Statistical Inference for the Occurrence Probability of Rare Events-Rule of Three and Related Topics-

Abstract

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, n k 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.