Combining independent, weighted P-values: achieving computational stability by a systematic expansion with controllable accuracy.

Given the expanding availability of scientific data and tools to analyze them, combining different assessments of the same piece of information has become increasingly important for social, biological, and even physical sciences. This task demands, to begin with, a method-independent standard, such as the P-value, that can be used to assess the reliability of a piece of information. Good's formula and Fisher's method combine independent P-values with respectively unequal and equal weights. Both approaches may be regarded as limiting instances of a general case of combining P-values from m groups; P-values within each group are weighted equally, while weight varies by group. When some of the weights become nearly degenerate, as cautioned by Good, numeric instability occurs in computation of the combined P-values. We deal explicitly with this difficulty by deriving a controlled expansion, in powers of differences in inverse weights, that provides both accurate statistics and stable numerics. We illustrate the utility of this systematic approach with a few examples. In addition, we also provide here an alternative derivation for the probability distribution function of the general case and show how the analytic formula obtained reduces to both Good's and Fisher's methods as special cases. A C++ program, which computes the combined P-values with equal numerical stability regardless of whether weights are (nearly) degenerate or not, is available for download at our group website http://www.ncbi.nlm.nih.gov/CBBresearch/Yu/downloads/CoinedPValues.html.