Genetic Studies of Multivariate Traits

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1 Genetic Studies of Multivariate Traits Heping Zhang Collaborative Center for Statistics in Science Yale University School of Public Health Presented at DIMACS, University of Rutgers May 17, 2013 Heping Zhang (C 2 S 2, Yale University) DIMACS 1 / 50

2 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 2 / 50

3 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 3 / 50

4 Comorbidity Multiple disorders or illnesses occur in the same person, simultaneously or sequentially Source: Heping Zhang (C 2 S 2, Yale University) DIMACS 4 / 50

5 Comorbidity Source: aasets.lifehack.org Heping Zhang (C 2 S 2, Yale University) DIMACS 5 / 50

Comorbidity Is there a relationship between childhood ADHD and later drug abuse? See page 2.

It is also a topic about which we have insufficient information, so it remains a research priority for NIDA. This Research Report provides information on the state of the science in this area.

* To help explain this comorbidity, we need to first recognize that drug addiction is a mental illness.

6 Comorbidity Is there a relationship between childhood ADHD and later drug abuse? See page 2. from the director: Comorbidity is a topic that our stakeholders patients, family members, health care professionals, and others frequently ask about. It is also a topic about which we have insufficient information, so it remains a research priority for NIDA. This Research Report provides information on the state of the science in this area. Although a variety of diseases commonly co-occur with drug abuse and addiction (e.g., HIV, hepatitis C, cancer, cardiovascular disease), this report focuses only on the comorbidity of drug use disorders and other mental illnesses.* To help explain this comorbidity, we need to first recognize that drug addiction is a mental illness. It is a complex brain disease characterized by compulsive, at times uncontrollable drug craving, seeking, and use despite devastating consequences behaviors that stem from drug-induced changes in brain structure and function. These changes occur in some of the same brain areas that are disrupted in other mental disorders, such as depression, anxiety, or schizophrenia. It is therefore not surprising that population surveys show a high rate of co-occurrence, or comorbidity, between drug addiction and other mental illnesses. While we cannot always prove a connection or causality, we do know that certain mental disorders are established risk factors for subsequent drug abuse and vice versa. It is often difficult to disentangle the overlapping symptoms of drug addiction and other mental illnesses, making diagnosis and treatment complex. Correct diagnosis is critical to ensuring appropriate and effective treatment. Ignorance of or failure to treat a comorbid disorder can jeopardize a patient s chance of recovery. We hope that our enhanced understanding of the common genetic, environmental, and neural bases of these disorders and the dissemination of this information will lead to improved treatments for comorbidity and will diminish the social stigma that makes patients reluctant to seek the treatment they need. Nora D. Volkow, M.D. Director National Institute on Drug Abuse Comorbidity: Addiction and Other Mental Illnesses What Is Comorbidity? W hen two disorders or illnesses occur in the same person, simultaneously or sequentially, they are described as comorbid. Comorbidity also implies interactions between the illnesses that affect the course and prognosis of both. continued inside *Since the focus of this report is on comorbid drug use disorders and other mental illnesses, the terms mental illness and mental disorders will refer here to disorders other than substance use disorders, such as depression, schizophrenia, anxiety, and mania. The terms dual diagnosis, mentally ill chemical abuser, and co-occurrence are also used to refer to drug use disorders that are comorbid with other mental illnesses. Dr. Volkow, Director, NIDA: Comorbidity is a topic that our stakeholders patients, family members, health care professionals, and others frequently ask about. It is also a topic about which we have insufficient information, so it remains a research priority for NIDA. Source: Published 12/2008. Revised 9/2010. Heping Zhang (C 2 S 2, Yale University) DIMACS 6 / 50

7 Common Etiology for Comorbidity { mental disorder Common etiology drug use disorder "Overlapping genetic vulnerabilities: Mounting evidence suggests that common genetic factors may predispose individuals to both mental disorders and addiction or to having a greater risk of the second disorder once the first appears." "Overlapping environmental triggers: Stress, trauma (e.g., physical or sexual abuse), and early exposure to drugs are common factors that can lead to addiction and to mental illness, particularly in those with underlying genetic vulnerabilities." Source: Heping Zhang (C 2 S 2, Yale University) DIMACS 7 / 50

