Conditional properties of likelihood-based significance tests

This paper considers the conditional properties of the unconditional significance tests based on standardized versions of the maximum likelihood estimate and the score statistic for the case of a simple null hypothesis and a one-parameter model. The properties of the tests are assessed by considering the conditional level of the test given the observed value of a statistic that is locally ancillary near the null model. It is shown that for the test based on the maximum likelihood estimate the conditional level is equal to the nominal level of the test, ignoring terms of order O(n^-1) and higher, provided that the inverse of the observed Fisher information is used as the variance estimate. If the inverse of the expected Fisher information is used, the extent to which the conditional level of the test depends on the observed value of the locally ancillary statistic is shown to depend on the 'statistical curvature' of the model as defined by Efron (1975). Hence, a new interpretation of statistical curvature is given in terms of conditional inference. Similar results are established for the test based on the score statistic.