Cross-phenotype association analysis using summary statistics from GWAS

Abstract

For over a decade, genome-wide association studies (GWAS) have been a major tool for detecting genetic variants underlying complex traits. Recent studies have demonstrated that the same variant or gene can be associated with multiple traits, and such associations are termed cross-phenotype (CP) associations. CP association analysis can improve statistical power by searching for variants that contribute to multiple traits, which is often relevant to pleiotropy. In this chapter, we discuss existing statistical methods for analyzing association between a single marker and multivariate phenotypes, we introduce a general approach, CPASSOC, to detect the CP associations, and explain how to conduct the analysis in practice.

1. Introduction

In the last decade, Genome-wide association studies (GWAS) have identified more than 8500 associations with complex traits such as heart disease, diabetes, auto-immune diseases and psychiatric disorders, etc, and have provided a wealth of novel insights into the etiology of human diseases and traits (https://www.genome.gov/gwastudies/). More and more empirical evidence suggests that many detected genetic loci harbor variants that are associated with multiple, sometimes seemingly distinct, traits (1, 2). Such associations are termed cross-phenotype (CP) associations and are potential evidence for pleiotropy, i.e. that the same variants are associated with multiple traits. For example, a gene desert located on chromosome 8q24 is associated with multiple cancer types, and multiple GWAS have demonstrated that this region is associated with prostate, breast, colon, ovarian, and bladder cancers and chronic lymphocytic leukemia (3–7). Recent studies (8–11) also show that type I diabetes (T1D), rheumatoid arthritis, systemic lupus erythematosus (SLE) and Graves disease share a functional variant in PTPN22, which encodes the lymphoid PTP Lyp gene (12). Those CP associations indicate the sharing of the same susceptibility genes and alleles, suggesting a role for pleiotropic genetic effects (13).

In general, pleiotropy is defined as a genetic variant truly affecting multiple phenotypes and is one of the underlying causes for an observed CP association (13). A CP association can occur in the context of: 1) biological pleiotropy, a SNP or a gene affects multiple phenotypes; 2) phenotype causal relationships, often called mediated pleiotropy or referred to as the situation where the genetic effect on phenotype B is mediated by phenotype A, which is causally related to phenotype B; 3) spurious associations caused by misclassifications or ascertainment bias. CP association implies potential pleiotropy where a variant is associated with multiple traits regardless of the underlying cause. Thus, CP association is more general than pleiotropy.

To date, GWAS are usually performed one trait at a time, although multiple related phenotypes with a common physiological process are often collected and studied. When cross-phenotype effects exist, focusing on single traits will miss the opportunity to systematically integrate the phenome-wide data available for genetic association analysis. On the other hand, analyzing multiple phenotypes can further improve statistical power in detecting significant genetic loci and provide a biological interpretation. Statistical methods to detect CP effects have been developed from different angles. They can be broadly classified into multivariate analyses and univariate analyses (summarized in Table 1).

Table 1.

Methods for detecting CP associations.

Methods	References	Input	Allow Overlapping Subjects	Combine data across multiple studies	Account for correlation	Allow heterogeneity effects
GEE	(16, 17)	individual-level data	Yes	No	Yes	No
PC analysis	(21, 22)	individual-level data	Yes	No	Yes	No
CCA	(23)	individual-level data	Yes	No	Yes	No
Fisher’s p value	(40)	P value	No	Yes	No	Yes
CPMA	(27)	P value	No	Yes	No	Yes
Fixed and random effects meta-analysis	(41)	Summary statistics	No	Yes	No	No (fixed effects) Moderate level (Random effects)
subset-based meta-analysis	(28)	Summary statistics	No	Yes	No	Yes
Extensions to O’Brien’s method	(31, 32)	individual-level data	Yes	No	Yes	Yes
CPASSOC	(8, 26, 38)	Summary statistics	Yes	Yes	Yes	Yes

1.1. Multivariate approaches

The joint analysis of multiple phenotypes within a cohort has recently become popular for improving statistical power to detect genetic linkage and association. In such studies, correlated phenotypes, or a multivariate phenotype with several components, may be measured at the same time. For example, a hypertension study often measures systolic blood pressure (SBP), diastolic blood pressure (DBP), and hypertension status (HTN) (14, 15). Most multivariate methods are based on a multivariate regression framework and require both genotypes and phenotypes at the individual level, with an assumption of approximately multivariate normally distributed phenotypes. Linear mixed effects models (LME) are applied using fixed effects for the genetic variants and random effects to account for correlations among the multivariate phenotypes (16, 17). Extensions to allow for non-normally distributed phenotypes and categorical phenotypes have also been developed based on generalized estimating equations (GEEs), ordinal regression, and a Bayesian framework (18–20).

