When there is uncertainty in sibling relationship,the classical affected sib-pair(ASP) linkage tests may be severely biased.This can happen,for example,if some of the half sib-pairs are mixed with full sib-pairs.The genomic control method has been used in association analysis to adjust for population structures.We show that the same idea can be applied to ASP linkage analysis with uncertainty in sibling relationship.Assuming that,in addition to the candidate marker,null markers that are unlinked to the disease locus are also genotyped,we may use the information on these loci to estimate the proportion of half sib-pairs and to correct for the bias and variance distortion caused by the heterogeneity of sibling relationship.Unlike in association studies,the null loci are not required to be matched with the candidate marker in allele frequency for ASP linkage analysis.This makes our approach flexible in selecting null markers.In our simulations,using a number of 30 or more null loci can effectively remove the bias and variance distortion.It is also shown that,even the null loci are weakly linked to the disease locus,the proposed method can also provide satisfactory correction.
The multivariate extension of the Cox model proposed by Wei,Lin and Weissfeld in 1989 has been widely used for analyzing multivariate survival data.Under the model assumption,failure times from an individual are assumed to marginally follow their respective proportional hazards regression relation,leaving the joint distribution completely unspecified.This paper presents a simple approach to efficiency improvement through segmentation of stochastic integrals in the marginal estimating equations and incorporation of the limiting covariance structure.It is shown that when partition of the time interval is done at a suitable rate,the resulting estimator is consistent and asymptotically normal.Through the reproducing kernel Hilbert space arising from the covariance function of the limiting Gaussian process,it is also shown that the proposed estimator is asymptotically optimal within a reasonable class of estimators under marginal specification.Simulations are conducted to assess the finite-sample performance of the proposed method.
