On tests of rate ratio under standard inverse sampling

Inverse sampling suggests one continues to sample subjects until a pre-specified number of rare events of interest is observed. It is generally considered to be more appropriate than the usual binomial sampling when the subjects come sequentially, when the response probability is rare, and when maximum likelihood estimators of some epidemiological measures are undefined under binomial sampling. Reliable but conservative exact conditional procedure for the ratio of the response probabilities of subject without the attribute of interest has been studied. However, such a procedure is inapplicable to the risk ratio (i.e., ratio of the response probabilities of subject with the attribute of interest). In this paper, we investigate various test statistics (namely Wald-type, score and likelihood ratio test statistics) for testing non-unity risk ratio under standard inverse sampling scheme, which suggests one continue to sample until the predetermined number of index subjects with the attributes of interest is observed. Both asymptotic and numerical approximate unconditional methods are considered for P-value calculation. Performance of these test procedures are evaluated under different settings by means of Monte Carlo simulation. In general, the Wald-type test statistic is preferable for its satisfactory and stable performance with approximate unconditional procedures. The methodologies are illustrated with a real example from a heart disease study.