|The Clopper-Pearson interval has coverage probabilities bounded below by the nominal confidence level, but the typical coverage probability is much higher than that level. The score and adjusted Wald intervals can have coverage probabilities lower than the nominal confidence level, yet the typical coverage probability is close to that level. In forming a 95% confidence interval, is it better to use an approach that guarantees that the actual coverage probabilities are at least 0,95 yet typically achieves coverage probabilities of about 0,98 or 0,99, or an approach giving narrower intervals for which the actual coverage probability could be less than 0,95 but is usually quite close to 0,95? For most applications, we would prefer the latter. The score and adjusted Wald confidence intervals for p provide shorter intervals with actual coverage probability usuallyu nearer the nominal confidence level. In particular, even though the score and adjusted Wald intervals leave something to be desired in terms of satisfying the usual technical definition of “95% confidence”, the operational performance of those methods is better than the exact interval in terms of how most practitioners interpret that term.|
Results similar to those in this article also hold in other discrete problems. For instance, similar comparisons apply for score, Wald, and exact confidence intervals for a Poisson parameter µ, based on an observation X from that distribution. Figure 5 illustrates, plotting the actual coverage probabilities when the nominal confidence level is 0,95. Here, the score interval for µ results from inverting the approximately normal test statistic [X], the Wald interval results from inverting [X], and the ENDPOINTS OF THE EXACT INTERVAL, (1/2)(X22X, 0,025, X22(X+1), 0,975), result from equating tail sums of null Poisson probabilities to 0,025 (Garwood 1936; for n independent Poisson observations, X1, ... Xn, the same formulas apply if one lets X = ƩXi e µ = E(X) = nE(Xi)). For another discrete example, ver Mehta e Walsh (1992) para uma comparação dos intervalos exatos com os de confiança de P médio for odds ratios or for a common odds ratio in several 2x2 contingency tables.
Exact inference has an important place in statistical inference of discrete data, in particular for sparse contingency table problems for which large-sample chi-squared statistics are ofter unreliable. However, approximate results are sometimes more useful than exact results, because of the inherent conservativeness of exact methods.