May and Johnson compared the coverage probabilities and average interval lengths for three competing methods of constructing con"dence intervals for di!erences in correlated proportions, the Wald intervals, the modi"ed Wald intervals and the Quesenberry and Hurst intervals, through a Monte Carlo simulation, and they concluded that the Quesenberry and Hurst interval performs the best in terms of coverage probability and average length. However, several results for the Wald intervals and the Quesenberry and Hurst intervals shown in their Figures 1}3 were very conservative, which seems to me to indicate something wrong. I asked Dr. Johnson about this problem via e-mail and he replied 0we have redone the simulations and our new results are more line with what you are ,nding1, but 0the Quesenberry and Hurst interval seems to perform the best overall as stated in our original paper1.
However, regarding the coverage probabilities, or Monte Carlo simulation study, which is similar to, but more precise than, May and Johnson's, indicates that (a) the Quesenberry and Hurst interval is constantly anticonservative in most cases and (b) the modi"ed Wald interval is sometimes conservative but has better empirical coverage probabilities than the Quesenberry and Hurst interval. Our result seems to suggest the reason for their simulation results that the average length of the Quesenberry and Hurst intervals is consistently shorter than either form of the =ald intervals (p. 2133 of their paper). Considering that the coverage probability is the primary criterion for evaluating the goodness of con"dence intervals, the modi"ed Wald interval will be ranked as the best among the three methods based on our simulation study.
Furthermore, when the e$cient score method that I have recently proposed was added into this Monte Carlo simulation, the score method had the best performance of the four methods. These results are shown in Figures 1}4.
The details of our Monte Carlo simulation study are as follows. We generated 100,000 samples (note: May and Johnson generated 10,000 samples) of size n"25, 50, 75, 100 for the same set of values of the expected cell proportions for the discordant cells as that of May and Johnson's, which is shown in Table I. We constructed 95 per cent con"dence intervals for the di!erence in the discordant proportions. The standard error of the 95 per cent coverage rate for these simulations was 0)00069, and, because we made four comparisons at each combination of cell proportions, we used a Bonferroni adjustment for multiple comparison as was done by May and Johnson. Thus, we considered con"dence intervals as conservative if the empirical coverage probability was greater than 0)9517 and anticonservative if less than 0)9483. Figures 1}4 show the results of the simulations giving empirical coverage probabilities and control limits of the con"dence intervals.
Results are summarized as follows: