Likelihood-based inference for the genetic relative risk based on affected-sibling-pair marker data

Using genetic marker data from affected sibling pairs, we study likelihood-based linkage analysis under quasi-recessive, quasi-dominant, and general single-locus models. We use an epidemiologic parameterization under a model where the marker locus is closely linked to the putative disease susceptibility gene. This model and parameterization allow inferences about the relative risk associated with the susceptible genotype. We base inferences on approximate likelihoods that focus on the affected siblings in the sibship and, using these likelihoods, we derive closed-form maximum likelihood estimators for model parameters and closed-form likelihood ratio statistics for tests that the relative risk associated with the susceptible genotype is one. Under the general single-locus model, our likelihood ratio test is the same as the iteratively computed triangle test proposed by Holmans (1993, American Journal of Human Genetics 52, 362-374) for the case where marker identity-by-descent is known; our derivation gives a closed form for the test statistic. We present quartiles of the distribution of parameter estimates and critical values for the exact null distribution of our likelihood ratio test statistics; we also give large-sample approximations to their null distributions. We show that the powers of our likelihood ratio tests exceed the powers of more commonly used nonparametric affected-sibling-pair tests when the data meet the inheritance model assumptions used to derive the test; we also show that our tests' powers are robust to violation of model assumptions. We conclude that our model-based inferences provide a practical alternative to more common affected-sibling-pair tests when investigators have some knowledge about the mode of inheritance of a disease and that our methods may sometimes be useful for comparing the genetic relative risk with environmental relative risks.