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A-Collapsibility of Distribution Dependence and Quantile Regression Coefficients

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.