β j β j+1 ) 2 + ζ j 1 ( β j 1 β j ) 2 ] = C + a j β 2 j + b j β j + c j β j, j = 2,..., p 1, where r i( j) = y i β 0 k j x ikβ k the partial residual for fitting β j, C is a term free of β j, β j+1 and β j 1 are current estimates of β j+1 and β j 1, respectively, and a, b, and c are determined as follows:
25 For β j < γλ 1, ( ) a j = 1 1 n x 2 ij + λ 2 (ζ j 1 + ζ j ) 1, 2 n γ i=1 b j = 1 n x ij r i( j), n i=1 and c j = λ 1 λ 2 ( β j+1 ζ j + β j 1 ζ j 1 ). (2.11) For β j γλ 1, ( a j = 1 1 2 n j n j ) x 2 ij + λ 2 (ζ j 1 + ζ j ), i=1 c j = λ 2 ( β j+1 ζ j + β j 1 ζ j 1 ), while b j remains the same as in the previous situation. Note that function R(β j ) is defined for j 1, p. It can be defined for j = 1 by setting β j 1 = 0 and for j = p by setting β j+1 = 0 in the above two situations. Overall, minimizing R(β j ) with respect to β j is equivalent to minimizing a j β 2 j + b j β j + c j β j, or equivalently, ( a j β j + b ) 2 j + c j β j. 2a j The solution is the same as the one given in expression (2.9). The algorithm amounts to the following steps: Initialize vector of residuals r i = y i ỹ i,where ỹ i = p j=1 x (0) ij β j. 1. Set s = 0.
26 2. Inner loop: (a) Set j = 1 and calculate a j (b) Update b j and c j according to expression (2.11). (c) Update β (s+1) j according to expression (2.9). (s+1) (s) (d) Update r i r i x ij ( β j β j ) for i = 0,..., n and j j + 1 (e) Repeat (b) to (d) until j = p and evaluate the criterion. (f) If criterion matched, go to step 3. Else, repeat (a) to (e) in step 2. 3. Update s s + 1. 4. Repeat step 2 and 3 until convergence. To make the computation more efficient, here we write the term b j in details. b j = 1 n x ij (y i x il βl ) n i=1 l j n = y i x ij + n ( x ij x il ) β l i=1 l j i=1 Hence, we can use inner product between x j s to represent b j. It will help us to get solution path more efficiently. Starting from λ max, β j s come into the model gradually. Once we calculate the inner product between x j and x l, we save and store it for the use of smaller λ. This step will save a large amount of computing time and storage space, since only a small number of β parameters will get into the model.
27 Wwe are only interested in the correlations between the selected predictors and all other predictors. Here we want to discuss the properties of the coefficients a j,b j and c j and how they will affect the update of the β j. Suppose, in the most recent step, β j 1 was updated and compared to its last (s) (s 1) update: set δ = β j 1 β j 1. We assume at the current step (s 1), only β j 1 is non-zero and is positive and δ is positive. We now go into the step of updating β j. a j is constant in that the design matrix and weights ζ j s are fixed. It only affects the scale of the parameter estimates. If corr(x j, x j 1 ) > 0, then ζ j 1 = corr(x j, x j 1 ). We have b (s) j = b (s 1) j δζ j 1 and c (s) j = c (s 1) j λ 2 δζ j 1. One should note that b (s 1) j is negative and thus b (s) j is getting larger. By inspecting on equation (2.9), we know that the probability that β j will be non-zero is getting larger, since c (s) j becomes smaller but b (s) j turns larger. It is similar when corr(x j, x j 1 ) < 0. Thus incorporating the smoothed penalty enables the adjacent SNPs with high correlation to be selected together. In practice, initial values of β j, j = 1,..., p are set to 0. Each β j is then updated in turn using the cyclical coordinate descent algorithm described above. One iteration completes when all β j s are updated.
