RE: “APPROPRIATE ASSESSMENT OF NEIGHBORHOOD EFFECTS ON INDIVIDUAL HEALTH: INTEGRATING RANDOM AND FIXED EFFECTS IN MULTILEVEL LOGISTIC REGRESSION”

Larsen and Merlo (1) propose some novel transformations to represent effects in multilevel logistic regression. These measures, the “median odds ratio” and “interval odds ratio,” are functions of the estimated slope coefficients and the estimated cluster-level variance (i.e., random effect), and they have attractive interpretations for the effects of cluster-level predictors. Irrespective of the merits of these new measures, I find the explanation provided for why traditional approaches are inadequate to be somewhat less persuasive.

I'll consider binary predictors (exposed coded as 1, and unexposed coded as 0) to simplify this discussion, although generalization to any interval scale exposure is straightforward. The usual approach for effect estimation in logistic regression is to exponentiate the estimated slope coefficient and to interpret this as a disease odds ratio (i.e., odds of disease in the exposed divided by odds of disease in the unexposed). This associational measure (i.e., comparing two distinct groups of individuals) can also have a causal interpretation (i.e., comparing one group of individuals under two exposure scenarios, one or both of which may be counterfactual). It is the causal interpretation that motivates the use of regression modeling in public health research, since this provides evidence on which to base policy decisions. In the case of logistic regression, this causal interpretation relies on not only an assumption of minimal uncontrolled confounding (2, pp. 363–365) but also a study design or outcome prevalence that allows the odds ratio to approximate the relative risk (2, pp. 52–53).

Larsen and Merlo argue that, in a multilevel logistic regression, the random effects (i.e., cluster-specific) interpretation for cluster-level predictors is problematic because it conditions on the random effect, as well as on all the other covariates that vary within the cluster. They assert that “… it is impossible to make comparisons within-cluster [because] all comparisons must be made between persons belonging to two different clusters” (1, p. 87). The imperative to interpret the effect measure as comparing two individuals with identical covariates but contrasting exposures, however, describes the associational interpretation, not the causal interpretation that is fundamentally of interest.

For illustration, let Z_j represent a binary indicator of whether or not community j has a hospital (coded 1 if a hospital exists in community j, 0 otherwise).

Larsen and Merlo suggest that exp(γ₀₁) cannot be readily interpreted as an odds ratio for the effect of presence of a hospital on outcome Y because the cluster-specific model conditions on μ_0j, and it is therefore very unlikely that one would find two individuals with the same values of μ_0j and X but with different values of Z. For the causal estimate, however, in which one or both of the exposure states may be counterfactual, conditioning interpretation on the community is unproblematic. The quantity exp(γ₀₁) estimates a disease odds ratio in which the numerator is the odds of outcome Y in community j if this community were to have a hospital, and the denominator is the odds of outcome Y in community j if this community were to have no hospital. If the factual condition from some community j is that there is no hospital present, then this estimated causal effect is interpretable as the effect of the hypothetical action of putting one there. In contrast, if one were to instead fit a population-averaged model, then the quantity exp(γ₀₁) estimates a causal disease odds ratio in which the numerator is the odds of outcome Y in the population if all communities were modified to have hospitals, and the denominator is the odds of outcome Y in the population if hospitals were to be removed from all communities. In nonlinear models such as logistic regression, these cluster-specific and population-averaged effect estimates will generally differ (3), although it is noteworthy that if one estimates the relative risk directly by using a natural log link in place of a logit link, then the effect estimates have both a marginal and a conditional interpretation (4).

There are a number of reasons why a causal estimate from a regression model may be biased, including unmeasured confounding, misclassification, and so forth. Whether a given model can support a causal interpretation is an issue of model specification, variable measurement, and conceptual validity. Furthermore, when there is poor overlap between the exposed and unexposed across the multivariate distribution of covariates, model assumptions such as linearity may become particularly consequential in obtaining the causal effect estimate, which is a precarious situation (5). Nonetheless, the cluster-specific effect in a multilevel logistic model does not seem to have any unique disadvantage in this regard. Other dilemmas of this model form have been pointed out previously (6), but it seems that the epidemiologic interpretability of the cluster-level predictor in a cluster-specific model is not fundamentally inadequate, as Larsen and Merlo suggest.

Conflict of interest: none declared.

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