- Open Access
Kernel score statistic for dependent data
© Malzahn et al.; licensee BioMed Central Ltd. 2014
Published: 17 June 2014
The kernel score statistic is a global covariance component test over a set of genetic markers. It provides a flexible modeling framework and does not collapse marker information. We generalize the kernel score statistic to allow for familial dependencies and to adjust for random confounder effects. With this extension, we adjust our analysis of real and simulated baseline systolic blood pressure for polygenic familial background. We find that the kernel score test gains appreciably in power through the use of sequencing compared to tag-single-nucleotide polymorphisms for very rare single nucleotide polymorphisms with <1% minor allele frequency.
Lately, much interest has been focused on global multilocus procedures that test for overall association of sets of markers. This is appealing if the loci carry rare variants or are thought to belong to a specific gene set, such as a pathway or genes with similar functions. The kernel score statistic is a specific type of such a global covariate-adjusted multilocus association test. Depending on context, it has received different names, such as SKAT (sequence kernel association test) . Kernel methods provide flexible semiparametric regression models of multimarker genetic influence on expected trait outcome and can conveniently be implemented within the linear mixed-model formalism [2, 3]. The kernel expresses genetic correlation between subjects. It allows us to test a whole model class by defining a prior probability in function space in a bayesian manner . In contrast, standard regression models are limited to just 1 a priori stated submodel of such a class.
Until very recently, kernel methods were only applied to independent data (see Ref.  for a review). An extension to include familial dependence was made for trait prediction , but not for testing genetic influence. This changed with independent Genetic Analysis Workshop 18 (GAW18) contributions by ourselves and others (Huang et al , Chen et al , Dufresne et al , and Schifano et al ). Huang et al  applied the kernel method to trios. The other 4 contributions (including ours) use a similar kernel extension to families with different study aims. We generalize the kernel score statistic to adjust for dependency in metric trait data. We apply it to real and simulated GAW18 baseline systolic blood pressure (SBP) and contrast its performance on the sequenced panel in comparison to tag-single-nucleotide polymorphisms (SNPs) of a genome-wide association study (GWAS).
GAW18 provided blood pressure trait data from Mexican Americans participating in the San Antonio Family Heart Study or the San Antonio Family Diabetes/Gallbladder Study, and in the Type 2 Diabetes Genetic Exploration by Next-generation sequencing in Ethnic Samples (T2D-GENES) Project 2 study . The sample is enriched for type 2 diabetes and for rare variants. Complete dense SNP allele dosage data were provided for 959 subjects from 20 large pedigrees with 22 to 86 members. Data are based on genome-wide tag-SNP genotyping of all subjects, whole-genome sequencing of 464 selected subjects, and imputation of all missing dosages for the remaining SNPs and subjects, exploiting kinship within pedigrees . We considered only subjects with known baseline SBP, sex, and age, who were not on blood pressure medication: 706 subjects with real SBP, excluding the first listed twin of monozygotic twin pairs and 740 to 781 subjects with simulated SBP (subject numbers vary for 200 simulated study replicates as a result of inclusion criteria). We tested candidate genes on chromosome 3 only (real SBP: 5 genes selected based on previous association reports , simulated SBP: gene MAP4). From group discussions, we knew that MAP4 associates with simulated SBP and that simulated trait Q1 mimics the segregation of genes within families, but otherwise represents the genetic null hypothesis.
Kernel score statistic with adjustment for random confounders
with matrix Vo = s2I+s2famZZT. With fixed effects estimate b o =(XTVo−1X)−1XTVo−1Y, we rewrite test statistic (3) as quadratic form Q=RTMR of standard normally distributed residuals R= Po1/2Y with matrix M=(Po1/2K Po1/2)/2 and null projection matrix P o =V o −1 - V o −1 X(X T V o −1 X) −1 X T V o −1 . The p values for test statistic (3) can be calculated by the Davies exact method  with the R package CompQuadForm from sample estimates Q and all eigenvalues of matrix M. We adjusted for polygenic familial background based on the kinship coefficient matrix Φkin=ZZT using R-packages kinship2 and coxme with R-function lmekin for genetic null model (2). We employed a linear kernel on allele dosages. This allows for additive genetic models on tested markers. The kernel matrix entry Kij=giTgj for 2 subjects i, j is the scalar product of their subject-specific vectors gi, gj of allele dosages for NSNP SNP marker. SNPs were included for testing if their Hardy-Weinberg equilibrium test p values ≥10−5 (rounding imputed dosages for this purpose only). We contrasted test performance for the same subjects on 2 SNP panels, sequence (allele dosage data) versus GWAS (allele dosage data reduced to intersection with GWAS SNPs).
Permutation testing demonstrated overall test validity: 50,000 permutations of subject-genotype assignment (permuting across families) yielded type I error 0.049 ± 0.0019, 0.00046 ± 0.00019 at significance level 0.05, 0.0005 for simulated SBP on all sequenced rare MAP4 SNPs. Simulated trait Q1 was used to test the quality of polygenic adjustment. We analyzed Q1 analogously to SBP on gene MAP4 and obtained p values ≤0.05 with rate 0.02 (all SNPs, and SNPs with minor allele frequency (MAF) >5%), rate 0.05 (for MAF 1% to 5%), and 0.06 (for MAF <1%). This is within 95% tolerance limits (0.02, 0.08) for true value 0.05 on 200 replicates.
