The Odds Ratios of a 2 × 2 Contingency Table

We 1 have earlier reported on the feasibility for setting exact limits on the underlying odds ratio of a 2 X 2 contingency table. We then used the wellestablished principle that conditional on all marginals of the 2 X 2 table, the distribution of the table, i.e. of any cell entry in the table, depended only on the true odds ratio. Here, we intend to review some other aspects of the all-marginals-fixed distribution, making contrasts with the cases variously of no fixed quantities, only grand total fixed, or only one set of marginals fixed. Whichever conditioning or lack of conditioning is made, a maximum likelihood requirement in parameter estimation turns out to be that the expected table should equal the observed table. The odds ratio of the fitted expected table is thus equal to that of the observed table, yet in the instance of the doubly-conditioned distribution the observed table odds ratio is not the maximum likelihood estimate. More general]y, the ratio of crossproduct expectations, E(A) E(D) E(B) E(C), where expectatioins are based on a specified value for the odds ratio, fails to yield that specified value for the odds ratio in the doubly-conditioned situation. For all sampling situations, as we shall bring out, E(AD) E(BC) does equal the parametric odds ratio. (The quantity E(AD/BC) is not appropriate to consider since it is infinite in non-asymptotic situations for which there is a positive probability for a zero cell outcome.) In connection with the doubly-conditioned situation we shall give calculations for chi square alternative to those we previously published. We begin by identifying the appearance of a sample 2 X 2 table outcome with row totals, N1 and N2, column totals MII, and A12, cell entries, A, B, C, and D, grand total T, the table taking the form