A General Framework for Association Tests With Multivariate Traits in Large-Scale Genomics Studies

Calculating Score Statistics and Their Covariance Matrix

We consider a single study with a total of n unrelated individuals, K (potentially correlated) traits, and p covariates (including the unit component). For and , let be the kth trait of the ith individual. For , let be the p-vector of covariates for the ith individual, and let denote the number of minor alleles (or the imputed dosage) the ith individual carries at a particular test locus.

We assume that the marginal distribution of is related to and through a generalized linear model with mean and dispersion parameter , where is a specific function, and and are unknown regression parameters. We adopt natural link functions such that for continuous traits and for binary traits.

To accommodate missing data, we let indicate, by the values 1 versus 0, whether is observed or missing, and let indicate, by the values 1 versus 0, whether is observed or missing. It is assumed that the covariates have no missing values. (We recommend to exclude the covariates with substantial missingness and to replace the missing values with their sample means for the remaining covariates.)

The score function for takes the form

Thus, the score statistic for testing the null hypothesis that is

where solves the equation

and is a sample estimator of . For continuous traits,

for binary traits, . Note that the construction of the s makes full use of the available data by estimating and from all individuals with nonmissing trait values and is more efficient than the traditional complete-case analysis.

By taking the Taylor series expansion of at and applying the law of large numbers, we can show that is asymptotically equivalent to the following sum of n-independent terms:

where and are the limits of and , respectively, and . Define the score vector

It follows from the multivariate central limit theorem that U is asymptotically K-variate normal with mean 0 and with a covariance matrix that can be estimated by

where

and

Note that and do not depend on the SNP genotypes and thus need to be calculated only once (before looping through all the SNPs). Note also that, given the s, the calculations of U and V do not involve solving any equations. Thus, the implementation of the proposed score-type statistics is orders of magnitude faster than that of the conventional Wald statistics. In addition, the score-type statistics are numerically more stable and statistically more accurate than the Wald statistics, especially when the minor allele frequency (MAF) is low [Lin and Tang, 2011].

We now extend the above results to family studies. Suppose that we have a total of n families, with members in the ith family. For , and , let denote the kth trait for the jth member of the ith family, denote the p-vector of covariates for the jth member of the ith family, and denote the number of minor alleles (or the imputed dosage) which the jth member of the ith family carries at a particular test locus. We assume that the marginal distribution of is related to and through the same marginal generalized linear regression model as in the case of unrelated individuals.

Let indicate whether is observed or missing, and let indicate whether is observed or missing. It is assumed that there are no missing values in the covariates. Under the independence working assumption [Liang and Zeger, 1986], the (pseudo-likelihood) score statistic for testing the null hypothesis that is

where solves the equation

for continuous traits, and for binary traits. Again, define . It follows from the above arguments for the case of unrelated individuals that U is asymptotically K-variate normal with mean 0 and a covariance matrix that can be estimated by , where

Note that the relatedness of family members is accounted for through the empirical correlations of the s.

Performing Multivariate Association Tests

To test the global null hypothesis that , we calculate the quadratic form

which is asymptotically chi-squared with K degrees of freedom. This is a global test statistic that is consistent (i.e., having the power of 1 as the sample size tends to ∞) against any alternative hypotheses.

To enhance power against alternative hypotheses under which genetic effects are similar among the K studies, we calculate a test statistic with one degree of freedom along the lines of O'Brien [1984]. Specifically, let Z be the standardized version of U and let R be the correlation matrix of U. That is, and . We then calculate

where . This test statistic is asymptotically standard normal.

The test statistic T maximizes the noncentrality parameter among all linear combinations of the s. Note that the score test statistic is asymptotically equivalent to the Wald test statistic, i.e., the estimate of divided by its standard error. Thus, T is optimal if the limits of the s or the standardized s are the same. To detect alternative hypotheses under which the original s are the same, we define and . Note that the limit of is approximately . We then calculate

where and . This test statistic is also asymptotically standard normal. When using this test statistic, it is important to use comparable scales for the K traits such that it is plausible for the to be equal. When using either T or , it is important to code the trait values in such a way that the genetic effects on the K traits are plausibly in the same direction.

If the effects of the SNP are similar among the K traits, then T and will tend to be more powerful than Q. If the effects are very different, then Q will likely be more powerful than T and . In the Appendix, we derive the asymptotic distributions of Q, T, and under alternative hypotheses for the important special case of two continuous traits.