Real SBP and candidate genes on chromosome 3
Kernel score statistic (linear kernel on allele dosage data)
MAF 1% to 5%
- 4 consecutive windows
- 5 consecutive windows
The kernel score statistic is a global covariance component test. P values for genes are dominated by their common SNPs (see Table 1: all SNPs vs. MAF >5%). Thus, common and rare SNP sets should be tested separately. The kernel method offers a simple and flexible way to perform multimarker analysis and genetic interaction analyses by means of the kernel choice. It improves power compared to single-marker tests by exploiting SNP correlation , and if jointly tested SNP sets tag multiple independent associated loci. The test statistic by Liu et al and Tzeng et al [2, 3] had 17% power at the 0.01 level for independent subjects to find MAP4 association with simulated SBP in 200 replicates. Our extension to families increased this to 98.5% power, used 84% more subjects with twice as many polymorphic SNPs (sequence), without type I error inflation. Ignoring familial dependency would inflate type I error to 16%, 6% at nominal 0.05, 0.01 levels (applying the independent subjects test statistic [2, 3] to families for MAP4 on polygenic trait Q1). We found that sequencing can improve test power appreciably, particularly for MAF <1% variants (Figure 1, right). For more frequent SNPs, the comparison was not decisive and tag-SNPs proved to be strong competitors. With sequencing, even a single, large gene may provide more SNP variants than available study subjects. The kernel score test remains valid because of the Bayesian nature of the underlying nonparametric genetic model . However, one might expect power loss. The kernel score statistic may be susceptible to random confounders. For example, simplistic adjustment of familial and common cultural effects by a familial random intercepts model (instead of the polygenic model) yields valid permutation test results, but shows up as inflated type I errors on simulated polygenic trait Q1 (p ≤0.05 on 10% of 200 study replicates for all MAP4 SNPs, inflation increases with decreasing MAF). This, in turn, would yield similar but overly optimistic estimates for Figure 1 and Table 1. With such a liberal test, AGTR1 p values in Table 1 would all drop below 0.05 for all sequence SNP sets. We would like to emphasize that other random confounder effects, such as different trait variability between study subgroups, can be adjusted in exact analogy to the familial polygenic background adjustment.
This work was supported by the Deutsche Forschungsgemeinschaft (grant Klinische Forschergruppe [KFO] 241: TP5, BI 576/5-1) and the German Federal Ministry of Education and Research BMBF (German National Genome Research Net NGFN grant 01GS0837). The GAW18 whole genome sequence data were provided by the T2D-GENES Consortium, which is supported by 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 Genetic Analysis Workshop 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.
- Wu MC, Lee S, Cai T, Li Y, Boehnke M, Lin X: Rare-variant association testing for sequencing data with the sequence kernel association test. Am J Hum Genet. 2011, 89: 82-93. 10.1016/j.ajhg.2011.05.029.PubMed CentralView ArticlePubMedGoogle Scholar
- Liu D, Lin X, Ghosh G: Semiparametric regression of multidimensional genetic pathway data: least-squares kernel machines and linear mixed models. Biometrics. 2007, 63: 1079-1088. 10.1111/j.1541-0420.2007.00799.x.PubMed CentralView ArticlePubMedGoogle Scholar
- Tzeng J-Y, Zhang D: Haplotype-based association analysis via variance-components score test. Am J Hum Genet. 2007, 81: 927-938. 10.1086/521558.PubMed CentralView ArticlePubMedGoogle Scholar
- Rasmussen CE, Williams CK: Gaussian Processes for Machine Learning. 2006, Cambridge, MA, MIT PressGoogle Scholar
- Schaid DJ: Genomic similarity and kernel methods I: advancements by building on mathematical and statistical foundations. Hum Hered. 2010, 70: 109-131. 10.1159/000312641.View ArticlePubMedGoogle Scholar
- Gianola D, van Kaam JB: Reproducing kernel Hilbert spaces regression models for genomic assisted prediction of quantitative traits. Genetics. 2008, 178: 2289-2303. 10.1534/genetics.107.084285.PubMed CentralView ArticlePubMedGoogle Scholar
- Huang J, Chen Y, Swartz M, Ionita-Laza I: Comparing the power of family-based association test for sequence data with applications in the GAW18 simulated data. BMC Proc. 2014, 8 (suppl 1): S27-PubMed CentralView ArticlePubMedGoogle Scholar
- Chen H, Choi SH, Hong J, Lu C, Milton JN, Allard C, Lacey SM, Lin H, Dupuis J: Rare genetic variant analysis on blood pressure in related samples. BMC Proc. 2014, 8 (suppl 1): S35-PubMed CentralView ArticlePubMedGoogle Scholar
- Dufresne L, Oualkacha K, Forgetta V, Greenwood CMT: Pathway analysis for genetic association studies: To do, or not to do, that is the question. BMC Proc. 2014, 8 (suppl 1): S103-PubMed CentralView ArticlePubMedGoogle Scholar
- Schifano ED, Epstein MP, Bielak LF, Jhun MA, Kardia SL, Peyser P, Lin X: SNP set association analysis for familial data. Genet Epidemiol. 2012, 36: 797-810.PubMed CentralPubMedGoogle Scholar
- Almasy L, Dyer TD, Peralta JM, Jun G, Fuchsberger C, Almeida MA, Kent JW, Fowler S, Duggirala R, Blangero J: Data for Genetic Analysis Workshop 18: Human whole genome sequence, blood pressure, and simulated phenotypes in extended pedigrees. BMC Proc. 2014, 8 (suppl 1): S2-PubMed CentralView ArticlePubMedGoogle Scholar
- SNPedia August 2012. [http://www.snpedia.com]
- Blom G: Statistical Estimates and Transformed Beta Variables. 1958, New York, John Wiley & SonsGoogle Scholar
- Davies R: Algorithm as 155: the distribution of a linear combination of chi-2 random variables. J R Stat Soc Ser C. 1980, 29: 323-333.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.