  The Yule-Simpson paradox notes that an association between random variables
can be reversed when averaged over a background variable. Cox and Wermuth
(2003) introduced the concept of distribution dependence between two random
variables X and Y, and developed two dependence conditions, each of which
guarantees that reversal cannot occur. Ma, Xie and Geng (2006) studied the
collapsibility of distribution dependence over a background variable W, under a
rather strong homogeneity condition. Collapsibility ensures the association
remains the same for conditional and marginal models, so that Yule-Simpson
reversal cannot occur. In this paper, we investigate a more general condition
for avoiding effect reversal: A-collapsibility. The conditions of Cox and
Wermuth imply A-collapsibility, without assuming homogeneity. In fact, we show
that, when W is a binary variable, collapsibility is equivalent to
A-collapsibility plus homogeneity, and A-collapsibility is equivalent to the
conditions of Cox and Wermuth. Recently, Cox (2007) extended Cochran's result
on regression coefficients of conditional and marginal models, to quantile
regression coefficients. The conditions of Cox and Wermuth are sufficient for
A-collapsibility of quantile regression coefficients. If the conditional
distribution of W, given Y = y and X = x, belong to one-dimensional natural
exponential family, they are also necessary. Some applications of
A-collapsibility include the analysis of a contingency table, linear regression
models and quantile regression models.