28 2.4 Methods 2.4.1 Tuning parameter selection Selecting appropriate values for tuning parameters is important. It affects not only the number of selected variables but also the estimates of model parameters and the selection consistency. There are various methods that can be applied, which include AIC (Akaike, 1974), BIC (Chen and Chen, 2008, Schwarz, 1978), crossvalidation and generalized cross-validation. However, they are all based upon the the performance of prediction error. In GWAS, disease markers may not be in the set of SNP markers. Practically it is rare that disease markers are part of SNPs data, which consequently results in non-true model for SNPs data. Hence, the methods mentioned above may be inadequate in GWAS. Wu et al. (2009) used a predetermined number of predictors to select the tuning parameter and implement a combination of bracketing and bisection to search for the optimal tuning parameter. We adopt Wu et al. (2009) method to select tuning parameters. For this purpose, tuning parameters λ 1 and λ 2 are re-parameterized through τ=λ 1 +λ 2 and η=λ 1 /τ. The value of η is fixed beforehand. When η = 1, the SMCP method becomes the MCP method. The optimal value of τ that selects the predetermined number of predictors is determined through bisection as follows. Let r(τ) denote the number of predictors selected under τ. Let τ max be the smallest value for which all coefficients are 0. τ max is the upper bound for τ. From expression (2.6), τ max = max j n j i=1 x ijy ij /(n j η). To avoid undefined saturated linear models, τ can not be 0 or close to 0. Its lower bound, denoted by τ min, is set at τ min = ɛτ max for preselected ɛ. Setting ɛ = 0.1 seems
29 to work well with the SMCP method. Initially, we set τ l = τ min and τ u = τ max. If r(τ u ) < s < r(τ l ), then we employ bisection. This involves testing the midpoint τ m = 1(τ 2 l + τ u ). If r(τ m ) < s, we replace τ u by τ m. If r(τ m ) > s, we replace τ l by τ m. This process repeats until r(τ m ) = s. From simulation study, we find that regularization parameter γ also has an important impact on the analysis. Based on our experience, γ = 6 is a reasonable choice for the SMCP method. 2.4.2 P -values for the selected SNPs The use of p-value is a traditional way to evaluate the significance of estimates. However, there are no straightforward ways to compute standard error of penalized linear regression estimates. Wu et al. (2009) proposed a leave-one-out approach for computing p-values by assessing the correlations among the selected SNPs in the reduced model. We use the multi-split method proposed by Meinshausen et al. (2009) to obtain reproducible p-values. This is a simulation-based method that automatically adjusts for multiple comparisons. In each iteration, the multi-split method proceeds as follows: 1. Randomly split the data into two disjoint sets of equal size: D in and D out. The case:control ratio in each set is the same as in the original data. 2. Fit the SMCP method with data in D in. Denote the set of selected SNPs by S. 3. Assign a p-value P j to SNP j in the following way:
30 (a) If SNP j is in set S, set P j to be its p-value on D out in the regular linear regression where SNP j is the only predictor. (b) If SNP j is not in set S, set P j = 1. 4. Define adjusted p-value by P j = min P j S, 1, j = 1,..., p, where S is the size of set S. This procedure is repeated B times for each SNP. Let P (b) j denote the adjusted p-value for SNP j in the bth iteration. For π (0, 1), let q π be the π-quantile on P (b) j /π; b = 1,..., B. Define Q j (π) = min1, q π. Meinshausen et al. (2009) proved that Q j (π) is an asymptotically correct p-value, adjusted for multiplicity. They also proposed an adaptive version that selects a suitable value of quantile based on the data: Q j = min1, (1 log π 0 ) inf π (π 0,1) Q j (π), where π 0 is chosen to be 0.05. It was shown that Q j, j = 1,..., p, can be used for both FWER (family-wise error rate) and FDR control (Meinshausen et al., 2009). 