Combining Results From Multiple Studies

We wish to combine results from L-independent studies. For , let and denote the score vector and its (estimated) covariance matrix from the lth study. Then the overall score vector is , and its covariance matrix is estimated by . Note that is the (pseudo-likelihood) score statistic in the joint analysis of the individual-level data of the L studies (allowing nuisance parameters to be different among the studies). Thus, meta-analysis of score statistics is equivalent to the joint analysis of individual-level data. When there are multiple studies, K pertains to the total number of distinct traits, some of which may not be measured in certain studies. (For , we may have four traits that are common between the two studies, two traits that are measured only in the first study, and three traits that are measured only in the second study. Then .) Given U and V, we can calculate Q, T, and in the same manner as in the case of a single study.

Determining Genome-Wide Significance

Suppose that we have a total of m SNPs. For , let be the value of Q for the jth SNP. If the critical value q₀ for the m test statistics satisfies

then the family-wise error rate will be α. We estimate q₀ by Monte Carlo simulation. At each test locus, we calculate

where are independent standard normal random variables. Let and be the values of and V for the jth SNP. Define

The joint distribution of can be approximated by that of [Lin, 2005]. Thus, we determine q₀ by the following equation

We simulate the normal random sample 10,000 times while holding the observed data fixed and set q₀ to be the 10,000th largest value of the resulting 's. We may convert the critical value q₀ to the P-value threshold p₀ by referring q₀ to the chi-squared distribution with K degrees of freedom. We can determine the genome-wide significance thresholds for T, and in a similar manner.

Simulation Studies

We conducted simulation studies to evaluate the performance of the proposed test statistics. We set G to be the number of minor alleles for a SNP with MAF of 0.4 and set X to be normal with mean 0.1G and unit variance. We generated two continuous traits under the bivariate linear model: and where ε₁ and ε₂ are zero-mean normal with variances and , respectively, and with correlation . We set α to 10⁻⁴. To evaluate the type I error, we simulated 10 million data sets under . To evaluate the power, we simulated 10,000 data sets under various combinations of γ₁ and γ₂. Each simulated data set consists of 1,000 unrelated individuals. In addition to the three multivariate test statistics, and , we considered two versions of univariate tests, Uni-B and Uni-corr, which adjust for multiple testing (between the traits) by adopting the Bonferroni correction (i.e., dividing the nominal significance level by the number of traits) and by accounting for the correlation between Z₁ and Z₂ (using the multivariate normal distribution of U), respectively. We also included the TATES method (van der Sluis et al., 2013).

The results are summarized in Table 1. The type I error for the TATES is inflated by about 12%. The type I errors for the other five tests are below the nominal significance level. The Q test has reasonable power against all 12 alternatives. As expected, is more powerful than the other tests when γ₁ is close to γ₂, and T is more powerful than the others when and are close to each other. The Uni-B is expected to have lower power than Uni-corr, but the two tests perform very similarly due to the relatively weak correlation between the two traits. The differences between the two tests become more pronounced as the correlation increases; see supplementary Table SI. The TATES has slightly higher power than Uni-B and Uni-corr but also has inflated type I error, especially when the correlation is high.

Table 1. Type I error and power of univariate versus multivariate tests for two continuous traits ()

		Uni-B	Uni-corr	TATES	Q	T
(0, 0)	(0, 0)	8.6	8.7	1.12	9.0	9.0	8.8
(0.3, 0)	(0.21, 0)	0.6861	0.6865	0.714	0.854	0.095	0.003
(0.3, 0.1)	(0.21, 0.1)	0.6865	0.6869	0.715	0.638	0.487	0.187
(0.25, 0.18)	(0.18, 0.18)	0.5714	0.5723	0.606	0.594	0.700	0.641
(0.3, 0.25)	(0.21, 0.25)	0.9358	0.936	0.945	0.942	0.967	0.966
(0.2, 0.2)	(0.14, 0.2)	0.6222	0.6229	0.651	0.594	0.632	0.702
(0.2, 0.25)	(0.14, 0.25)	0.9054	0.9056	0.917	0.881	0.828	0.922
(0.25, 0.25)	(0.18, 0.25)	0.9128	0.913	0.925	0.906	0.917	0.947
(0, 0.25)	(0, 0.25)	0.9034	0.9036	0.915	0.977	0.197	0.701
(0, 0.3)	(0, 0.3)	0.9902	0.9903	0.992	0.999	0.402	0.914
(0.1, 0.25)	(0.07, 0.25)	0.9034	0.9036	0.915	0.907	0.523	0.841
(0.1, 0.3)	(0.07, 0.3)	0.9902	0.9903	0.992	0.994	0.742	0.968
(0.2, 0.3)	(0.14, 0.3)	0.9903	0.9904	0.992	0.987	0.940	0.990

We also considered the mixture of a binary trait and a continuous trait. We simulated the binary trait under the logistic regression model and simulated the continuous trait under the linear model , where ε is normal with mean 2Y₁ and unit variance. (The Pearson correlation between the two traits is about 0.65.) As shown in Table 2, Q tends to have higher power than the two univariate tests. As expected, is more powerful than the other tests when , and T outperforms the others when the means of Z₁ and Z₂ are similar. Again, the TATES has higher power than Uni-B and Uni-corr but at the expense of inflated type I error.