Disorders, Genes and Covariates Covariates: interact or confound genetic effects Failure to account for covariates: bias or reduced power Null hypothesis: no

8 Disorders, Genes and Covariates Covariates: interact or confound genetic effects Failure to account for covariates: bias or reduced power Null hypothesis: no association between marker alleles and any linked locus that influences traits conditional on covariates Heping Zhang (C 2 S 2, Yale University) DIMACS 8 / 50

9 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 9 / 50

10 An Example Trait Questionnaire Fagerstrom Test for Nicotine Dependence Heping Zhang (C 2 S 2, Yale University) DIMACS 10 / 50

11 Genotypes and Covariates Source: en.wikipedia.org; 2010.census.gov Heping Zhang (C 2 S 2, Yale University) DIMACS 11 / 50

12 Study Design Population-based studies Family-based studies Heping Zhang (C 2 S 2, Yale University) DIMACS 12 / 50

13 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 13 / 50

14 Multivariate Distributions t n t! a n t,a! a ˆp nt,a t,a exp { 1 2 (x µ) Σ 1 (x µ)} (2π) n Σ Heping Zhang (C 2 S 2, Yale University) DIMACS 14 / 50

15 Challenge How do we model a hybrid of continuous, ordinal, and/or binary responses??? Heping Zhang (C 2 S 2, Yale University) DIMACS 15 / 50

16 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 16 / 50

17 Notation and Hypothesis n study subjects, from a population-based study or family-based study For each subject: A vector of traits T = (T (1),..., T (p) ) Marker genotype M Parental marker genotypes M pa (only available in a family-based study) A vector of covariates Z = (Z (1),..., Z (l) ) Null hypothesis: no association between marker alleles and any linked locus that influences traits T, conditional on Z Heping Zhang (C 2 S 2, Yale University) DIMACS 17 / 50

18 Kendall s Tau A nonparametric statistic measuring the rank correlation between two variables Pairs of observations: {(X i, Y i ) : i = 1,..., n} (X i, Y i ) and (X j, Y j ): Concordant, if X i X j and Y i Y j have the same sign Disconcordant, if X i X j and Y i Y j have the different sign Kendall s tau: τ = 2(A B)/{n(n 1)} A and B: numbers of concordant and disconcordant pairs Or τ = ( ) n 1 sign{(x i X j )(Y i Y j )} 2 i<j Heping Zhang (C 2 S 2, Yale University) DIMACS 18 / 50

19 Generalized Kendall s Tau F ij = {f 1 (T (1) i T (1) j ),..., f p (T (p) i T (p) j )} f k ( ): identity function for a quantitative or binary trait f k ( ): sign function for an ordinal trait D ij = C i C j. C: number of any chosen allele in marker genotype M Genaralized Kendall s tau (Zhang, Liu and Wang, 2010): U = ( ) n 1 D ij F ij 2 i<j Special cases: FBAT-GEE (Lange et al. 2003) Test for a single ordinal trait (Wang, Ye and Zhang, 2006) Heping Zhang (C 2 S 2, Yale University) DIMACS 19 / 50

20 Weighted Test A weight function w(z i, Z j ) imposes a relatively large weight when Z i is close to Z j, and a relatively small weight when Z i and Z j are far away Weighted U-statistic: S = Weighted test statistic: ( ) n 1 D ij F ij w(z i, Z j ) 2 i<j χ 2 τ = {S E 0 (S)} Var 1 0 (S){S E 0(S)} Heping Zhang (C 2 S 2, Yale University) DIMACS 20 / 50

21 Weight Function I: Distance Write Z = (Z co, Z ca ), with Z co for the continuous covariates and Z ca for the categorical covariates For example, w(z i, Z j ; h, q) = W h ( Z co i Z co j )W q {I(Z ca i Z ca j )} W h (u) = exp( u 2 /2h 2 ), h > 0, W q (v) = (1 q)i(v = 0) + qi(v = 1), 0 q 0.5 Weighted U-statistic (called fixed-(h, q) U-statistic): S(h, q) = ( ) n 1 D ij F ij w(z i, Z j ; h, q) 2 i<j Heping Zhang (C 2 S 2, Yale University) DIMACS 21 / 50

