A likelihood ratio test for nested proportions

For policy and medical issues, it is important to know if the proportion of an event changes after an intervention is administered. When the later proportion can only be calculated in a portion of the sample used to compute the previous proportion, the two proportions are nested. The motivating example for this work comes from the need to test whether admission rates in emergency departments are different between the first and a return visit. Here, subjects who contribute to the admission rate at the return visit must be included in the first rate and also return, but not vice versa. This conditionality means that existing methods, including the basic test of equality of two proportions, longitudinal data analysis methods, and recurrent event approaches are not directly applicable. Currently, researchers can only explore this question by the use of descriptive statistics. We propose a likelihood ratio test to compare two nested proportions by using the product of conditional probabilities. This test accommodates the conditionality, subject dependencies, and cluster effects and can be implemented in SAS PROC NLMIXED allowing for the proposed method to be readily used in an applied setting. Simulation studies showed that our approach provides unbiased estimates and reasonable power. Moreover, it generally outperforms the two-sample proportion z-test, in the presence of heterogeneity, and the Cochran-Mantel-Haenszel test. An example based on readmission rates through an emergency department is used to illustrate the proposed method.