Table 2. Type I error and power of univariate versus multivariate tests for one binary and one continuous traits

	*	Uni-B	Uni-corr	TATES	Q	T
(0, 0)	—	8.7	9.0	1.02	8.7	9.7	9.3
(0.3, 0)	(3.07, 2.09)	0.171	0.173	0.171	0.141	0.141	0.026
(0.3, 0.1)	(3.07, 3.65)	0.377	0.380	0.398	0.334	0.410	0.382
(0.25, 0.18)	(2.54, 4.52)	0.685	0.687	0.709	0.664	0.488	0.758
(0.3, 0.25)	(3.07, 5.88)	0.973	0.973	0.977	0.975	0.845	0.986
(0.2, 0.2)	(2.02, 4.49)	0.676	0.678	0.700	0.709	0.372	0.764
(0.2, 0.25)	(2.02, 5.24)	0.892	0.893	0.907	0.938	0.534	0.937
(0.25, 0.25)	(2.54, 5.56)	0.943	0.944	0.952	0.958	0.707	0.967
(0, 0.25)	(0.02, 4.02)	0.489	0.493	0.519	0.880	0.046	0.655
(0, 0.3)	(0.02, 4.79)	0.782	0.784	0.808	0.987	0.104	0.888
(0.1, 0.25)	(1.01, 4.62)	0.722	0.725	0.750	0.899	0.210	0.832
(0.1, 0.3)	(1.01, 5.37)	0.917	0.919	0.930	0.990	0.347	0.960
(0.2, 0.3)	(2.02, 5.97)	0.979	0.979	0.984	0.994	0.688	0.992

• * and are the sample means of Z₁ and Z₂, respectively.

We also considered four continuous traits with a compound-symmetry correlation structure for the error terms. The results are shown in supplementary Table SII. The basic conclusions remain the same. We added the MANOVA method implemented in R to the case of no covariates. As shown in supplementary Table SIII, MANOVA has slightly higher type I error and power than the Q test. This is consistent with the general phenomenon that the likelihood ratio test is more liberal than the score test [Lin and Zeng, 2011]. Finally, we considered family studies with 250 families (two parents and two children in each family) and two continuous traits. As shown in supplementary Table SIV, the conclusions are similar to the case of unrelated subjects.

Cardiovascular Studies

We analyzed the GWAS data on the Caucasian samples from the Atherosclerosis Risk in Communities (ARIC) study, the Coronary Artery Risk Development in Young Adults (CARDIA) study, the Cardiovascular Health Study (CHS), the Multi-Ethnic Study of Atherosclerosis (MESA), and the Framingham Heart Study (FHS), the sample sizes being 9,068, 1,433, 3,892, 2,286, and 2,789, respectively. The FHS is a family study, and the others consist of unrelated individuals. Each individual was genotyped on 250,000 SNPs. We considered four cardiovascular traits: diabetes status, high-density lipoprotein (HDL), low-density lipoprotein (LDL), and triglycerides; the first trait is binary whereas the other three are continuous. These traits are major players in the development of coronary artery diseases and metabolic syndrome [Grundy, 2012; Holmes et al., 1981]. We aimed to identify genetic factors underlying these traits. Since the HDL is “good” cholesterol, we used negative values of HDL in the analysis.

We performed single-SNP analysis with the following covariates: age, gender, study centers, and the top 10 principle components for ancestry. We calculated the score-type statistics and their covariance matrices for each study and then combined the results of the five studies. The (unadjusted) P-values of the univariate and multivariate tests are displayed in Figures 1 and 2, respectively. The genome-wide significance thresholds based on the Bonferroni correction and the Monte Carlo procedure are marked in both figures.

We first examine the results based on the Bonferroni correction. For the four traits, more than 10 regions are above the Bonferroni threshold in the Uni-B test (Fig. 1). Compared to Uni-B, the Q test identifies one new signal that is located on chromosome (Chr) 9 (Fig. 2). The signal on Chr9 identified by the Q test is an accumulation of weak/moderate signals for individual traits. The gene at this locus encodes a protein called ABCA1, which is involved in cellular cholesterol removal (Lawn et al., 1999). This gene was previously found to be associated with metabolic syndrome (Avery et al., 2011). Table 3 lists all the loci discovered by the Q test. (The P-values from the univariate-trait analysis are also shown.) The T and tests did not identify any additional signals that achieve genome-wide significance, but the two tests yielded more extreme P-values for several SNPs than the univariate tests.

