A general condition for avoiding effect reversal after marginalization

The paper examines the effect of marginalizing over a possibly unobserved background variable on the conditional relation between a response and an explanatory variable. In particular it is shown that some conclusions derived from least squares regression theory apply in general to testing independence for arbitrary distributions. It is also shown that the general condition of independence of the explanatory variable and the background ensures that mono- tonicity of dependence is preserved after marginalization. Relations with effect reversal and with collapsibility are sketched.