Informative Allocation and Consistent Treatment Selection

When data in the form of a 2×2 treatment by response table are available, the better out of the two treatments is often selected using the odds ratio (cross product ratio). Such decisions do not depend on the allocation of the observations in the different treatment categories and may exhibit a counter-intuitive reversal property, called Simpson’s paradox. In cases when those receiving different treatments are potentially different, as in observational studies or in designed experiments with dropout or noncompliance, decisions taking into account the difference in observed allocations may be useful. Using an approach that postulates certain desirable properties of decision functions and derives further characteristics from these, a new decision procedure based on the cross sum ratio is investigated. This procedure is not only sensitive to allocation but also turns out to be the only selection procedure that avoids Simpson’s paradox. In addition to these logical advantages, the probability of wrong decision when using the cross sum ratio tends to be smaller than when using the cross product ratio. The application of the new decision procedure is illustrated by the re-analysis of data sets, some of which exhibit Simpson’s paradox when analyzed using the cross product ratio. Finally, generalizations of the decision procedures to 2×J decision tables are considered.