  This paper deals with the concept of equivalence between direct and indirect
effects of a treatment on a response using two sets of intermediate variables
and covariates. First, we provide criteria for testing whether two sets of
variables can estimate the same direct and indirect effects. Next, based on the
proposed criteria, we discuss the variable selection problem from the viewpoint
of estimation accuracy of direct and indirect effects, and show that selecting
a set of variables that has a direct effect on a response cannot always improve
estimation accuracy, which is contrary to the situation found in linear
regression models. These results enable us to judge whether different sets of
variables can yield the same direct and indirect effects and thus help us
select appropriate variables to estimate direct and indirect effects with cost
reduction or estimation accuracy.