Other approaches have been developed based on a dimension reduction technique on the phenotypes, such as principal-component (PC) analysis and canonical correlation analysis (CCA) (21–23). Variable reduction approaches are in general only applicable to multivariate phenotypes consisting of all continuous phenotypes that are approximately normally distributed. These approaches derive a single or a few new phenotypes that are linear combinations of the original phenotypes. For example, a few top PCs are constructed to preserve the multivariate trait variation, but this may not be able to capture sufficient association (24). In practice, typically not all subjects have been measured for all traits, and when different traits have been investigated in different studies, the individual subject data may not even be available.

1.2. Univariate approaches

An alternative way to analyze multivariate phenotypes is to perform univariate phenotype-genotype association tests for each phenotype individually and then combine the test statistics from the univariate analyses. A key feature of such approaches is the ability to derive the output statistics from SNP summary statistics, hence making it possible to perform systematic meta-analysis-type comparisons across multiple GWAS datasets.

The Fisher’s combined P-value method is a step in this direction. However, it suffers from several limitations, such as an inability to provide the summary effect of a genetic variant and difficulty in addressing genetic heterogeneity. Moreover, it is designed for independent studies and therefore it is not straightforward when aggregating P-values of different but correlated phenotypes within the same cohort, which may result in inflated type I error.

The success of GWAS has shown that there is considerable benefit in being able to derive association tests on the basis of summary statistics through meta-analysis (25) and many association tests have been successfully extended to combine association across multiple phenotypes (26–28). The techniques developed for meta-analysis can be applied to summary data, thus diminishing the limitations that are imposed by restrictions on sharing individual-level data. Meta-analytical approaches aggregate summary statistics from individual studies into one statistic to test for CP effects, the evidence for association is combined across studies of multiple phenotypes. The null hypothesis is therefore that there is no association between a genetic marker and any of the phenotypes, and the alternative hypothesis is that the genetic maker is associated with at least one of the studied phenotypes.

The cross-phenotype meta-analysis (CPMA) statistic aims to detect multiple associations at a marker across different diseases that may share a common genetic background or involve a common biological process (27). The null hypothesis is that the expected P-values are uniformly distributed, and hence the values of $-ln\left(P\right)$ are exponentially decaying with a decay rate $\lambda =1$. The test statistic is expressed as a likelihood ratio test: $CPMA=-2\frac{P\left[Data|\lambda =1\right]}{P\left[Data|\lambda =\widehat{\lambda }\right]}$ and is asymptotically distributed as ${\chi }^2$ with one degree of freedom. It determines evidence for the hypothesis that each independent SNP has multiple phenotypic associations, rather than directly evaluating the aggregated association evidence between a SNP and multiple phenotypes. When one of the traits is not associated with a SNP, CPMA will have less statistical power. Moreover, CPMA assumes that the P-values used for the individual traits come from different non-overlapping cohorts, and it does not allow for overlapping or correlated samples among studies.

Fixed and random effects meta-analysis methods can also be used to detect CP effects. Fixed effects meta-analysis assumes the genetic variant has the same effect across multiple phenotypes. Random effects meta-analysis allows a moderate level of effect heterogeneity, but nevertheless it is not well suited to situations where a genetic variant has opposite effects on different phenotypes, because of loss of power (13, 29). An extension of fixed effects meta-analysis is the subset-based meta-analysis, which allows for opposite effects and is able to test association to a subset of phenotypes (28). This method exhaustively searches all possible phenotype subsets and identifies the subset of traits with the strongest association, but at the cost of exponentially increased multiple tests. In addition, the method does not allow for heterogeneity either across cohorts or for the same phenotype.

O’Brien’s linear combination test has been developed based on a weighted linear combination of the univariate test statistics (30). The power of O’Brien’s method depends on the assumption of homogeneous genetic effect across phenotypes. Some authors have proposed extensions to O’Brien’s approach to allow for heterogeneity among individual test statistics (31, 32). However, these approaches were specifically developed for correlated traits measured on the same individuals.

