More Powerful Genome-wide Association Methods for Case-control Data Robert C. Elston, PhD Case Western Reserve University Cleveland Ohio

SINGLE-MARKER AND TWO-MARKER ASSOCIATION TESTS FOR UNPHASED CASE-CONTROL GENOTYPE DATA, WITH A POWER COMPARISON Kim S, Morris NJ, Won S, Elston RC Genetic Epidemiology, in press 2

Introduction A genome-wide association study with case-control data aims to localize disease susceptibility regions in the genome A genome-wide association study with case-control data aims to localize disease susceptibility regions in the genome Single Nucleotide Polymorphism (SNP) markers, which are usually diallelic, have been used to cover the whole genome Single Nucleotide Polymorphism (SNP) markers, which are usually diallelic, have been used to cover the whole genome Two categories of tests have been applied to these data Two categories of tests have been applied to these data single marker association tests, which examine association between affection status and the SNP data one SNP at a time single marker association tests, which examine association between affection status and the SNP data one SNP at a time multi-marker association tests, which examine association between affection status and multiple SNP data simultaneously multi-marker association tests, which examine association between affection status and multiple SNP data simultaneously 3

Allele HWDLD Association Analysis Information for association g. phase-known genotype- based test a bc de f g a.Allele frequency trend test b.HWD trend test c.LD contrast test d. genotype frequency test e.haplotype-based test with HWE f. ???

The allele frequency, HWD and LD contrast tests are typically developed in what has been termed a retrospective context; i.e. case-control status is considered fixed and the genotypes are considered random The allele frequency, HWD and LD contrast tests are typically developed in what has been termed a retrospective context; i.e. case-control status is considered fixed and the genotypes are considered random For case-control data, epidemiologists typically take advantage of the properties of the odds ratio and use the prospective logistic regression model, making the case-control status the random variable dependent on the predictors For case-control data, epidemiologists typically take advantage of the properties of the odds ratio and use the prospective logistic regression model, making the case-control status the random variable dependent on the predictors Prospective modeling tends to allow for greater flexibility, especially when adjusting for covariates Prospective modeling tends to allow for greater flexibility, especially when adjusting for covariates It also provides a natural way to adjust for any correlations between the tests or other covariates, and can be extended to quantitative traits It also provides a natural way to adjust for any correlations between the tests or other covariates, and can be extended to quantitative traits 5

Notation and Assumptions We suppose there are two diallelic SNP markers, A and B having alleles {A 1,A 2 } and {B 1,B 2 }, respectively, where A 1 and B 1 are the minor alleles We suppose there are two diallelic SNP markers, A and B having alleles {A 1,A 2 } and {B 1,B 2 }, respectively, where A 1 and B 1 are the minor alleles X = 1 for A 1 A 1 1 for B 1 B 1 0 for A 1 A 2, Y = 0 for B 1 B 2 for A 2 A 2 for B 2 B 2 I case and I ctrl denote the sets of cases and controls I case and I ctrl denote the sets of cases and controls We make minimal assumptions about the general population sampled; in particular, we do not assume HWE in the population We make minimal assumptions about the general population sampled; in particular, we do not assume HWE in the population μ X, and σ XY denote the expected value of X, the variance of X and the covariance of X and Y, respectively μ X, and σ XY denote the expected value of X, the variance of X and the covariance of X and Y, respectively 6

The HWD parameter for marker A is given by The HWD parameter for marker A is given by The HWD parameter can be expressed as The HWD parameter can be expressed as This means that the HWD parameter, d A, is half the deviation of the variance from the variance expected under HWE This means that the HWD parameter, d A, is half the deviation of the variance from the variance expected under HWE The composite LD parameter for alleles A 1 and B 1 of markers A and B is The composite LD parameter for alleles A 1 and B 1 of markers A and B is 7

Probabilities for unphased genotypes 8

The joint test of allele frequency and HWD contrasts between cases and controls tests the null hypothesis H 0 : (p A|case d A|case ) = (p A|ctrl d A|ctrl ) The joint test of allele frequency and HWD contrasts between cases and controls tests the null hypothesis H 0 : (p A|case d A|case ) = (p A|ctrl d A|ctrl ) Let Z i = (X i )’; the sample mean Z is a sufficient statistic for (p A d A )’ Let Z i = (X i )’; the sample mean Z is a sufficient statistic for (p A d A )’ The Allelic-HWD contrast test can be performed by comparing Z case and Z ctrl. The T 2 statistic for this test is The Allelic-HWD contrast test can be performed by comparing Z case and Z ctrl. The T 2 statistic for this test is _ _ _ 9

