Re: "Estimating the Relative Risk in Cohort Studies and Clinical Trials of Common Outcomes"

In cohort studies and in randomized trials that address a short risk period, the parameter of interest is commonly the adjusted risk ratio (presumed to be constant across levels of the relevant covariate(s)). In 1998, Zhang and Yu (1) published a formula for the derivation of the adjusted risk ratio, RR, based on the adjusted odds ratio, OR, as estimated by the traditional logistic model and the incidence proportion of the outcome among the unexposed, P0: RR = OR/[(1 − P0) + (P0 × OR)]. However, McNutt et al. later pointed out that deriving an adjusted risk ratio estimate using this method “is incorrect and will produce a biased estimate when confounding is present” (2, p. 941) and suggested several alternative approaches for estimating the adjusted risk ratio (such as the stratified analysis, the log-binomial model, and the Poisson model). Although the examples provided by McNutt et al. clearly demonstrate the bias in the estimates of the adjusted risk ratio derived by using the Zhang and Yu method, the exact reason for the bias has been left without explication. However, understanding and correcting the underlying flaw in this method could be of theoretical and practical interest, especially given that the alternative approaches to estimation of the adjusted risk ratio are themselves subject to various limitations. Furthermore, the cautionary note by McNutt et al. (2) does not seem to have been heeded in the epidemiologic community, as reflected by the ongoing use of the Zhang and Yu method in scientific publications and by the fact that this formula is presented in a popular textbook of epidemiology (3). The flaw in the Zhang and Yu method is that the 2 inputs on which it is based are not mutually coherent. Specifically, the first input—the adjusted odds ratio (estimated using the traditional logistic model)—is conditional on the covariates at issue, whereas the second input—P0—is the population-averaged, marginal incidence proportion (among the unexposed). However, the odds ratio is a noncollapsible measure of association, so the conditional odds ratio is generally not equal to the marginal odds ratio (except under the null hypothesis) (4–6). Thus, if the researcher wishes to estimate a marginal adjusted risk ratio, then both of the inputs (i.e., odds ratio and P0) into the derivation of such an estimate must be marginal and not specific to any given covariate level. Accordingly, the necessary revision of the original Zhang and Yu formula requires that the conditional odds ratio be replaced as one of its inputs with the marginal odds ratio, which can be estimated using a marginal structural logistic model (4, 7). To demonstrate the validity of the revised Zhang and Yu method, we used the data from the hypothetical examples provided in Table 1 of the article by McNutt et al. (2). Specifically, we fitted a marginal structural logistic model estimated by inverse probability weighting, where the weights were the inverse of the probability of being exposed (for those who were exposed) or nonexposed (for those who were unexposed) (7). Because the weighting effectively creates a pseudopopulation with a balanced distribution of covariates across exposure categories, the incidence proportion of the outcome among the unexposed was estimated in this pseudopopulation, so as to make this input into the revised Zhang and Yu formula coherent with the second input (i.e., the unconfounded marginal odds ratio in the pseudopopulation). The results of this analytic approach, presented in Table 1, show that application of the revised Zhang and Yu method produces unbiased estimates of the marginal adjusted risk ratio for all of the scenarios considered. However, the only scenario in which the original Zhang and Yu method was valid is the one in which the true risk ratio is 1, because only for this scenario are the marginal and conditional odds ratios identical. Comparison of the Original and Revised Zhang and Yu Method for Estimating the Adjusted Risk Ratio in Studies of Acute Effects Abbreviations: aOR, adjusted odds ratio; aRR, adjusted risk ratio; D, disease; E, exposure; MSLM, marginal structural logistic model; RR, risk ratio; TLM, traditional logistic model. a Values when using the original Zhang and Yu method. b Values when using the revised Zhang and Yu method. c Data are from Table 1 in McNutt et al. (2). d In these columns, 1 indicates the presences of the exposure or disease and 0 indicates absence. Comparison of the Original and Revised Zhang and Yu Method for Estimating the Adjusted Risk Ratio in Studies of Acute Effects Abbreviations: aOR, adjusted odds ratio; aRR, adjusted risk ratio; D, disease; E, exposure; MSLM, marginal structural logistic model; RR, risk ratio; TLM, traditional logistic model. a Values when using the original Zhang and Yu method. b Values when using the revised Zhang and Yu method. c Data are from Table 1 in McNutt et al. (2). d In these columns, 1 indicates the presences of the exposure or disease and 0 indicates absence. I. Karp is a Fonds de la Recherche en Santé du Québec Junior 1 Scholar and Canadian Institutes of Health Research New Investigator. Conflict of interest: none declared.

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