On approximate conditioning and higher order asymptotics for 2x2 tables

Chris Lloyd, Melbourne Business School

Abstract

For testing canonical parameters in a continuous exponential family, P-values based on higher order asymptotic formulas such as p* approximate the exact conditional P-value with great accuracy. For discrete models, the conditional distribution can be extremely discrete or even degenerate which raises the questions (a) should one try to approximate the conditional P-value, (b) what does p* approximate? Pierce and Peters (1999) have argued that p* approximates an approximately conditional P-value and that this approximately conditional P-value is an inferentially sensible quantity worth approximating. Their arguments and numerical results are oriented towards problems where the conditioning variable has 3 or 4 dimensions. We investigate the performance and logic of approximately conditional P-values for the case of 2x2tables, as well as the extent to which p* functions as an approximation to these P-values. We conclude that approximately conditional P-values have rather erratic properties and suffer from a logical flaw. We also find that the mid-P value approximates them as well or better than p*, but that neither approximations work well when the observed data is near the boundary of the sample space.