22 Weight Function II: Propensity Score Propensity score: probability of a unit being assigned to a particular treatment given a set of covariates Causal effect analysis: match subjects according to their propensity scores (Rosenbaum and Rubin, 1984) Genomic propensity score: p(z) = {p 1 (z), p 2 (z)}, p c (z) = P(C = c Z = z) Genetic association analysis: match subjects according to their genomic propensity scores Weight function: w(z i, Z j ) = W h { p(z i ) p(z j ) }, with W h (u) = exp( u 2 /2h 2 ), h > 0 Heping Zhang (C 2 S 2, Yale University) DIMACS 22 / 50

23 Asymptotic Distribution: Null Hypothesis When n, Var 1/2 D 0 {S(h, q)}[s(h, q) E 0 {S(h, q)}] N(0, I p ) Fixed-(h, q) test statistic: Mean and variance: χ 2 τ (h, q) D χ 2 p E 0 {S(h, q)} = Var 0 {S(h, q)} = 2 n 1 n i=1 4 (n 1) 2 ū i E 0 (C i M pa i, Z i ), n i=1 n i=1 ū i ū jcov 0 (C i, C j M pa i, M pa j, Z i, Z j ), with ū i = n 1 n j=1 F ijw(z i, Z j ; h, q) Heping Zhang (C 2 S 2, Yale University) DIMACS 23 / 50

24 Asymptotic Distribution: Power Under the alternative hypothesis, χ 2 τ (h, q) p e i χ 2 1(φ i ) i=1 µ = µ 1 µ 0 E 1 {S(h, q)} E 0 {S(h, q)} Σ 0 = Var 0 {S(h, q)} Σ 1 = Var 1 {S(h, q)} e 1 e p 0: eigenvalues of Σ 1/2 1 Σ 1 0 Σ1/2 1 φ i = µ 2 i µ i : ith component of µ = QΣ 1/2 1 µ Q: an orthonormal matrix, QΣ 1/2 1 Σ 1 0 Σ1/2 1 Q = diag(e 1,..., e p ) Heping Zhang (C 2 S 2, Yale University) DIMACS 24 / 50

25 Factors Determining the Power The conditional power P: P = P { p i=1 e iχ 2 1 (φ i) q χ 2 p (1 α) } Taking a family-based study as an example, µ 1 = Σ 1 = 2 n 1 n i=1 4 (n 1) 2 ū i E(C i T i, Z i, M pa i ) n i=1 n j=1 ū i ū jcov(c i, C j T i, T j, Z i, Z j, M pa i, M pa j ) By Bayes theorem, P(C = c T, Z, M pa ) = P(T C=c,Z)P(C=c Mpa ) c P(T C=c,Z)P(C=c M pa ) Penetrance: P(T C = c, Z) Allele frequency: P(C = c M pa ) Heping Zhang (C 2 S 2, Yale University) DIMACS 25 / 50

26 Power Approximation Using the result from Liu et al. (2009), we have Theorem P P{χ 2 l (ν) q }, where l, ν, and q depend on µ 1 and Σ 1. Heping Zhang (C 2 S 2, Yale University) DIMACS 26 / 50

27 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 27 / 50

28 Maximum Weighted Test Fixed-(h, q) test: how to choose optimal parameters h and q? Choose a grid of h and q values and maximize the weighted test statistic over those choices {h 1,..., h L1 }: pre-specified grid points of h {q 1,..., q L2 }: pre-specified grid points of q χ 2 τ,max = max 1 l 1 L 1,1 l 2 L 2 χ 2 τ (h l1, q l2 ) Approximate the optimal weighting scheme, yielding the strongest association measure Heping Zhang (C 2 S 2, Yale University) DIMACS 28 / 50

