Knapp et al. [14] proved that the ``mean IBD'' statistic 
is uniformly most powerful (in 
) for a single diallelic disease locus with strict-recessive mode of inheritance and the usual assumptions of random mating and HWE. The proof is based on the fact that for such models, the IBD probabilities at the marker are on the Hardy-Weinberg curve:
The log-likelihood ratio for testing 
vs. 
is:
In general, the ``mean IBD'' statistic is UMP whenever the IBD probabilities are on the HW curve.
In another paper, Knapp et al. [15] show how the ``mean IBD'' statistic is equivalent to lod score analysis for an assumed strict-recessive mode of inheritance with random mating at the DS locus, regardless of the true mode of inheritance of the disease. They do this by maximizing the strict recessive likelihood with respect to 
and showing that it is a monotone function of the ``mean IBD'' statistic.