Methods for Coherent and Exact Inference

Abstract

This dissertation provides methods for coherent and exact inference for two types of problems in statistics. The first problem provides coherent criteria for the testing of nested interval null hypothesis. The second and third problems examine exact inference for meta-analysis of proportions and 2Ã 2 tables, respectively.Criticism of using p-values as measures of support is well-documented in the literature. In particular, one setting where uniformly most powerful unbiased (UMPU) test p-values for null hypotheses that are nested intervals can be incoherent; we may accept a smaller null but reject a larger one in which it is nested (see Schervish, P values: what they are and what they are not, 1996). In order to avoid this incoherence, the Bayesian paradigm offers guarantees of certain forms of coherence. Using Bayesian decision theory, we establish straightforward conditions that ensure coherence. From these, we establish novel frequentist criteria - different to Type I error rate, that give tests that are coherent. Meta-analysis is a practice that utilizes multiple studies and seeks inference on some overall effect. Meta-analysis of proportions, for example, seeks inference on some overall proportion of successes-failures, where the multiple studies estimate a binary outcome. Un- der common effect models, which assume each study is estimating the same underlying truth, exact inference has long been available. However, under a more reasonable fixed- effects models, exact inference is not readily available. Instead, non-exact methods are used which can be challenging to interpret. We present methods for exact tests and confidence intervals for fixed-effects meta-analysis of proportions that maintain interpretability of the parameter of interest and are easily implemented. Another area of meta-analysis examines 2x2 tables. These are common when synthesiz- ing the results of multiple placebo-controlled trials for a binary outcome. In these analyses, we seek inference on some overall comparison of the outcome between two groups, such as the odds ratio. We provide an approach that is exact by extending the method we used in meta-analysis of proportions that provides inference on an overall odds ratio.