Abstract
Inverse sampling for McNemars test is studied. Sampling is conducted until a pre-specified number of discordant pairs is observed instead of sampling until a pre-specified total number of pairs is observed. The joint likelihood is decomposed into a product of a negative binomial distribution for the number of pairs required to observe r discordant pairs, a binomial distribution for the number of successes in the concordant observations, and a binomial distribution for the number of successes in the discordant observations. Since inference in this problem is based on the discordant observations, inverse sampling controls the type II error when small numbers of discordant observations are observed and the exact binomial test is required. The control results from fixing the sample size for the exact binomial test. Standard sampling instead lets the sample size for the exact binomial test vary and then performs the test conditionally on the observed number of discordant pairs.
Highlights
This paper focuses on planning an experiment on paired binary responses
The control results from fixing the sample size for the exact binomial test
The number of discordant pairs, r = 23, is the smallest sample size that gives an exact binomial test under inverse sampling with 90% power
Summary
This paper focuses on planning an experiment on paired binary responses. We want our hypothesis test comparing the probability of success in the two margins to control both the type I error and power. We speak of power of a discrete test as being the expected power over all the possible tables that can result in the experiment This metric is okay if we wish to describe the average performance of the test when used repeatedly by the scientific community. The paper examines conditional power for different sampling schemes and test statistics for the paired binary experiment. Inverse sampling controls the type II error when small numbers of discordant observations are observed and the exact binomial test is used. Standard sampling lets the sample size for the exact binomial test vary and performs the test conditionally on the observed number of discordant pairs. The paper first presents probability models, estimators and test statistics for paired binary experiments. Tests are compared in terms of the percent of tables with conditional power less than nominal power and the standard deviation of the conditional power
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