A measurement error explanation of the biased QTL effect estimates when pi-hat is used as the IBD probability for untyped sib pairs²

A. Leo Beem¹ and Dorret I. Boomsma¹

¹ Department of Biological Psychology, Free University, Amsterdam, The Netherlands

In variance components models for QTL linkage analysis, the IBD status of sib pairs can be modeled by a mixture (or weighted likelihood) approach or by using pi-hat, the estimated expected proportion of alleles shared IBD. As QTL effects are often quite small and resources limited, marker data are mostly obtained for a selected sample of sib pairs, which are extreme concordant, discordant or both. Results from analyses that do not take the selection into account will usually be invalid. Several methods have been proposed for the analysis of such data (P.C. Sham, J.H. Zhao, S.S. Cherny and J.K. Hewitt, 2000, Genet. Epidemiol. 19, S22-S28). The analyses may include only the subjects for which marker data are available or both subjects with and without marker data. If the mixture approach is used in the full sample, the weights for the untyped subjects are the a priori probabilities for IBD zero, one and two. With the pi-hat approach, the a priori expected number of alleles shared IBD is substituted for the untyped subjects. The latter approach, which is attractive because of its simplicity, has been shown in simulations to give seriously biased estimates of the QTL effect (C.V. Dolan, D.I. Boomsma and M.C. Neale, 1999, Am. J. Hum. Genet. 64, 268-280). However, the reason for this bias remained unclear. Here the imputation of IBD probabilities will be approached as a measurement problem, in which true IBD status is estimated with various amounts of error. Obtaining consistent estimators then requires corrections for attenuation (as it is called in psychometrics). It will be demonstrated that the pi-hat approach yields the wrong correction for attenuation.