2.5 Simulation and Empirical Studies 2.5.1 Simulation Studies and Results To make the LD structure as realistic as possible, genotypes are obtained from the rheumatoid arthritis (RA) study provided by the Genetic Analysis Workshop (GAW) 16. This study involves 2062 individuals. Four hundred of them are randomly chosen. Five thousand SNPs are selected from chromosome 6. For individual i, its
31 phenotype y i is generated as follows: y i = x iβ + ɛ i, i = 1,..., 400, where x i (a vector of length 5000) represents the genotype data of individual i, β is the vector of genetic effect whose elements are all 0 except that (β 2287,..., β 2298 ) = (0.1, 0.2, 0.1, 0.2, 1, 0.1, 1, 0.1, 1, 0.1, 0.6, 0.2) and (β 2300,..., β 2318 ) = (0.1, 0.6, 0.2, 0.3, 0.1, 0.3, 0.4, 1.2, 0.1, 0.3, 0.7, 0.1, 1, 0.2, 0.4, 0.1, 0.5, 0.2, 0.1). ɛ i is the residual sampled from a normal distribution with mean 0 and standard deviation 1.5. The loss function g(β) is given in expression (2.4). To evaluate the performance of SMCP, we use false discovery rate (FDR) and false negative rate (FNR) which are defined as follows. Let ˆβ j denote the estimated value of β j, and FDR = # of SNPs with ˆβ j 0 but β j = 0 # of SNPs with ˆβ 0 FNR = # of SNPs with ˆβ j = 0 but β j 0. # of SNPs with β 0 Table 2.1 and Table 2.2 show the mean and standard deviation of true positive, FDR and FNR for various values of η for the SMCP method and a LASSO method over 100 replications under marginal model and joint model respectively. For all replications, the number of feature selected in the model is 50. The loss function used in LASSO is the one in expression (2.4). FDR and FNR perform in the same direction, since the number of the selected predictors is fixed. As the number of true positive increases, the number of false negative and the number of false positive
32 Table 2.1: Simulation results for marginalized linear model with SMCP. Method η True Positive FDR FNR SMCP 0.05 29.12(0.78) 0.418(0.016) 0.061(0.025) 0.06 28.75(0.87) 0.425(0.017) 0.073(0.028) 0.08 28.24(0.81) 0.435(0.016) 0.089(0.026) 0.1 27.87(0.75) 0.443(0.015) 0.101(0.024) 0.2 27.02(0.67) 0.460(0.013) 0.128(0.021) 0.3 26.14(0.45) 0.477(0.009) 0.157(0.015) 0.4 25.97(0.36) 0.481(0.007) 0.162(0.012) 0.5 25.71(0.61) 0.486(0.012) 0.171(0.020) 0.6 25.42(0.61) 0.492(0.012) 0.180(0.020) 0.7 25.15(0.56) 0.497(0.011) 0.189(0.018) 0.8 24.82(0.55) 0.504(0.012) 0.199(0.018) 0.9 24.66(0.65) 0.507(0.013) 0.205(0.021) 1 24.20(0.84) 0.516(0.017) 0.219(0.027) LASSO 24.31(0.83) 0.514(0.016) 0.216(0.027) Table 2.2: Simulation results for joint linear model with SMCP. Method η True Positive FDR FNR SMCP 0.05 22.07(1.68) 0.559(0.035) 0.288(0.054) 0.08 20.73(1.73) 0.585(0.035) 0.331(0.056) 0.1 20.03(1.87) 0.599(0.037) 0.354(0.060) 0.2 16.88(1.92) 0.662(0.038) 0.455(0.062) 0.3 14.53(1.83) 0.709(0.036) 0.541(0.059) 0.4 11.65(1.66) 0.767(0.033) 0.624(0.054) 0.5 9.77(1.37) 0.804(0.027) 0.684(0.044) 0.6 8.62(1.15) 0.828(0.023) 0.722(0.037) 0.7 7.67(1.11) 0.847(0.022) 0.753(0.036) 0.8 6.96(1.12) 0.861(0.022) 0.775(0.036) 0.9 6.39(1.07) 0.872(0.022) 0.794(0.035) 1 6.19(0.97) 0.876(0.019) 0.800(0.031) LASSO 11.62(1.21) 0.768(0.024) 0.625(0.039)
