From Angel (Nov. 17, 2003):

At the last week's seminar, there was a question as to why the multivariate Haseman-Elston method worked better for two traits that had a strong negative residual correlation. I completely forgot about the article Steph had pointed out that talked about this situation until after the seminar and after going back to my notes:

David M. Evans "The Power of Multivariate Quantitative-Trait Loci Linkage Analysis Is Influenced by the Correlation between Variables" Am. J. Hum. Genet. 70:1599-1602, 2002.

Basically, Evans works out the non-centrality parameter (NCP) of the linkage test based on IBD-sharing (for variance components, but the situation is probably similar for Haseman-Elston). The NCP is basically the expectation of the 2*ln(likelihood ratio test) which is expectation of {2*ln(likelihood there is linkage) - 2*ln(likelihood no linkage)}. We can view the NCP as expected evidence for linkage, and the power to detect linkage will increase as NCP increases.

The NCP was written out in terms of shared and unshared correlations between genetic and residual effects, respectively. From the NCP formula, as these correlations become more negative, then the NCP increases which suggests an increase in power to detect linkage. The increase in power or NCP is more sensitive to negative correlations among unshared effects (ie, the residual correlations) than shared effects as seen in the Figure 2 of this reference.