Efficient and exact tests of the risk ratio in a correlated 2×22×2 table with structural zero

For a correlated 2×22×2 table where the (01) cell is empty by design, the parameter of interest is typically the ratio of the probability of secondary response conditional on primary response to the probability of primary response, also known as a risk ratio. It is common to test whether or not the risk ratio equals one. One method of obtaining an exact PP-value is to maximise the tail probability of the test statistic over the nuisance parameter. It is argued that better results are obtained by first replacing the nuisance parameter by its profile estimate in the calculation of its exact significance followed by maximisation—termed an E+ME+MPP-value. We consider four standard approximate test statistics with and without the common correction of adding 12 to each count. From a complete enumeration of the distributions of these PP-values (for sample sizes 50 and 100), we recommend E+ME+MPP-values based on the uncorrected Wald statistic for testing the greater than alternative and on the corrected Wald statistic on the log-scale for testing the less than alternative. A good compromise statistic for both kinds of alternatives is the likelihood ratio statistic.