Abstract: | Random variables A and B, whose joint distribution depends on factors (x,y), are selectively influenced by x and y, respectively, if A and B can be represented as functions of, respectively, (x,SA,C) and (y,SB,C), where SA,SB,C are stochastically independent and do not depend on (x,y). Selective influence implies selective dependence of marginal distributions on the respective factors: thus no parameter of A may depend on y. But parameters characterizing stochastic interdependence of A and B, such as their mixed moments, are generally functions of both x and y. We derive two simple necessary conditions for selective dependence of (A,B) on (x,y), which can be used to conduct a potential infinity of selectiveness tests. One condition is that, for any factor values x,x′ and y,y′, sxy≤sxy′+sx′y′+sx′y, |