29 Resampling Approach An intuitive approach through computation... Population-based studies: restricted permutation in Yu et al. (2010) Family-based studies: children s genotypes solely determined by their parents marker alleles, resample the children s genotype by Mendelian laws Calculate M resampling test statistics χ 2 τ,max,1,..., χ2 τ,max,m using M resampled data Resampling p-value: the proportion of the resampling test statistics that exceed our observed test statistic, i.e., M 1 M m=1 I( χ2 τ,max,m χ 2 τ,max) Computation too intensive! Heping Zhang (C 2 S 2, Yale University) DIMACS 29 / 50

30 Asymptotic Distribution: Joint Distribution An theoretical approach through approximation... Equivalently, χ 2 τ,max = max R l1,l l 1 L 1,1 l 2 L 2 R = Var 1/2 0D (S){S E 0(S)} S = {S (h 1, q 1 ),..., S (h L1, q L2 )} Var 0D (S) = diag[var 0 {S(h 1, q 1 )},..., Var 0 {S(h L1, q L2 )}]: the diagonal blocks of Var 0 (S) Var 1/2 0 (S){S E 0 (S)} D N(0, I pl1 L 2 ) R = Var 1/2 0D (S)Var1/2 0 (S)G, G N(0, I pl1 L 2 ) Heping Zhang (C 2 S 2, Yale University) DIMACS 30 / 50

31 Asymptotic Distribution: Uniform Approximation Theorem Assume that the eigenvalues of Var 0D (S) and Var 0 (S) are uniformly bounded from both above and below, i.e., there exist two positive numbers c and C such that c λ min {Var 0D (S)} λ max {Var 0D (S)} C and c λ min {Var 0 (S)} λ max {Var 0 (S)} C uniformly for all n, where λ min and λ max denote the smallest and largest eigenvalues respectively. Then for any x R, as n, ( ) ( ) sup P χ 2 τ,max x P max R l1,l 2 2 x 0. 1 l 1 L 1,1 l 2 L 2 x R Heping Zhang (C 2 S 2, Yale University) DIMACS 31 / 50

32 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 32 / 50

33 WTCCC Bipolar Disorder Data Collected by Wellcome Trust Case-Control Consortium (WTCCC, 2007, Nature) Phenotype: 1998 cases/3004 controls of bipolar disorder Genotype: genotyped by Affymetrix GeneChip 500K arrays Covariates: gender, age at recruitment Our method: weighted test using propensity score approach (h = 1) Methods for comparison: non-weighted test and logistic regression Strong association: p-value < ; moderate association: < p-value < 10 5 Heping Zhang (C 2 S 2, Yale University) DIMACS 33 / 50

34 Manhattan Plot: Comparison of Three Methods Heping Zhang (C 2 S 2, Yale University) DIMACS 34 / 50

35 GWAS Results Chr. SNP Position Non-weighted Weighted Logistic Regression 6 rs e e e-9 16 rs e e e-9 16 rs e e e-6 16 rs e e e-6 16 rs e e e-6 17 rs e e e-7 20 rs e e e-7 20 rs e e e-7 Heping Zhang (C 2 S 2, Yale University) DIMACS 35 / 50

36 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 36 / 50

37 Collaborative Studies on Genetics of Alcoholism A large scale study to map alcohol dependence susceptible genes Heping Zhang (C 2 S 2, Yale University) DIMACS 37 / 50

38 COGA Data The data include 143 families with a total of 1,614 individuals Multiple Traits: ALDX1 (the severity of the alcohol dependence): pure unaffected, never drunk, unaffected with some symptoms, and affected MaxDrink (maximum number of drinks in a 24 hour period): 0-9, 10-19, 20-29, and more than 30 drinks TimeDrink (spent so much time drinking, had little time for anything else): no, yes and lasted less than a month, and yes and lasted for one month or longer Genotypes: markers on chromosome 7 Covariates: age at interview and gender Heping Zhang (C 2 S 2, Yale University) DIMACS 38 / 50

39 Results P-values between one of the three traits and markers on Chromosome 7 with covariates unadjusted. Heping Zhang (C 2 S 2, Yale University) DIMACS 39 / 50

