Abstract :
Given a finite population characterized by two attributes A and B, a factor C with n levels, one case of Simpsonʹs paradox (SP) occurs when A and B are positively associated within each level of C, but they are negatively associated or independent in the population. Given an attribute K, let be its complement. Assume the conditional proportions of the combinations of attributes , respectively, within each level of C are known to the analyst, but the proportions of the n subpopulations (corresponding to the n levels of C) in the population are not known to the analyst. The problem is to find conditions under which SP occurs, and find the probability of SP. The first part of the problem is solved completely for all n 2 using properties of copositive matrices, and the theorems of Cottle, Habetler, and Lemke (1970) and of Pereira (1972). The second part of the problem is solved partially.
