  Many practical studies rely on hypothesis testing procedures applied to data
sets with missing information. An important part of the analysis is to
determine the impact of the missing data on the performance of the test, and
this can be done by properly quantifying the relative (to complete data) amount
of available information. The problem is directly motivated by applications to
studies, such as linkage analyses and haplotype-based association projects,
designed to identify genetic contributions to complex diseases. In the genetic
studies the relative information measures are needed for the experimental
design, technology comparison, interpretation of the data, and for
understanding the behavior of some of the inference tools. The central
difficulties in constructing such information measures arise from the multiple,
and sometimes conflicting, aims in practice. For large samples, we show that a
satisfactory, likelihood-based general solution exists by using appropriate
forms of the relative Kullback--Leibler information, and that the proposed
measures are computationally inexpensive given the maximized likelihoods with
the observed data. Two measures are introduced, under the null and alternative
hypothesis respectively. We exemplify the measures on data coming from mapping
studies on the inflammatory bowel disease and diabetes. For small-sample
problems, which appear rather frequently in practice and sometimes in disguised
forms (e.g., measuring individual contributions to a large study), the robust
Bayesian approach holds great promise, though the choice of a general-purpose
"default prior" is a very challenging problem.
