The exposition above assumes that the causal effect of interest does not vary across observations, that is, that formula_4 is a constant. Generally, different subjects will respond in different ways to changes in the "treatment" "x". When this possibility is recognized, the average effect in the population of a change in "x" on "y" may differ from the effect in a given subpopulation. For example, the average effect of a job training program may substantially differ across the group of people who actually receive the training and the group which chooses not to receive training. For these reasons, IV methods invoke implicit assumptions on behavioral response, or more generally assumptions over the correlation between the response to treatment and propensity to receive treatment.