A linkage strategy for detection of human quantitative-trait loci .1. Generalized relative risk ratios and power of sib pairs with extreme trait values

We generalize the concept of the relative risk ratio (lambda) to the case of quantitative traits, to take into account the various trait outcomes of a relative pair. Formulas are derived to express the expected proportion of genes shared identical by descent by a sib pair, in terms of the generalized lambda's for sib pairs (lambda(s)), parent-offspring pairs (lambda(o)), and monozygotic twins (lambda(M)) and in terms of the recombination fraction, with the assumption of no residual correlations. If residual correlations are nonzero among relative pairs, we assume that they are the same among sib pairs, parent-offspring pairs, and monozygotic twins, and we employ a slightly different definition for the generalized lambda so that the same set of formulas still hold. The power (or, the sample size necessary) to detect quantitative-trait loci (QTLs) by use of extreme sib pairs (ESPs) is shown to be a function of the three generalized lambda's. Since lambda(M) can be derived by use of values of lambda(s) and lambda(o), estimates of the latter two lambda's will suffice for the analysis of power and the necessary sample sizes of ESPs, for a QTL linkage study.