A Kernel of Truth

Abstract

Statistical genetic analysis of quantitative traits in large pedigrees is a formidable computational task due to the necessity of taking the non-independence among relatives into account. With the growing awareness that rare sequence variants may be important in human quantitative variation, heritability and association study designs involving large pedigrees will increase in frequency due to the greater chance of observing multiple copies of rare variants amongst related individuals. Therefore, it is important to have statistical genetic test procedures that utilize all available information for extracting evidence regarding genetic association. Optimal testing for marker/phenotype association involves the exact calculation of the likelihood ratio statistic which requires the repeated inversion of potentially large matrices. In a whole genome sequence association context, such computation may be prohibitive. Toward this end, we have developed a rapid and efficient eigensimplification of the likelihood that makes analysis of family data commensurate with the analysis of a comparable sample of unrelated individuals. Our theoretical results which are based on a spectral representation of the likelihood yield simple exact expressions for the expected likelihood ratio test statistic (ELRT) for pedigrees of arbitrary size and complexity. For heritability, the ELRT is: −∑lntrue[1+ĥ2false(λgi−1false)true], where ĥ2 and λgi are respectively the heritability and eigenvalues of the pedigree-derived genetic relationship kernel (GRK). For association analysis of sequence variants, the ELRT is given by ELRTtrue[hq2>0:unrelatedstrue]−true(ELRTtrue[ht2>0:pedigreestrue]−ELRTtrue[hr2>0:pedigreestrue]true), where ht2,hq2, and hr2 are the total, quantitative trait nucleotide, and residual heritabilities, respectively. Using these results, fast and accurate analytical power analyses are possible, eliminating the need for computer simulation. Additional benefits of eigensimplification include a simple method for calculation of the exact distribution of the ELRT under the null hypothesis which turns out to differ from that expected under the usual asymptotic theory. Further, when combined with the use of empirical GRKs—estimated over a large number of genetic markers— our theory reveals potential problems associated with non positive semi-definite kernels. These procedures are being added to our general statistical genetic computer package, SOLAR.

Genetic background and the risk of otitis media

“…Cases with missing values were excluded from frequency tables, frequency and mean tests. The genetic component of OM, RAOM and COME was estimated by calculating the narrow-sense h 2 using SOLAR software, version 4.2.7 [24], which is based on variancecomponent linkage methods that can be used for pedigrees of arbitrary size and complexity [25]. The narrow-sense h 2 is defined as the ratio of variances of additive genetic effects (s 2 A ) and the observed phenotype (s 2 P ), h 2 = s 2 A /s 2 P [19].…”

Genetic Prediction in the Genetic Analysis Workshop 18 Sequencing Data

“…Let ${\mathrm{boldX}}_i={(x_{i1}^{\prime },\dots ,x_{i{T}_i}^{\prime })}^{\prime }$ be the T i × p matrix of covariates, and family member t = 1,..., T i . The basic LMM is given by $${\mathrm{boldy}}_i={\mathrm{boldX}}_i\beta +{\mathrm{boldG}}_i\alpha +{\mathrm{boldu}}_i+{bold\epsilon }_i,$$ where X i β denotes the fixed component of covariates, such as environmental variables, where β is the p -dimensional parameter vector of interest; G i α is a fixed genetic marker component, where α is the q -dimensional parameter vector of interest and G i is of dimension T i × q ; ${\mathrm{boldu}}_i\sim N(bold0,{\sigma }_u^2{\mathbf{K}}_i)$ is the polygenic component; ${bold\epsilon }_i\sim N(bold0,{\sigma }_{\epsilon }^2\mathrm{boldI})$ is the error term; K i reflects the correlation within pedigree i on the polygenic component, and it is twice the expected kinship matrix (i.e., K i = 2 Φ i ) ( K i has been termed the genetic relationship matrix or genetic relationship kernel (GRK) by Blangero et al [2013]); ${\sigma }_u^2$ is the variance of the polygenic component, which is assumed to be homoskedastic; and ${\sigma }_{\epsilon }^2$ is the variance of the random error, which is assumed to be independent and homoskedastic. With these assumptions y i is normally distributed with mean X i β and covariance matrix ${bold\Sigma }_i={\sigma }_u^2{\mathrm{boldK}}_i+{\sigma }_{\epsilon }^2\mathbf{I}$.…”

“…(3) under the assumption that phenotypes are standardized, that is, $$\mathrm{Var}\left(y_{it}\right)={\sigma }_y^2={\sigma }_u^2+{\sigma }_{\epsilon }^2=1$$ [Blangero et al, 2013]. In this case the covariance matrix Σ i = Var( y i ) is identical to h 2 K i + (1 – h 2 ) I , where $h^2={\sigma }_u^2∕({\sigma }_u^2+{\sigma }_{\epsilon }^2)$ denotes the narrow sense heritability of the polygenic component, and the log-likelihood function reduces to $$L_i=-\frac{1}{2}\sum _{t=1}^{T_i}true[1+h^2({\lambda }_{it}-1)true]-\frac{1}{2}\sum _{t=1}^{T_i}\frac{{\tilde{y}}_{it}^2}{1+h^2({\lambda }_{it}-1)},$$ where ${\tilde{\mathbf{y}}}_i={\mathrm{boldU}}_i^{-1}({\mathbf{y}}_i-{\mathbf{X}}_{i}bold\beta )$ is the vector of transformed residuals having components ỹ it .…”

“…Blangero et al [2013] and Diego et al [unpublished data] have shown that for this model the expected likelihood ratio statistic (LRT), when the estimate ĥ 2 exceeds 0, is given by $${\phantom{true\mid }E\left(\mathrm{LRT}\right)\mid }_{{\hat{h}}^2>0}=-\sum _{t=1}^{T_i}\ln true[1+h^2({\lambda }_{it}-1)true].$$ This means that the expected value of the test statistic depends on the eigenvalues λ it of the GRK, that is, the kinship matrix. Eigenvalues of selected relative pairs are displayed in Table 2; for other relative pairs and expected LRT values for varying heritabilities, see Blangero et al [2013].…”

“…Eigenvalues of selected relative pairs are displayed in Table 2; for other relative pairs and expected LRT values for varying heritabilities, see Blangero et al [2013]. It is important to note that the eigenvalues for a monozygotic twin pair are either 0 or 2.…”