33 decrease. The SMCP method outperforms the LASSO in terms of true positive and FDR. It also suggests that η=0.05 gives acceptable true positive and FDR at the same time. To investigate further the performance of the SMCP method and the LASSO method for genome-wide screening purpose, we use a simulated data set. 50 SNPs are selected by the SMCP and the LASSO method (Table 2.3). For both methods, p-values for selected SNPs are computed using the multi-split method. From Table 2.3, it is apparent the SMCP method selects much higher true positive than the marginal LASSO method. SMCP selects 25 out of 31 true disease associated SNPs while LASSO selects 21. Note that multi-split method can effectively produce p-values for the selected SNPs. It assigns insignificant p-values for all non-disease associated SNPs for the SMCP method. Note that SNP 2320 and 2321, they are both selected by the SMCP and the LASSO method. However, they have insignificant p-values for the SMCP method but not the LASSO. From this point, all real data analysis below are conducted using marginal model. 2.5.2 Application to GAW 16 RA Data Rheumatoid arthritis (RA) is a complex human disorder with a prevalence ranging from around 0.8% in Caucasians to 10% in some native American groups (Amos, 2009). Its risk is generally higher in females than in males. Some studies have identified smoking as a risk factor. Genetic factors underlying RA have been mapped to the HLA region on region 6p21 (Newton et al., 2004), PTPN22 locus at
Table 2.3: List of 50 SNPs selected by the marginalized SMCP method and marginalized LASSO method for a simulated data set. SMCP LASSO Regression SNP ˆβ p-value ˆβ p-value ˆβ p-value 2110-0.042 1-0.944 4.4e-04 2112 0.042 1 0.944 4.4e-04 2118-0.001 1-0.077 1-0.925 2.5e-04 2120 0.002 1 0.071 1 0.920 2.7e-04 2181-0.002 1-0.080 1-1.037 2.7e-04 2240 0.045 1 0.241 1 1.103 1.6e-05 2241 0.059 1 0.251 1 1.175 1.8e-05 2242 0.046 1 0.158 1 1.103 8.6e-05 2247-0.010 1-0.101 1-0.941 1.7e-04 2269-0.059 1-0.481 1-1.627 1.6e-06 2270 0.034 1 1.136 0.002 2272-0.003 1-0.089 1-0.979 2.2e-04 2279-0.019 1-0.181 1-1.506 1.3e-04 2281-0.037 1-1.145 5.2e-04 2284-0.037 1-1.310 5.5e-04 2286-0.167 1-0.163 1-1.165 9.1e-05 2287 0.621 0.006 0.816 0.008 1.642 9.5e-12 2288 0.618 0.006 0.812 0.008 1.640 1.2e-11 2289-0.896 0.324-0.890 0.191-2.223 1.5e-08 2290 0.467 0.002 1.040 5.1e-04 1.884 1.2e-14 2291 0.068 1 0.569 0.383 2293 0.108 0.012 0.808 0.003 1.625 9.1e-12 2294 0.083 1 0.815 0.002 2295-0.061 0.660-0.413 0.405-1.299 7.0e-07 2299-0.132 1-0.079 0.815 2300 0.580 0.003 1.004 0.002 1.836 2.6e-14 2301-0.782 0.003-1.084 0.015-2.086 8.5e-13 2302 0.687 2.7e-04 1.205 6.3e-05 2.039 1.7e-17 2303 1.221 1 0.722 1.9e-01 2304-0.856 0.001-1.089 1.92e-04-1.933 2.3e-15 2305-0.892 8.2e-06-1.395 1.19e-05-2.239 1.2e-20 2306 0.824 0.030 0.724 0.014 1.527 8.1e-11 2307-0.914 0.159-0.684 0.203-1.709 1.5e-08 2308 0.740 1 0.429 0.705 1.328 5.7e-07 2309 0.738 1.1e-04 1.321 1.51e-05 2.182 8.3e-19 2310-0.910 0.252-0.853 0.133-2.139 1.6e-08 2311 0.477 1 0.1554 0.642 2312 0.717 9.4e-04 1.390 6.30e-05 2.412 2.6e-16 2313 1.029 1 0.036 1 1.525 5.8e-04 2314-0.762 0.019-0.916 0.004-1.776 1.3e-12 2315 0.786 0.019 0.916 0.004 1.776 1.3e-12 2316-0.831 0.006-0.960 0.006-1.853 9.8e-13 2317-0.757 0.251-0.458 0.161-1.285 1.4e-07 2318 0.986 0.001 1.393 1.03e-04 2.442 7.0e-16 2319 9.3e-05 1 0.348 0.198 2320 0.399 0.073 0.928 0.014 2.031 3.2e-10 2321-0.388 0.066-0.911 0.017-2.016 4.9e-10 2332-0.010 1-0.133 1-1.046 1.2e-04 2337-0.049 1-0.439 1-1.733 6.1e-06 2343 0.007 1 0.133 1 1.009 1.1e-04 2346-0.033 1-0.310 1-1.414 1.6e-05 2360-0.015 1-1.052 6.6e-04 2363-0.020 1-0.273 1-1.127 8.4e-06 2371-3.2e-04 1-0.059 1-0.916 3.3e-04 2772 0.035 1 0.872 4.6e-04 4421-0.001 1-0.077 1-1.109 3.0e-04 4628-4.15e-04 1-1.013 7.8e-04 Computed using the multi-split method Single SNP analysis, not corrected for multiple testing. 34
