Exact inference for categorical data: recent advances and continuing controversies

Methods for exact small-sample analyses with categorical data have been increasingly well developed in recent years. A variety of exact methods exist, primarily using the approach that eliminates unknown parameters by conditioning on their sufficient statistics. In addition, a variety of algorithms now exist for implementing the methods. This paper briefly summarizes the exact approaches and describes recent developments. Controversy continues about the appropriateness of some exact methods, primarily relating to their conservative nature because of discreteness. This issue is examined for two simple problems in which discreteness can be severe--interval estimation of a proportion and the odds ratio. In general, adjusted exact methods based on the mid-P-value seem a reasonable way of reducing the severity of this problem.