1.3. CPASSOC

CPASSOC was first applied to the Continental Origins and Genetic Epidemiology Network (COGENT) African ancestry samples for three blood pressure traits (26). In this study CPASSOC identified four loci associated with HTN at genome wide significance level that were missed by a single trait analysis in the original report. All the four loci, CHIC2, HOXA-EVX1, IGFBP1/IGFBP3 and CDH17 have been shown by others to be associated with HTN-related traits in European ancestry GWAS (15, 33–37), suggesting that our method reliably identifies true signals. Six additional loci with suggestive association evidence (P<5.0×10⁻⁷) were also observed, including CACNA1D and WNT3. CPASSOC was also applied to anthropometric traits, such as height, BMI, and waist-to-hip ratio adjusted for BMI (WHRadjBMI), where the summary statistics were obtained from the GIANT consortium (38). Those applications strongly suggest that analyzing multiple phenotypes can improve statistical power and that such an analysis can be executed with the summary statistics from GWAS.

CPASSOC uses summary-level data from single SNP-trait associations from GWAS to detect which variant is associated with at least one trait. This method improves statistical power by analyzing multiple phenotypes. CPASSOC provides two statistics, S_Hom and S_Het. S_Hom is similar to the fixed effect meta-analysis method (39) but accounting for the correlations of summary statistics among traits and among cohorts induced by correlated traits, potential overlapping or related samples.

In brief, we assume we have summary statistics of GWAS from J cohorts with K phenotypic traits. In each cohort, single SNP-trait association has been analyzed for each trait separately. Let $T_{jk}$ be a summary statistic for a SNP, derived from the $j^{th}$ cohort for the $k^{th}$ trait. Let $T={\left(T_{11},\cdots ,T_{J1},\cdots ,T_{1K},\cdots ,T_{JK}\right)}^T$ represent a vector of test statistics for testing the association of a SNP with K traits. We use a Wald test statistic $T_{jk}=\frac{{\widehat{\beta }}_{jk}}{{\widehat{s}}_{jk}}$, where ${\widehat{\beta }}_{jk}$ and ${\widehat{s}}_{jk}$ are the estimated coefficient and corresponding standard error for the k^th trait in the j^th cohort. S_Hom is then defined as

$$S_{Hom}=\frac{e^T{\left(RW\right)}^{-1}T{\left(e^T{\left(RW\right)}^{-1}T\right)}^T}{e^T{\left(WRW\right)}^{-1}e},$$ (1)

which follows a ${\chi }^2$ distribution with one degree of freedom, where $e^T=\left(1,\dots ,1\right)$ has length $J\times K$ and $W$ is a diagonal matrix of weights for the individual test statistics. We used the sample sizes for the weights, i.e., $w_{jk}=\sqrt{n_j}$ for the sample size $n_j$ of the j^th cohort.

To define S_Het, we first let

$$S\left(\tau \right)=\frac{e^T{\left(R\left(\tau \right)W\left(\tau \right)\right)}^{-1}T\left(\tau \right){\left(e^T{\left(R\left(\tau \right)W\left(\tau \right)\right)}^{-1}T\left(\tau \right)\right)}^T}{e^{T}W{\left(\tau \right)}^{-1}R{\left(\tau \right)}^{-1}W{\left(\tau \right)}^{-1}e},$$

where $T\left(\tau \right)$ is the subvector of $T$ satisfying $\left|T_{jk}\right|>\tau$ for a given $\tau >0$, $R\left(\tau \right)$ is a sub-matrix of $R$ representing the correlation matrix, and $W\left(\tau \right)$ is the diagonal submatrix of $W$, corresponding to $T\left(\tau \right)$. Here we let $w_{jk}=\sqrt{n_j}\times sign\left(T_{jk}\right)$. Then the test statistic is $S_{Het}={\max }_{\tau >0}S\left(\tau \right)$.

The asymptotic distribution of S_Het does not follow a standard distribution but can be evaluated using simulation. S_Het is an extension of S_Hom that improves power when the genetic effect sizes vary for different traits. The distribution of S_Het under the null hypothesis can be obtained through simulations or approximated by an estimated beta distribution.

1.4. Discussion

We have reviewed several existing methods for detecting cross phenotype associations. Choosing an appropriate statistical approach depends on the study design, phenotypes, data availability and genetic heterogeneity. Systematic simultaneous analysis of multiple traits could improve the quality of inferences from analysis of outcomes that all relate to the biological construct of interest. Compared to multivariate approaches that model all outcomes simultaneously, combining statistics from the univariate analysis has the advantages of involving fewer assumptions about the relationships among individual phenotypes. For most published GWAS, obtaining summary data is substantially easier than accessing individual-level phenotype and genotype data. Compared with existing methods, CPASSOC has multiple advantages for identifying cross-phenotype associations. It can accommodate opposite risk effects and different types of phenotypic traits, either correlated, independent, continuous, or binary traits. It is well suited for overlapping or related subjects within and among different studies or cohorts. Since this approach accounts for the correlations of test statistics among traits or cohorts, S_Hom and S_Het are able to control the effect of cryptic relatedness occurring among cohorts.

