No distinction between maternal and paternal sharing

Suppose we have IBD data at a marker for ASPs, in the form of counts , i=0,1,2, for the number of ASPs sharing DNA IBD at the marker on i chromosomes. Similarly for DSPs and USPs. Under Assumptions 5 and 6, the log-likelihood of the IBD data conditional on phenotypic information is

where the 's are defined in Section 5. Hence the sib-pair score statistic is given by

For sib-pairs, we showed in Section 5 that

where and . Hence, derivatives of the 's with respect to are obtained by computing the derivatives of the transition matrix .

It turns out that

but

Consequently,

Note:

• The score statistic for each particular type of sib-pair is the usual ``mean IBD'' statistic, , regardless of the genetic model (since ).
• The preceding finding is consistent with the fact that, regardless of the genetic model, as the conditional IBD probabilities approach the null point along the line . is the score statistic for alternatives along the line .
• The score statistic for combining sib-pairs is a linear combination of the ``mean IBD'' statistic for each type of sib-pair, with weights depending on the parameters of the genetic model, .
• The weight for each type of sib-pair is given by , the expected value of the when , i.e. when we are ``on top of'' the DS locus. The greater , the greater the ``distance'' from the null, the greater the weight.
• For each type of sib-pair, the ``mean IBD'' statistic is locally most powerful in , i.e. it maximizes the second derivative of the power function at .