Family-based association test using normal approximation to gene dropping null distribution
© Jiang et al.; licensee BioMed Central Ltd. 2014
Published: 17 June 2014
We derive the analytical mean and variance of the score test statistic in gene-dropping simulations and approximate the null distribution of the test statistic by a normal distribution. We provide insights into the gene-dropping test by decomposing the test statistic into two components: the first component provides information about linkage, and the second component provides information about fine mapping under the linkage peak. We demonstrate our theoretical findings by applying the gene-dropping test to the simulated data set from Genetic Analysis Workshop 18 and comparing its performance with existing population and family-based association tests.
When testing genotype-phenotype association using individuals from extended families, one has to account for correlations in genotypes and/or phenotypes between related individuals. One simple and effective method to account for genotype correlations is to simulate the null genotype distribution by gene dropping , which is simulating founder alleles according to estimated allele frequencies and dropping these alleles down the pedigrees according to random segregation of gametes (i.e., Mendel's first law). The gene-dropping method is straightforward to implement (e.g., implemented in by Allen-Brady et al ) and applies to all pedigree structures, but it is computationally intensive and thus is impractical to use when dealing with millions of single-nucleotide polymorphisms (SNPs).
In this article, we derive the analytical mean and variance of the score test statistic under the gene-dropping setting and approximate the gene-dropping null distribution of the test statistic by a normal distribution with the analytically derived mean and variance. Using this normal approximation, the gene-dropping test becomes computationally efficient and can be easily applied to millions of SNPs.
Furthermore, we provide insights into the gene-dropping test by decomposing the test statistic into two components: the first component resembles a quantity frequently used in variance-component based linkage tests and provides information for linkage, and the second component provides information for fine mapping under the linkage peak. Rabinowitz and Laird , among others, have pointed out the subtle distinction between two types of null hypotheses in family-based association analysis: the null hypothesis of no linkage and no association versus the null hypothesis of no association in the presence of linkage. To test the latter, one needs to condition on the inheritance vector at the test locus . Our decomposition provides an explicit separation of linkage and association information in a family-based study.
We compare the performance of the gene-dropping test (using normal approximation) to association tests using only unrelated individuals and to the family-based association test in the software program FBAT  by analyzing Genetic Analysis Workshop 18 (GAW18) simulated data set.
Preprocessing of genotype data
We analyzed SNPs from chromosome 3 only. At each of the SNPs, we performed Pearson's chi-squared test for the Hardy-Weinberg equilibrium using 142 unrelated individuals. We excluded SNPs that yielded a p-value smaller than 10−4 from our analysis. In the gene-dropping test, we excluded SNPs with estimated minor allele frequency (MAF) smaller than 0.001.
Preprocessing of phenotype data
We focused on the analysis of the quantitative trait systolic blood pressure (SBP) in the simulated data set 1. The true simulation model was known to us . When testing association between genotype doses and trait values (see later discussion), we include factors AGE, SEX, and AGE by SEX interaction as covariates ('s in equation ). Including BPMED as a covariate will overcompensate because BPMED is a consequence of SBP level. Instead, we estimated the effect of BPMED from a regression model with only individuals with hypertension. Because BPMED was randomly assigned to individuals with hypertension, the BPMED effect estimated this way will not be biased by its correlation with SBP. We then adjusted the trait values by subtracting the estimated BPMED effect.
Score tests of genotype-phenotype association using unrelated individuals
and test the null hypothesis . In equation (1), is the vector of trait values (SBP adjusted for the BPMED effect), is a constant vector of baseline mean trait values, coefficients represent the effects of the covariates (e.g., AGE, SEX and AGE by SEX interaction) on trait values, is the vector of genotype doses (the number of minor alleles possessed by each individual) at locus , and the coefficient represents the effect size of a single allele. The fitted value of will reflect the collective effect of all causal SNPs that are in linkage disequilibrium (LD) with the test SNP .
where and is the sample variance of the residual trait values ( is a vector of ones) . To test association, is compared with a distribution.
Family-based association test by gene dropping
When related individuals are used to compute the score test statistic , components of can be dependent, and the variance estimator (2) is no longer valid. One can account for correlations between components in by simulating the null distribution of using gene dropping. We now derive the analytical mean and variance of under the gene-dropping setting. In the score test using unrelated individuals, we treat as random, and can be viewed as either random or fixed. In a gene-dropping simulation, is held fixed, and is random.
where is a matrix of all ones. Because for residuals from a linear regression model with an intercept, the variance of under gene dropping is if unconditional on the inheritance vector , and is if conditional on the inheritance vector (holding fixed). We can approximate the gene-dropping null distribution of by a normal distribution with mean and variance , and compute the gene-dropping p-value by comparing with a distribution. To test association in the presence of linkage, one needs to condition on the inheritance vector at  and use . In practice, is not observable, but we estimate by drawing Markov chain Monte Carlo (MCMC) samples of based on observed genotypes in the pedigrees using MORGAN (http://www.stat.washington.edu/thompson/Genepi/MORGAN/Morgan.shtml) .