2. Methods

In this section, we will illustrate the procedure for conducting CPASSOC analysis with the sex-specific summary statistics of the three traits: height, BMI and WHRadjBMI (38), which were downloaded from the GIANT consortium website (https://www.broadinstitute.org/collaboration/giant/index.php/GIANT_consortium_data_files).

The data we downloaded include sex-specific GWAS meta-analysis summary statistics from the discovery phase, including the SNP estimated effect sizes and their corresponding standard errors. For the discovery stage in sex-specific studies, 46 studies (up to 60,586 males, 73,137 females) were included on height and BMI, and 32 studies (up to 34,629 males, 42,969 females) were included on WHR. To apply CPASSOC for combining all the sex-specific summary statistics on height, BMI and WHRadjBMI, we chose 2,723,278 SNPs available for all three traits.

2.1. Step 1. Estimating the correlation matrix

The CPASSOC package is downloaded from http://hal.case.edu/zhu-web/. To perform CPASSOC analysis, a correlation matrix is required to account for the correlations among phenotypes or induced by overlapping or related samples from different cohorts. The correlations can be estimated from the summary statistics for all the independent SNPs in a genome-wide association study. Zhu et al. (26) suggested using a set of SNPs in linkage equilibrium to estimate the correlation coefficients. Table 2 gives an example of the estimated correlations among the three sex-specific traits: height, BMI and WHRadjBMI. Since only summary statistics are available in GIANT, we borrowed the linkage disequilibrium (LD) pattern from the ARIC European American (EA) data (downloaded from dbGaP http://www.ncbi.nlm.nih.gov/gap). The SNP set is obtained from applying pairwise LD pruning with r²=0.2 ARIC EA data using the software PLINK (http://pngu.mgh.harvard.edu/purcell/plink/). SNPs with large effect sizes may represent true association, and consequently may inflate correlations among summary statistics. Therefore, we removed SNPs whose summary statistics Z scores were greater than 1.96 or less than −1.96 (See Note 1). Below is an example of R code showing how to estimate the correlation matrix.

# read the data: the summary statistics data file (“HeightBMIWHRadjBMI.txt”) for CPASSOC includes seven columns (SNP ID, height Z-score for male, height Z-score for female, BMI Z-score for male, BMI Z-score for female, WHRadjBMI Z-score for male, WHRadjBMI Z-score for female. The SNP list file after pruning based on ARIC data is “ARICdot2.prune.in”.
Dt<-read.table(“HeightBMIWHRadjBMI.txt”,header=T)
# extract the SNP set obtained from ARIC EA data.
pruned.lst<-read.table(“ARICdot2.prune.in”)
Dtcorr<-Dt[Dt$MarkerName%in%pruned.lst$V1,]
# Remove SNPs with Z-score larger than 1.96 or smaller than −1.96, assuming the summary statistics for 6 sex-specific traits are named Z1,Z2,…,Z6.
Index<-which(abs(Dtcorr$Z1)>1.96 | abs(Dtcorr$Z2)>1.96 |abs(Dtcorr$Z3)>1.96 | abs(Dtcorr$Z4)>1.96 |abs(Dtcorr$Z5)> 1.96 | abs(Dtcorr$Z6)> 1.96)
Dtcorr<-Dtcorr[-Index,]
corMatrix<-diag(x=1,ncol=6,nrow=6)
for(i in 1:5){
for(j in (i+1):6)
corMatrix[i,j]=corMatrix[j,i]=cor(Dtcorr[,i+1],Dtcorr[,j+1])
}

Table 2. Correlations for sex-specific cohorts for height, BMI, and waist-to-hip ratio adjusted for BMI.

The SNP sets for correlation estimation include 81,322 SNPs for height, 82,012 SNPs for BMI, and 81,130 SNPs for WHRadjBMI from the GIANT consortium studies.

		Height		BMI		WHRadjBMI
		Male	Female	Male	Female	Male	Female
Height	Male	1	0.128	−0.038	−0.007	−0.004	−0.002
	Female	0.128	1	−0.012	−0.056	0.007	−0.003
BMI	Male	−0.038	−0.012	1	0.096	0.010	−0.007
	Female	−0.007	−0.056	0.096	1	0.011	0.001
WHRadjBMI	Male	−0.004	0.007	0.010	0.011	1	0.031
	Female	−0.002	−0.003	−0.007	0.001	0.031	1

2.2. Step 2. Perform CPASSOC analysis

The input file for CPASSOC software is an $M\times K$ matrix of summary statistics, where $M$ is the number of SNPs and $K$ is the number of summary statistics to be combined, with M = 2,723,278 and N = 6 in our example with anthropometric traits. Each column consists of summary statistics of $M$ SNPs for one trait, SNP with missing values should be excluded (see Note 2). If multiple traits have been analyzed in a cohort, each phenotype’s summary statistics will be put in one column. Each row represents a SNP. An example of the input file is presented in Table 3.

