Attributable effects in case2-studies

In an effort to determine whether a particular treatment causes a particular outcome event, data are obtained from a database system that records events when they occur, and for such events, the system records exposure to the treatment. That is, the system records information about cases. The system provides no information about events that might have occurred but did not, that is, about units which are not cases. Roughly speaking, we know the number of successes for two proportions, treated and control, but not the numbers of trials or units for these proportions; indeed, the concept of a "trial" may be somewhat vague. With no further information, the situation is quite hopeless. However, an interesting strategy that is sometimes used entails identifying two types of cases whose origin is entirely different so that it is known the cases of the second type were definitely not affected by the treatment under study. This strategy--the case-case or case2-study--seems to have been reinvented independently many times, and has recently been offered as a general strategy for infectious disease epidemiology by McCarthy and Giesecke (1999, International Journal of Epidemiology 28, 764-768). Can this strategy permit estimation of the number of cases caused by the treatment? Using attributable effects in a new way, a method of exact inference is proposed, along with a large sample approximation. Two examples are discussed: one concerning the effects of daytime running lights (DRLs) on the risk of multivehicle accidents; the other concerning the origin of a Salmonella infection. A counterexample with superficially similar appearance is also discussed concerning suicide rates following the publication of Final Exit; here, the treatment may alter the outcome, or it may alter the type, and the attributable effect cannot be estimated.