For the occurrence of a rare event A such as a severe adverse drug reaction, there exists the “Rule of Three” to remind practitioners that “absence of evidence is not evidence of absence.” The Rule of Three actually says that even if the event A was not observed among n patients it would be quite possible to observe three events among other n patients. The present paper examines this useful rule in detail and also extends it to a testing problem for occurrence probability of A. First, the Rule of Three is extended to the case that the number of the event observed among the first n patients is more than zero. We give rules that when k (> 0) events were observed among n patients, nk events would be possibly observed among other n patients. Next, a testing procedure is introduced to examine whether the occurrence probabilities of A for two populations are the same under the condition that k events were observed among n patients for one population. It will be shown that the relevant probability distribution is a negative binomial, and then critical regions for small k's are given. For a possible application of the procedure, we mention the signal detection for spontaneous reporting system of adverse drug reaction.
The sign test is a best known statistical technique to make inferences about the population median, and has been applied in various research fields. A confidence interval is also easily obtained by inverting the test. In spite of their wide applicability the sign test and the confidence interval still have several problems. One notable difficulty is the conservativeness of confidence intervals in the sense that the actual confidence coefficients are greater than the nominal value. The inconsistency between testing and estimation in the presence of ties is another problem. The aim of the present paper is to propose a modified procedure to reduce such difficulties. In constructing confidence intervals, the mid-P value is also taken into account in addition to the usual P-value. Consequently, our modified intervals have actual confidence coefficient closer to the nominal value than those obtained by the conventional method. A modification of usual sign test is also introduced, which is consistent with the interval estimation. Modified confidence intervals obtained by inverting the Wilcoxon signed rank test and the Wilcoxon-Mann-Whitney rank sum test are also discussed.
Recurrent events data such as epileptic seizures and recurrence of superficial bladder cancer are frequently encountered in medical researches when individuals may experience multiple events of the same type. The analysis of recurrent events is complicated because related recurrent events within a subject are correlated and we need to take into account the dependence of responses from the same subject to draw valid statistical inferences. In principle, statistical strategies are classified into two approaches. The one is we focus on the number of events occurring within defined time intervals and compare / model the event rate (number of events per unit of time). The other is the recurrence times are viewed as multivariate failure times and survival analysis methods are applied. According to this perspective, we review several statistical methods to analyze recurrent events data and illustrate the techniques with real medical applications. We recommend that the choice of the endpoint (effect measure) and the corresponding statistical analysis method should be determined by the study purpose. Robust methods for the assumption of event occurring process should be used especially for analyzing confirmatory studies.