Statistical genetic analysis of quantitative traits in large pedigrees is a formidable computational task due to the necessity of taking the nonindependence 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 among 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 eigen simplification 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 where h2 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 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 eigen simplification 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 nonpositive semidefinite kernels. These procedures are being added to our general statistical genetic computer package, SOLAR.
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