How can the score test be consistent?

TL;DR

This paper introduces a modified score test for comparing two binomial proportions that improves power and consistency, especially under alternative hypotheses, by allowing negative values and using a simple rejection rule.

Contribution

The authors propose a new modified score test that addresses the inconsistency and negative value issues of the traditional score test in binomial proportion comparisons.

Findings

The new test has higher power than traditional tests in simulations.

The modified test restores consistency under the alternative hypothesis.

Inference remains straightforward and always feasible with the new rule.

Abstract

The score test statistic using the observed information is easy to compute numerically. Its large sample distribution under the null hypothesis is well known and is equivalent to that of the score test based on the expected information, the likelihood-ratio test and the Wald test. However, several authors have noted that under the alternative this no longer holds and in particular the statistic can take negative values. Here we examine the score test using the observed information in the context of comparing two binomial proportions under imperfect detection, a common problem in ecology when studying occurrence of species. We demonstrate through a combination of simulations and theoretical analysis that a new modified rule which we propose that rejects the null hypothesis when the observed score statistic is larger than the usual chi-square cut-off or is negative has power that is mostly greater to any other test. In addition consistency is largely restored. Our new test is easy to use and inference is always possible.