Directionally collapsible parameterizations of multivariate binary distributions

TL;DR

This paper explores the concept of directionally collapsible parameters in multivariate binary distributions, demonstrating that only linear contrasts of cell probabilities can reliably prevent reversals in association direction, thus avoiding Simpson's paradox.

Contribution

It proves that under certain assumptions, no conditional distribution-dependent parameter can be directionally collapsible, and shows that linear contrasts are uniquely suited for this purpose.

Findings

Linear contrasts of cell probabilities are the only directionally collapsible parameters.

No conditional distribution-dependent parameter can be directionally collapsible under the assumptions.

Using linear contrasts prevents the reversal of association direction, avoiding Simpson's paradox.

Abstract

Odds ratios and log-linear parameters are not collapsible, meaning 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