A remark on rare variants.

The genetic architecture of a disease determines the epidemiological methods for its examination. Recently, Bodmer and Bonilla suggested that moderately strong, moderately rare variants contribute substantially to the genetic population attributable risk (PAR) of common diseases. In the first part of this communication, I provide a concise reconstruction of their deliberation. Variants contributing to human disease can be identified by linkage or by association tests. Risch and Merikangas analyzed the power of these tests by comparing the affected sib-pair linkage test (ASP) and the transmission disequilibrium association test (TDT). In the second part of this paper, I give an accessible reconstruction of this comparison and derive simple approximations in the low allele frequency range, directly showing that the linkage test is much more sensitive to a decrease of frequency or effect size. In the third part, I analyze a disease model whose genetic architecture is proportional to Kimura's infinite sites model. The relation between a variant's selection coefficient and its effect size in disease generation is assumed to be simple, and the number of contributing genetic variants is determined by the sum of their approximative PAR contributions. An association test (TDT) is finally applied to this disease model. For different ranges of effect size and allele frequency, I derive the minimal sample size necessary to detect at least one contributing variant. It turns out that, although the majority of contributing variants is not accessible with realistic sample sizes, a minimum of sample size may be given for moderately strong variants in the 1% frequency range.