Interaction as Departure from Additivity in Case-Control Studies: A Cautionary Note

Anders Skrondal ) 0 0 From the Division of Epidemiology, Norwegian Institute of Public Health , Oslo , Norway It has been argued that assessment of interaction should be based on departures from additive rates or risks. The corresponding fundamental interaction parameter cannot generally be estimated from case-control studies. Thus, surrogate measures of interaction based on relative risks from logistic models have been proposed, such as the relative excess risk due to interaction (RERI), the attributable proportion due to interaction (AP), and the synergy index (S). In practice, it is usually necessary to include covariates such as age and gender to control for confounding. The author uncovers two problems associated with surrogate interaction measures in this case: First, RERI and AP vary across strata defined by the covariates, whereas the fundamental interaction parameter is unvarying. S does not vary across strata, which suggests that it is the measure of choice. Second, a misspecification problem implies that measures based on logistic regression only approximate the true measures. This problem can be rectified by using a linear odds model, which also enables investigators to test whether the fundamental interaction parameter is zero. A simulation study reveals that coverage is much improved by using the linear odds model, but bias may be a concern regardless of whether logistic regression or the linear odds model is used. additivity; case-control studies; epidemiologic methods; interaction Abbreviations: AP, attributable proportion due to interaction; RERI, relative excess risk due to interaction; S, synergy index. - Logistic regression analysis is the workhorse of contemporary epidemiology. Consequently, assessment of interaction is often performed by simply introducing product terms into logistic risk models. This practice has been vehemently criticized by some epidemiologists, who argue that assessment of interaction should mainly be based on additive rate or risk models (17). For rare outcomes, this notion of interaction follows from probabilistic independence, as embodied in the classical toxicologic notion of simple independent action discussed by Finney (8). The purpose of this article is not to engage in the debate on how interaction should be conceptualized in epidemiology. Rather, I confine my investigation to the performance of suggested measures of interaction as departure from additivity. In cohort studies, the desired interaction assessment can easily be accomplished by fitting linear rate or risk models. However, the parameters of linear models cannot be validly estimated for case-control studies unless the sampling fractions for cases and controls are known or can be estimated. On the other hand, it is well known that odds ratios can be estimated in case-control studies. Furthermore, relative risks are often well approximated by odds ratios in case-control studies. On the basis of these observations, Rothman (1, 2) suggested a synergy index (S) which can be used in casecontrol studies to measure interaction as departure from additive risks. Moreover, Rothman considered statistical inference for the index, deriving confidence intervals using the delta method. Rothman presented several additional measures of interaction (3), including the relative excess risk due to interaction (RERI), renamed the ICR by Rothman and Greenland (6), and the attributable proportion due to interaction (AP), which is the focus in Rothmans latest book (7). Rothman furthermore pointed out (3, p. 324) that estimates of RERI, AP, and S are easily obtained from logistic regression analysis, as are Wald tests and confidence intervals (9). Alternatively, a likelihood ratio test of additive risks could be performed in the logistic regression model. Although this test would be expected to have better properties than the Wald test, it would be much harder to implement. Discussion of the measures advocated by Rothman is typically confined to the somewhat unrealistic situation in which there are two exposures but no additional covariates to control for confounding. An exception is Flanders and Rothman (10), who suggested a likelihood approach to estimating S from stratified case-control data. As Rothman acknowledged (3), their approach only handles one or possibly two additional covariates, because otherwise data in each stratum become too sparse. Hence, Rothman suggests invoking multivariate methods in estimating RERI, AP, and S when there are additional covariates. Specifically, Rothman states, Confounding factors can be controlled by including terms for those factors in the multiple logistic model (3, p. 324). This suggestion has been adhered to by epidemiologists (for instance, see Olsen et al. (11)). There has been a paucity of studies investigating the performance of RERI, AP, and S. The only paper I am aware of is that of Assmann et al. (12), where the investigation was limited to coverage of confidence intervals for RERI and AP in models without additional covariates. The primary concern in this article is the extent to which RERI, AP, and S are useful summary measures of interaction as departure from additive risks. In addition to the conventional approach based on logistic regression, I also suggest an alternative approach based on linear odds models. Attention is focused on the more realistic setting in which there are additional covariates. However, the concepts are best introduced in a setting with two exposures and no additional covariates. MODELS FOR TWO EXPOSURES Let Y be a dichotomous outcome variable with outcomes 1 and 0. Consider the case of two dichotomous exposure variables x1 and x2 with levels j = 0, 1 and k = 0, 1, respectively. Let Let Rjk P(Y = 1|xl, x2) be the conditional risk or probability that the outcome variable Y takes the value 1 given the values of the exposures. For all j and k, define risk differences as RDjk Rjk R00, relative risks as RRjk Rjk/ R00, odds as Ojk Rjk/(l Rjk), and odds ratios as ORjk Ojk/O00. The linear risk model A linear risk model is now specified as Rjk = a + b1x1 + b2x2 +b3x1x2, where it is assumed that a > 0, b1 > 0, and b2 > 0. It follows that a = R00, b1 = R10 R00 = RD10, and b2 = R0l R00 = RD0l. Hence, a is interpreted as the risk when there is no exposure, b1 as the excess risk under exposure x1 (compared with no exposure whatsoever), and b2 as the excess risk under exposure x2. The parameter b3 can be expressed as representing the excess risk due to interaction of the exposures. If b3 = 0, RD11 = RD01 + RD10, which is riskdifference additivity. According to Rothman (3, p. 320), b3 is the most fundamental epidemiologic measure of interaction. Unfortunately, the linear risk model cannot in general be validly estimated from case-control designs, unless the sampling fraction of cases and controls is known or can be estimated. Since this rarely appears to be the case, it follows that direct inferenc (...truncated)