35 1p13 (BEG, 2004), and the CTLA4 locus at 2q33 (Plenge, 2005). There are some other loci reported. These loci are at 6q (TNFAIP3), 9p13 (CCL21), 10p15 (PRKCQ), and 20q13 (CD40) and seem to be of weaker effects (Amos, 2009). GAW 16 RA data is from the North American Rheumatoid Arthritis Consortium (NARAC). It is the initial batch of whole genome association data for the NARAC cases (N=868) and controls (N=1194) after removing duplicated and contaminated samples. The total sample size is 2062. After quality control and removing SNPs with low minor allele frequency, there are 475672 SNPs over 22 autosomes, of which 31670 are on chromosome 6. The marginalized loss function implemented with SMCP is used for analyzing real data for GAW 16 and GAW 17. SNPs on the whole genome are analyzed simultaneously. By using different numbers for predetermined number of SNPs, we found that 800 SNPs along the genome are appropriate for the GAW 16 RA dataset. For the SMCP method, the optimal value for tuning parameter τ corresponding to this setting is 1.861 with η = 0.05. p-values of the selected SNPs are computed using multi-split method. The majority of the SNPs (539 out of 800) selected by the SMCP method are on chromosome 6, 293 of which are significant at significance level 0.05. The plot of log 10 (p-value) for the selected SNPs against their physical positions is shown in Fig. 2.3(a) for chromosome 6. For the LASSO method (i.e., η = 1 and γ = ), the same procedure is implemented to select 800 SNPs across the genome. The optimal value for tuning
36 parameter τ is 0.091. There are 537 SNPs selected on chromosome 6 and 280 of them are significant with multi-split p-value less than 0.05. The plot of log 10 (p-value) for chromosome 6 is shown in Fig. 2.3(b). We also analyzed the data using the MCP method (i.e., η = 1). It selects the same set of SNPs as the LASSO method. The difference of the LASSO and the MCP lies in the magnitude of estimates, since the MCP is unbiased under proper choice of γ but the LASSO is always biased. Two sets of SNPs selected by the SMCP and the LASSO, respectively, on chromosome 6 are both in the region of HLA-DRB1 gene that has been found to be associated with RA (Newton et al., 2004). There are SNPs on other chromosomes that are significant or close to be significant (Table 2.4). Particularly, association to rheumatoid arthritis at SNP rs2476601 in gene PTPN22 has been reported previously (BEG, 2004). Other noteworthy SNPs include SNP rs512244 in RAB28 region, 6 SNPs in TRAF1 region, SNP rs12926841 in CA5A region, SNP rs3213728 in RNF126P1 region, and SNP rs1182531 in PHACTR3 region. On chromosome 9, 6 SNPs in the region of TRAF1 gene are identified by the SMCP method and the LASSO method. Among them, 2 insignificant SNPs (rs10985073, rs10760130) have much smaller p-values for the SMCP method than the LASSO method. The estimates of βs obtained from SMCP, MCP, LASSO and the regular single-snp linear regression analysis are presented in Fig. 2.4 and Fig. 2.5. One can see from Fig. 2.4 and Fig. 2.5, the MCP method produce larger estimates than the LASSO method, but the estimates from the SMCP method are smaller than those from the LASSO. This is caused by the (side) shrinkage effect of the proposed
