On effect-measure modification: Relationships among changes in the relative risk, odds ratio, and risk difference

It is well known that the presence or absence of effect-measure modification depends upon the chosen measure. What is perhaps more disconcerting is that a positive change in one measure may be accompanied by a negative change in another. Therefore, research demonstrating that an effect is 'stronger' in one population when compared with another, but based on only one measure, for example, the odds ratio, may be difficult to interpret for researchers interested in another measure. The present article investigates relationships among changes in the relative risk, odds ratio, and risk difference from one stratum to another. Monte Carlo integration shows that the three measures change in the same direction for 78 or 89 per cent of the volume of the geometric space defined by the four underlying proportions, depending on whether the strata are presumed to share the same direction of effect or not. Analytic results are presented concerning necessary and sufficient conditions for the measures to change in opposite directions. In general, the conditions are seen to be quite complicated, though they do give way to some interesting results. For example, when exposure increases risk but all risks are less than 0.5, it is impossible for the relative risk and risk difference to change in the same direction but opposite to that of the odds ratio. Both data-analytic and hypothetical examples are presented to demonstrate circumstances under which the measures change in opposite directions.