40 Results D7S1790 D7S513 D7S1802 D7S629 D7S673 D7S1838 NPY2 D7S817 D7S2846 D7S521 D7S691 D7S478 D7S679 D7S665 D7S1830 D7S3046 D7S1870 D7S1797 D7S820 D7S821 D7S1796 D7S1799 D7S1817 D7S2847 D7S490 D7S1809 log(p value) D7S1804 D7S509 D7S1824 D7S794 D7S1805 adjusted unadjusted Distance (cm) P-values between the three traits and markers on Chromosome 7 with covariates adjusted or not. Heping Zhang (C 2 S 2, Yale University) DIMACS 40 / 50

41 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 41 / 50

42 Method Developed a nonparametric weighted test to adjust for covariates that accommodates multiple traits Provided its asymptotic distribution and analytical power calculation Refined the weighted test by proposing the idea of maximum weighting over the grid points of parameters Proposed an asymptotic approach to assessing its significance Heping Zhang (C 2 S 2, Yale University) DIMACS 42 / 50

43 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 43 / 50

44 Applications WTCCC bipolar disorder data: not only confirmed the results reported by the WTCCC (2007), but also identified another region at the genome-wide significance level The identified haplotype block is near the RPGRIP1L gene that was reported to be associated with bipolar disorder (O Donovan et al., 2008; Riley et al., 2009) COGA data: confirmed and strengthened the top signal; provided evidences for the advantage of maximum weighted test over non-weighted test Heping Zhang (C 2 S 2, Yale University) DIMACS 44 / 50

45 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 45 / 50

46 Acknowledgment Dr. Yuan Jiang, Oregon State University Dr. Ching-Ti Liu, Boston University Dr. Xueqin Wang, Sun Yat-Sen University, China Dr. Wensheng Zhu, Northeastern Normal University, China Heping Zhang (C 2 S 2, Yale University) DIMACS 46 / 50

47 Acknowledgment Supported by grant R01DA from National Institute on Drug Abuse. The SAGE data were obtained from dbgap ( The COGA data were provided by COGA. WTCCC data were provided by Wellcome Trust Case-Control Consortium. The views expressed here are those of the authors. Heping Zhang (C 2 S 2, Yale University) DIMACS 47 / 50

48 Outline 1 Background Comorbidity: Definition and Mechanisms Data and Study Design Challenge 2 Association Test Generalized Kendall s Tau Maximum Weighted Test over Grids 3 Data Analyses WTCCC Bipolar Disorder Data COGA Family Data 4 Conclusions and Acknowledgment Method Data Analysis Acknowledgment References Heping Zhang (C 2 S 2, Yale University) DIMACS 48 / 50

49 Related Readings W. Zhu and H. Zhang (2009) Why do we test multiple traits in genetic association studies? (with discussion) J. Korean Statist. Soc., 38:1-10. H. Zhang, C.-T. Liu and X. Wang (2010) An association test for multiple traits based on the generalized Kendall s tau. J. Amer. Statist. Assoc., 105: X. Chen, K. Cho, B. Singer, and H. Zhang (2011) The nuclear transcription factor PKNOX2 is a candidate gene for substance dependence in European-origin women. PLoS One, 27; 6:e Y. Jiang and H. Zhang (2011) Propensity Score-Based Nonparametric Test Revealing Genetic Variants Underlying Bipolar Disorder. Genet. Epidemiol., 35: H. Zhang (2011) Statistical Analysis in Genetic Studies of Mental Illnesses. Statistical Science, 26: W. Zhu, Y. Jiang, and H. Zhang (2012) Covariate-Adjusted Association Tests and Power Calculations Based on the Generalized Kendall s Tau. J. Amer. Statist. Assoc., 107:1-11. Heping Zhang (C 2 S 2, Yale University) DIMACS 49 / 50

50 For everything we did, there may be a better way!" David Banks (?) Heping Zhang (C 2 S 2, Yale University) DIMACS 50 / 50

Genetics Studies of Multivariate Traits

Genetics Studies of Multivariate Traits Heping Zhang Department of Epidemiology and Public Health Yale University School of Medicine Presented at Southern Regional Council on Statistics Summer Research