Table 3. The input file for CPASSOC.

This is the start of the $M\times K$ matrix of summary statistics (M = 2,723,278 and N = 6). Each column represents the summary statistics from $M$ SNPs for one trait. Each row represents a SNP.

rs10	0.404545	−0.00455	0.326087	−0.0087	−0.0087	−1.04348
rs1000000	1.486486	0.362319	−0.67532	−0.51351	−0.525	−0.60759
rs10000010	0.491803	−0.07018	0.3125	−0.72581	1.911765	2.575758
rs10000012	2.247191	0.313253	1.914894	1.222222	−0.61	0.387755
	…

CPASSOC preforms two tests: S_Hom and S_Het. S_Hom is more powerful when heterogeneity is not present, while S_Het allows for trait heterogeneity effects. The two tests can be performed using R code, as described below:

• 1Perform S_Hom test.

source(“FunctionSet.R”)
# Extract the M×K matrix of summary statistics.
Sumstat<-subset(Dt,select=c(“Z1”,”Z2”,”Z3”,”Z4”,”Z5”,”Z6”))
## Because each SNP may have a different sample size in the GIANT consortium data, we used the median sample size across SNPs for each trait (see Note 3).
Samplesize<-c(60491.2,73046.2,58603.5,67938.6,34596.1,42730.2)
Test.shom<-SHom(Sumstat,Samplesize,corMatrix)
# obtain P-values of SHom
p.shom<-pchisq(Test.shom, df = 1, ncp = 0, lower.tail = F)
res.shom<-data.frame(Dt[,”MarkerName”],p.shom)
names(res.shom)<-c(“MarkerName”,”p.Shom”)

• 2Perform S_Het test. The empirical distribution of S_Het is approximated by a gamma distribution with a mean shift and we use simulations to estimate this gamma distribution. The three parameters: shape k, scale theta and shift c, are dependent on the trait correlations and the corresponding summary statistics. When there are no missing values, they can be estimated by calling the function EstimateGamma. When the dimension of CorrMatrix is large, directly using the empirical distribution is desirable (see Note 4). If we deal with a large dataset, it will be very useful to split the data into small pieces first.

#Create 20 equally size folds
folds <- cut(seq(1,nrow(Sumstat)),breaks=20,labels=FALSE)
res.shet=list()
#estimate parameters of gamma distribution.
para = EstimateGamma (N = 1E4, Samplesize, corMatrix) ;
for(i in 1:20){
#Segement your data by fold using the which() function
Indexes<- which(folds==i,arr.ind=TRUE)
Test.shet<-SHet(Sumstat,Samplesize,corMatrix)
# obtain P-values of SHet using the estimated gamma parameters
p.shet = pgamma(q = Test.shet-para[3], shape = para[1], scale = para[2], lower.tail = F);
res.shet[[i]]<-data.frame(Sumstat[Indexes,”MarkerName”],p.shet)
}
res.shet <-do.call(rbind.data.frame,res.shet)
names(res.shet)<-c(“MarkerName”,”p.Shet”)

The output of both tests is a vector of P-values for $M$ SNPs. Because CPASSOC combines multiple traits, we can apply the significant level P = 5 × 10⁻⁸ as in GWAS. We examined loci that reached genome-wide significance (P < 5 × 10⁻⁸) by CPASSOC from sex-specific data, as presented in Fig 1. To do this, for a SNP reaching P < 5 × 10⁻⁸ by either S_Hom or S_Het, we examined the region within 500 kb of each side of the SNP. The SNP was considered to be identified only by CPASSOC if no SNPs reached P < 5 × 10⁻⁸ by the conventional meta-analysis from the discovery phase data in the 1.0 Mb region, and the SNP is not in LD with sentinel SNPs from the GIANT studies.

Figure 1. Manhattan plots of S_Hom and S_Het for combining three sex specific traits (38).

The loci reaching genome-wide significance by S_Hom and S_Het, but not by the conventional meta-analysis, are marked with corresponding SNP names. (A) S_Hom (B) S_Het.