On the validity of the likelihood ratio test and consistency of resulting parameter estimates in joint linkage and linkage disequilibrium analysis under improperly specified parametric models

It has been shown that parametric analysis of linkage disequilibrium conditional on linkage using an overly deterministic model can be optimal for family-based association analysis. However, if one applies this strategy carelessly, there is a risk of false inference. We analyse properties of such likelihood ratio tests when the assumed disease mode of inheritance is inaccurate. Under some conditions, problems result if one is not careful to consider what null hypothesis is being tested. We show that: (a) tests for which the null hypothesis assumes the absence of both linkage and association are independent of the true mode of inheritance; (b) likelihood ratio tests assuming either linkage or association under the null hypothesis may depend on the true mode of inheritance, leading to inconsistent parameter estimates, in particular under extremely deterministic models; (c) this problem cannot be eliminated by increasing sample size or adding population controls--as sample size increases, the chance of false positive inference goes to 100%; (d) this issue can lead to systematic false positive inference of association in regions of linkage. This is important because highly deterministic models are often used intentionally in model-based analyses because they can have more power than the true model, and are implicit in many model-free analysis methods.