37 Table 2.4: SNPs found significant other than chromosome 6 SMCP LASSO Gene Chr Position SNP name Estimates p-value Estimates p-value PTPN22 1 114089610 rs2476601-0.026 6e-05-0.061 2e-05 RAB28 4 12775151 rs512244 0.019 0.024 0.033 0.021 LOC392232 8 73406911 rs346617 0.026 0.074 0.032 0.051 TRAF1 9 120720054 rs1953126-0.021 0.025-0.031 0.053 TRAF1 9 120723409 rs10985073-0.026 0.110-0.025 0.184 TRAF1 9 120732452 rs881375-0.030 0.014-0.033 0.016 TRAF1 9 120769793 rs3761847 0.029 0.014 0.033 0.027 TRAF1 9 120781544 rs10760130 0.025 0.061 0.029 0.073 TRAF1 9 120785936 rs2900180-0.019 0.008-0.037 0.006 CA5A 16 86505516 rs12926841-0.031 0.002-0.042 0.002 RNF126P1 17 52478747 rs3213728 0.046 8-06 0.066 1e-06 PHACTR3 20 57826397 rs1182531 0.018 0.025 0.032 0.021 smoothing penalty. In terms of model selection, SMCP tends to select more adjacent SNPs that are in high LD. Quantitative traits anti-ccp and rheumatoid factor IgM are measured over case group in the study. Here we choose anti-ccp as response. Because it shows a great promise as a diagnostic marker of RA Khosla et al. (2004). Total sample size after removing missing anti-ccp is 867. After quality control and removing SNPs with low minor allele frequency, there are 444116 SNPs over 22 autosomes. Genome scans by using the SMCP, the MCP, the LASSO and regular linear regression are shown in Fig. 2.6 and Fig. 2.7 respectively. All chromosomes are analyzed at one time for the SMCP, the MCP and the LASSO. The number of SNPs selected is 500 for these methods. One can observe that the SMCP will have more clustered estimates. For example, see estimates in chromosome 12, the SMCP have three clustered spots, but the MCP and the LASSO are much more noisy. It is in
38 the same manner for other chromosomes. This is because the effect of the smoothing penalty, which takes into account the LD information. For both binary and quantitative traits, we can see that the MCP method produce larger estimates than the LASSO. However, they do not take into account LD information and thus noisier in estimates, especially in quantitative trait. 2.5.3 Application to GAW 17 Data The GAW17 data set consists of 24,487 SNP markers throughout the genome for 697 individuals. We analyse unrelated individual data with quantitative trait Q1 in replicate 1. More details can be found in (Liu et al., 2011). Fig. 2.8(a) shows the absolute lag one autocorrelation coefficients over the whole genome. Fig. 2.8(b) shows the proportion of the absolute lag one autocorrelation coefficients > 0.5 over non-overlapping 100-SNP windows. One can see that even for partially selected SNPs over genome, there exist strong correlations between adjacent SNPs. But the correlations are not as strong as a dense set of SNPs (Fig. 2.1). Although it may be more informative to use pair-wise correlations among SNPs, the computational burden makes it impossible to be implemented in real dataset. All SNPs are included in the analysis. We coded the seven population groups as dummy variables. The quantitative trait Q1 is first regressed on gender, age, smoking status and group dummy variables in order to remove their confounding effect. This procedure helps adjust for population stratification. Then, the residuals from this regression are used as the response and fitted by the SMCP model and
39 the LASSO model. Extended Bayesian information criteria (EBIC) (Chen and Chen, 2008, 2010) is used to select the tuning parameter. The selected tuning parameter τ is 1.655 for the SMCP with η=0.1 and 0.184 for the LASSO. Absolute values of estimates from simple linear regression are plotted in Fig. 2.9. The estimation results are presented in Table 2.5. Both the SMCP and the LASSO selected two SNPs (C13S522 and C13S523) from gene FLT1. For each method, these two SNPs have significant LOO p-values. The SMCP selected three more SNPs, one (C13S524) from gene FLT1 and the other two (C12S707 and C12S711) from gene PRR4. Only one SNP (C13S524) from gene FLT1 is significant. The box plots for these 5 SNPs selected by the SMCP and the LASSO are shown in Fig. 2.10. By the knowledge of the underlying model, we have computed the true positive rate and false positive rate for the SMCP, the LASSO and regular single SNP regression on trait Q1 using all the 200 replicates (Table 2.6). For regular single SNP regression, Benjamini-Hochberg method is used to control FDR and conduct multiple testing. The SMCP tends to select more SNPs than the LASSO with higher true positive rate and false positive rate. Although regular methods can select a higher true positive, its false positive is extremely higher than the SMCP and the LASSO. Further simulation studies can be found in (Liu et al., 2011). For trait Q1 in replicate 1, the SMCP selected 3 SNPs (C13S522, C13S523 and C13S524) from the associated gene FLT1 and 2 SNPs that are false positive. In comparison, the LASSO selected 2 SNPs (C13S522 and C13S523) both of which are true positive. We note that SNPs provided for GAW 17 are a small subset of the
40 Table 2.5: SNPs selected by the SMCP and LASSO for trait Q1 in replicate 1 Univariate Univariate SMCP LOO LASSO LOO SNP name Gene estimates p-value estimates p-value estimates p-value C12S707 PRR4 0.195 2.0e-07 0.002 0.143 C12S711 PRR4 0.202 7.6e-08 0.007 0.077 C13S522 FLT1 0.314 2.1e-17 0.188 2.4e-07 0.107 7.5e-09 C13S523 FLT1 0.357 2.6e-22 0.191 2.9e-10 0.150 9.4e-14 C13S524 FLT1 0.195 2.2e-07 0.069 0.004 Table 2.6: Mean and standard error of true positives and false positives for selected SNPs over 200 replicates for trait Q1 SMCP LASSO Regular Regression True Positive 3.35(1.52) 2.48(1.19) 7.03(1.81) False Positive 18.42(36.20) 8.64(21.53) 174.35(87.87) SNPs that are genotyped. The strength of LD for this set of SNPs has been greatly reduced. In addition, the GAW 17 data were simulated to mimic rare variants. The SMCP method is specially designed to map rare variants. Even so, the SMCP is able to select 3 SNPs, more than the LASSO is. In comparison, the results of the regular simple linear regression are much noisier.
41 (a) a j = 1, b j = 1, c j = 5 (b) a j = 1, b j = 1, c j = 5 (c) a j = 1, b j = 5, c j = 2 (d) a j = 1, b j = 5, c j = 2 Figure 2.2: Graph of Function R under different a j,b j and c j
42 (a) SMCP (b) LASSO Figure 2.3: Plot of log 10 (p-value) for SNPs on chromosome 6 selected by (a) the SMCP method and (b) the LASSO method for the rheumatoid arthritis data. These p-values are generated using the multi-split method. The horizontal line corresponds to significance level 0.05.
43 (a) SMCP (b) MCP Figure 2.4: Genome-wide plot of β estimates for the SMCP and the MCP methods on binary trait.
44 (a) LASSO (b) Regular Single-SNP Linear Regression Figure 2.5: Genome-wide plot of β estimates for the LASSO and regular single-snp linear regression on binary trait.
45 (a) SMCP (b) MCP Figure 2.6: Genome-wide plot of β estimates for the SMCP and MCP methods on quantitative trait.
46 (a) LASSO (b) Regular Single-SNP Linear Regression Figure 2.7: Genome-wide plot of β estimates for the LASSO and regular single-snp linear regression on quantitative trait.
47 (a) Absolute lag-one autocorrelation (b) Proportion of absolute lag-one autocorrelation coefficients > 0.5 for 100 SNPs per segment Figure 2.8: Correlation plots along the genome from Genetic Analysis Workshop 17 simulated dataset
48 (a) Univariate Linear Regression Figure 2.9: Genome-wide plot of β estimates for regular single-snp linear regression. (a) C13S522 Figure 2.10: Boxplots for comparison of residuals on genotypes of selected SNPs. Y axis is for residuals and x axis for genotypes