In a gene-dropping simulation, the analytical mean of the score statistic is . The variance of the score statistic is if conditional on the inheritance vector (i.e., holding the inheritance vector fixed during gene-dropping simulation) and is if unconditional on the inheritance vector. The normal approximation is justified by the central limit theorem because the test statistic is additive over pedigrees. Its performance depends on the number, sizes, and structure of pedigrees and on MAF at the test locus. The approximation may not be accurate for extremely small p-values. However, the rankings of the p-values will not change.
The first component can be used as a test statistic for detecting association in the presence of linkage (i.e., fine mapping under a linkage peak) because the denominator is the variance of conditional upon the observed IBD sharing. The second component provides information about linkage. The kinship coefficients in are determined by pedigree structure, so is a constant in a gene-dropping simulation. measures the correlation between trait value similarity and IBD sharing at locus across all pairs of individuals in a pedigree. This correlation is expected to be stronger if there is stronger linkage between and a true causal locus. Therefore, can be used as a test statistic to detect linkage, with null distribution obtained by gene-dropping simulations. In a gene-dropping simulation, the inheritance vectors are simulated as if they were from a marker unlinked to any potential causal loci. resembles similar quantities that are frequently used in linkage analysis methods such as the well-known Haseman-Elston regression  as well as many variance components or generalized estimating equation-based methods .
Ranks of truly influential single-nucleotide polymorphisms by genome-wide association studies, FBAT, and gene dropping
Relative rank (%)
Relative rank (%)
Relative rank (%)
We also examined adjusting for population stratification by fitting the first two principal components of genetic variation  as covariates in the regression model (1). The p-values resulting from this expanded model differed negligibly from the original model. The ranks in Table 1 were essentially unchanged by this adjustment.
Comparison between genome-wide association studies, FBAT and gene-dropping test
Correlation between log p-values of genome-wide association studies, FBAT, and gene dropping
It is also possible to derive the analytical mean and variance of the test statistic in the gene-dropping test where we permute the founder alleles rather than resimulate the founder alleles. FBAT is more robust to population stratification by conditioning on founder genotypes. The gene-dropping test can gain similar robustness by restricting permutations to founder alleles within each family.
It is somewhat surprising that the gene-dropping test did not outperform GWAS given that it uses more individuals. One possible interpretation is that the effect of LD is stronger when more individuals are used. As we can see in Figure 1, the signals detected by the gene-dropping test come in bigger clusters. In other words, many SNPs ranked high by the gene-dropping test might be in LD with one or more of the causal SNPs.
Separating linkage and association signals
The gene-dropping test captures both linkage and association signals. One can decompose the test statistic into a linkage component and an association component.
The association component corresponds to testing association in the presence of linkage, which requires one to condition on the true inheritance vector at the test locus. Our results through MCMC approximation show that whether or not to condition on the inheritance vector actually does not make a big difference for this data set because the variance of the test statistic with conditioning only differs slightly from the variance of the test statistic without conditioning. This conclusion might be dependent on the structure of the pedigree.
The linkage component, however, clearly provides valuable information. The linkage signal is stronger in most regions containing causal SNPs. It is obvious that the linkage curve can help eliminate many of the false association signals in this study. It would be interesting to investigate how to use the linkage information more effectively in the future.
We thank the GAW18 organizers. The GAW18 whole genome sequence data were provided by the Type 2 Diabetes Genetic Exploration by Next-generation sequencing in Ethnic Samples (T2D-GENES) Consortium, which is supported by National Institutes of Health (NIH) grants U01 DK085524, U01 DK085584, U01 DK085501, U01 DK085526, and U01 DK085545. The other genetic and phenotypic data for GAW18 were provided by the San Antonio Family Heart Study and San Antonio Family Diabetes/Gallbladder Study, which are supported by NIH grants P01 HL045222, R01 DK047482, and R01 DK053889. The GAW is supported by NIH grant R01 GM031575.
This article has been published as part of BMC Proceedings Volume 8 Supplement 1, 2014: Genetic Analysis Workshop 18. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcproc/supplements/8/S1. Publication charges for this supplement were funded by the Texas Biomedical Research Institute.
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