Directionally collapsible parameterizations of multivariate binary distributions

Odds ratios and log-linear parameters are not collapsible, which means that including a variable into the analysis or omitting one from it, may change the strength of association among the remaining variables. Even the direction of association may be reversed, a fact that is often discussed under the name of Simpson’s paradox. A parameter of association is directionally collapsible, if this reversal cannot occur. The paper investigates the existence of parameters of association which are directionally collapsible. It is shown, that subject to two simple assumptions, no parameter of association, which depends only on the conditional distributions, like the odds ratio does, can be directionally collapsible. The main result is that every directionally collapsible parameter of association gives the same direction of association as a linear contrast of the cell probabilities does. The implication for dealing with Simpson’s paradox is that there is exactly one way to associate direction with the association in any table, so that the paradox never occurs.