Statistical inference for the risk ratio in 2×22×2 binomial trials with structural zero

In some statistical analyses, researchers may encounter the problem of analyzing a correlated 2×22×2 table with a structural zero in one of the off diagonal cells. Structural zeros arise in situation where it is theoretically impossible for a particular cell to be observed. For instance, Agresti [2002. Categorical Data Analysis, 2nd ed. Wiley, Hoboken, New Jersey] provided an example involving a sample of 156 calves born in Okeechobee County, Florida. Calves are first classified according to whether they get a pneumonia infection within certain time. They are then classified again according to whether they get a secondary infection within a period after the first infection clears up. Because subjects cannot, by definition, have a secondary infection without first having a primary infection, a structural void in the cell of the summary table that corresponds with no primary infection and has secondary infection is introduced. For discussion of this phenomenon, see Tang and Tang [2002. Exact unconditional inference for risk ratio in a correlated 2×22×2 table with structural zero. Biometrics 58, 972–980], and Lui [1998. Interval estimation of the risk ratio between a secondary infection, given a primary infection, and the primary infection. Biometrics 54, 706–711].The risk ratio (RR) between the secondary infection, given the primary infection, and the primary infection may be a useful measure of change in the pneumonia infection rates of the primary infection and the secondary infection. In this paper, we first develop and evaluate the large sample confidence intervals of RR. In addition to the three confidence intervals in the literatures, we propose a confidence interval based on Rao's score test. The performance of these confidence intervals is studied by means of extensive simulation studies. We also investigate the tests of hypothesis for the RR and the power of these tests. Simulation studies are carried out to examine the performance of these tests in terms of their power. An example, from the literature, is also provided to illustrate these procedures. Finally, the four confidence intervals were compared with those obtained by corrected version of Tang et al. [2004. Confidence interval for rate ratio in a 2×22×2 table with structural zero: an application in assessing false-negative rate ratio when combining two diagnostic tests. Biometrics 60, 550–555; 2006. Sample size determination for 2-step studies with dichotomous response. Journal of Statistical Planning and Inference 136, 1166–1180].