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 introduced a new 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 studied the collapsibility of distribution dependence over a backround variable W, un- der 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 notion called 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 extended Cochran's result on regression coefficients of conditional and marginal models, to quantile re- gression coefficients. The conditions of Cox and Wermuth are also sufficient for A-collapsibility of quantile regression coefficients. Under a conditional completeness assumption, they are also necessary.
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