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Directionally Collapsible Measures of Association

The talk addresses possible characteristics of measures of association among k binary variables. There are various measures of association proposed in the literature, but the most often used one is the odds ratio, which plays a central role in log-linear models, most importantly, in various kinds of Markov models. But not every analyst is entirely satisfied with this choice. In particular, lack of collapsibility is often cited as an undesirable property. The fact that even the direction of association may change after collapsing, is seen as paradoxical by many, as shown by the huge literature on 'Simpson's paradox'. Directional collapsibility means, that such a reversal cannot occur.

While odds ratios are not directionally collapsible, they are variation independent from lower dimensional marginals, which is usually considered a very desirable property when measuring association. The talk raises the question, whether there is a measure of association for k-dimensional binary distribution, which is both directionally collapsible and variation independent from lower dimensional marginals.

The answer is negative, at least, when two very simple assumptions, which seem reasonable if the variables indicate the presence or lack of characteristics, are made. Variation independence from the lower dimensional marginals implies that the direction of association, for every distribution, is the same as the direction found by the odds ratio, thus there is no directional collapsibility. On the other hand, every directionally collapsible measure of association gives the same direction of association as a simple contrast of the cell probabilities does.

Thus, when data analysts choose a measure (= concept) of association, they cannot hope to have both variation independence from the lower dimensional marginals and directional collapsibility.