Table 3. P-values of the genetic loci identified by the Q test

					P-values of single-trait analysis
Chr	SNP	Gene	P-value of Q test		Diabetes	LDL	HDL	Trig
2	rs515135	APOB	6.3		4.0	3.4	5.8	3.1
2	rs1260326	GCKR	2.2		3.7	3.6	6.1	8.9
5	rs12916	HMGCR	1.7		8.2	3.0	2.0	9.4
6	rs10455872	LPA	1.1		9.3	8.7	1.8	8.7
7	rs7777102	MLXIPL	2.2		4.6	8.4	3.8	1.2
8	rs1011685	LPL	5.3		5.2	7.6	2.3	4.6
8	rs2954021	TRIB1	5.9		6.4	1.1	2.0	1.1
9	rs2575876	ABCA1	9.5		2.4	1.9	2.0	4.8
10	rs7903146	TCF7L2	8.4		5.8	8.0	8.3	6.4
11	rs174538	FEN1	3.5		1.9	2.9	3.2	6.6
11	rs964184	ZNF259	2.7		8.6	2.9	3.6	3.3
15	rs1077835	LIPC	1.1		2.2	2.9	5.2	3.1
16	rs247616	CETP	6.7		7.3	5.5	3.5	2.5
18	rs4121823	LIPG	8.0		5.3	7.3	2.4	8.2
19	rs6511720	LDLR	5.8		8.6	3.3	6.6	6.0
19	rs10401969	SUGP1	2.4		5.1	1.1	3.5	1.6
19	rs445925	APOC1	2.1		8.6	2.2	6.3	1.1

Not surprisingly, the Monte Carlo procedure reduced the genome-wide significance thresholds for all tests. For the univariate test on the HDL, one SNP on Chr20 becomes significant by the Monte Carlo criterion. For the T test, one SNP (rs5752792) on Chr22 is above the Monte Carlo threshold. This SNP resides near gene HSCB, which is mainly expressed in liver, muscle and heart [Sun et al., 2003] and is involved in the biogenesis of an elementary metabolic function unit [Rouault and Tong, 2008]. The expression pattern and biological function of HSCB strongly suggest that this gene is pleiotropic.

We have provided a very general and flexible approach to association testing with multivariate traits. An earlier version of this approach (focusing on the Q test for continuous traits) was recently used to successfully identify genes associated with metabolic syndrome [Avery et al., 2011]. The new application presented in this paper further demonstrates the usefulness of the proposed approach. It only took several hours to calculate all the P-values shown in Figures 1 and 2. We have posted our software online at http://dlin.web.unc.edu/software.

When the number of traits is very large, we recommend to reduce the dimension through principal component analysis [Avery et al., 2011]. Although we have focused on main effects of genetic variants, our approach can be easily modified to test gene–environment interactions. It can also be extended to perform burden tests on rare variants (Lin and Tang, 2011).

Univariate-trait analysis and multivariate-trait analysis are complementary to each other. The former is easier to implement and can be used to rapidly screen a large number of genetic variants. The multivariate-trait analysis provides a useful tool to uncover pleiotropic variants that have weak or moderate effects on individual traits. This is particularly important for dissecting the genetic basis of complex diseases, as most of the genetic variants with strong effects and high MAFs might have already been identified.

There is no uniformly most powerful test for analyzing multivariate traits. If the effects of a genetic variant are similar across the traits, then T and are generally preferable. If the effects are considerably different or even in opposite directions, then Q is preferable. The theoretical results of the Appendix offer useful insights into the relative power of the three test statistics and can be used to determine the power and sample size for future studies.

For family data, we adopted the marginal models with an independence working correlation matrix. A more efficient approach would be a random-effect model which utilizes the family relationships. We adopted marginal models instead of random-effects models for several reasons. First, the association tests under marginal models are more robust to model misspecification. Second, it is much faster to fit marginal models than random-effects models. Third, marginal models can easily handle mixtures of continuous and binary traits.

Adjustment for multiple testing is an important issue in genetic association analysis. The Monte Carlo procedure considered in this paper accounts for the correlations among the test statistics and is thus less conservative than the conventional Bonferroni correction. Some existing methods, such as the TATES, may yield inflated type I error. We have focused on determining the genome-wide significance threshold rather than calculating individual adjusted P-values. The former only requires several thousands Monte Carlo samples whereas the latter would entail millions of Monte Carlo samples to estimate extremely small P-values. If the number of SNPs is small, the joint distribution of the test statistics can be evaluated through numerical integration [Conneely and Boehnke, 2007, 2010].