Let Z i = (X i Y i X i Y i )’; is a sufficient statistic for (p A p B Δ) ’ Let Z i = (X i Y i X i Y i )’; is a sufficient statistic for (p A p B Δ) ’ Z _ The Allelic-LD contrast test can be performed using a version of Hotelling’s T 2 The Allelic-LD contrast test can be performed using a version of Hotelling’s T 2 The additional case-control differences can be captured by the HWD and LD contrast tests, given the allele frequency contrast(s) The additional case-control differences can be captured by the HWD and LD contrast tests, given the allele frequency contrast(s) The Allelic-HWD-LD contrast test can be constructed in a similar manner by contrasting the mean vector of Z i = (X i Y i X i Y i )’ between cases and controls The Allelic-HWD-LD contrast test can be constructed in a similar manner by contrasting the mean vector of Z i = (X i Y i X i Y i )’ between cases and controls 10

Single-marker and two-marker association tests with corresponding models and hypotheses 11

Multistage Tests “Self-replication” if the tests are independent “Self-replication” if the tests are independent Sequential tests Sequential tests E.g. The HWD contrast test adjusted for allele frequency information which is used in the first stage can be performed by the test of 12

Penetrance Model and True Marker Association Model Let D denote the disease genotype variable coded as Let D denote the disease genotype variable coded as D = 1 for D 1 D 1 0 for D 1 D 2 for D 2 D 2 We write the penetrance model as: We write the penetrance model as: 13

Constraints for disease models 14

Given the true disease model and the LD structure, we can set up the true single-marker association model between the phenotype and single-marker data X: Given the true disease model and the LD structure, we can set up the true single-marker association model between the phenotype and single-marker data X: This true association model has the same form as the penetrance model This true association model has the same form as the penetrance model When (1 – 2p D ) - ≠ 0, the coefficient of the When (1 – 2p D ) - ≠ 0, the coefficient of the quadratic terms generally approaches 0 faster than does that of the linear term quadratic terms generally approaches 0 faster than does that of the linear term γDγDγD2γD2γDγDγD2γD2 15

Power Computation T 2 test in a retrospective model and the score test and LRT in a prospective logistic model are expected to perform similarly T 2 test in a retrospective model and the score test and LRT in a prospective logistic model are expected to perform similarly The noncentrality parameter of the T 2 test for test 2-5 is The noncentrality parameter of the T 2 test for test 2-5 is 16 The noncentrality parameters for the other tests can be obtained by using the corresponding sub-matrices of (μ case – μ ctrl ) and (Σ case + Σ ctrl ) The noncentrality parameters for the other tests can be obtained by using the corresponding sub-matrices of (μ case – μ ctrl ) and (Σ case + Σ ctrl ) Then Then

Comparisons of theoretical and empirical power of test 1-2 For each of the four disease models, parameters were set as follows: p D = 0.2, p A = 0.3, K = 0.05, D XD = 0.048(D’ = 0.8), n = 2,000 (500 for recessive), α = 0.05/500,000 Empirical power is obtained by the ratio of the number of rejected replicates to the total 100,000 replicates. 17

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Power comparisons of two-marker tests 19

Power Comparisons on Real Data We estimated LD parameters and marker allele frequencies from the HapMap CEU population We estimated LD parameters and marker allele frequencies from the HapMap CEU population The data consist of 120 haplotypes estimated from 30 parent-offspring trios The data consist of 120 haplotypes estimated from 30 parent-offspring trios We split chromosome 11 into mutually exclusive consecutive regions containing 3 SNPs each We split chromosome 11 into mutually exclusive consecutive regions containing 3 SNPs each For each region we estimated the LD and allele frequency parameters For each region we estimated the LD and allele frequency parameters We excluded regions where the minor allele frequencies of three consecutive markers were less than 0.1, leaving 4,648 regions We excluded regions where the minor allele frequencies of three consecutive markers were less than 0.1, leaving 4,648 regions We chose the disease SNP to be the one with the smallest allele frequency We chose the disease SNP to be the one with the smallest allele frequency Parameters other than the allele frequency and LD parameters were set to be the same as before Parameters other than the allele frequency and LD parameters were set to be the same as before 20

Mean of power over chromosome 11 of CEU HapMap data 21

Conclusions The best two marker test always appear to be more powerful than either the best single- marker test or the haplotype-based test The best two marker test always appear to be more powerful than either the best single- marker test or the haplotype-based test It should be possible, by examining the LD structure of the markers, to predict which will be the best two-marker test to perform It should be possible, by examining the LD structure of the markers, to predict which will be the best two-marker test to perform We need to study > two marker tests We need to study > two marker tests 22