The Goodness of Match

Though the statistical techniques vary, the matching problem is essentially the same in each case and can be stated formally as follows: Given "observations on X,Y from one sample and on X,Z from another sample, when will it be true that by matching observations according to X, an artificial Y,Z sample will result whose distribution is the true joint Y,Z distribution?"(Sims,1972, p. 355). Though the imputed Y,Z distribution will, in general, be different from the true Y,Z distribution, the closeness of the two yields a natural criterion of the goodness of match. By making certain simplifying assumptions, we can make this criterion operational. The goodness of match depends on how much of the relation between Y and Z is transmitted through X - that is, on how X "mediates" between Y and Z. Since the functional form the lower and upper bounds on the true correlation between Y and Z takes depends on the number of X variables, we shall treat the problem in three stages: (a) The case of one mediating variable.(b) The case of two mediating variables. (c) The case of n mediating variables.