Effects of Omitting Non-confounding Predictors From General Relative-Risk Models for Binary Outcomes

Abstract

Background: The effects, in terms of bias and precision, of omitting non-confounding predictive covariates from generalized linear models have been well studied, and it is known that such omission results in attenuation bias but increased precision with logistic regression. However, many epidemiologic risk analyses utilize alternative models that are not based on a linear predictor, and the effect of omitting non-confounding predictive covariates from such models has not been characterized.

Methods: We employed simulation to study the effects on risk estimation of omitting non-confounding predictive covariates from an excess relative risk (ERR) model and a general additive-multiplicative relative-risk mixture model for binary outcome data in a case-control setting. We also compared the results to the effects with ordinary logistic regression.

Results: For these commonly employed alternative relative-risk models, the bias was similar to that with logistic regression when the risk was small. More generally, the bias and standard error of the risk-parameter estimates demonstrated patterns that are similar to those with logistic regression, but with greater magnitude depending on the true value of the risk. The magnitude of bias and standard error had little relation to study size or underlying disease prevalence.

Conclusions: Prior conclusions regarding omitted covariates in logistic regression models can be qualitatively applied to the ERR and the general additive-multiplicative relative-risk mixture model without substantial change. Quantitatively, however, these alternative models may have slightly greater omitted-covariate bias, depending on the magnitude of the true risk being estimated.

Keywords: bias; binary outcomes; general relative-risk model; generalized nonlinear model; logistic regression; omitted covariate.

Figure 1.. Simulated relative bias in the estimated excess relative risk (ERR) for exposure from a fit of model (2), as a function of true excess relative risk and various combinations of the relative risk (RR) of omitted covariates: body mass index (BMI) and parity. Cohort size N = 20,000, baseline prevalence p₀ = 0.05, and control:case sampling ratio 2:1.

Figure 2.. Simulated absolute bias: the difference between that for a true excess relative risk (ERR) model parameter (calculated with the generalized nonlinear model) and that for a true logistic regression model parameter (rescaled to ERR by [exp{log(RR)} − 1], where RR is relative risk), with risk at unit dose the same under each model. Includes all simulation configurations: cohort size N = 5,000 or 20,000; baseline prevalence p₀ = 0.015, 0.05, or 0.1; and control:case ratio m = 2 or 5 (although some configurations could not be evaluated, as described in the text). Results for three different pairs of magnitudes of omitted-covariate effect sizes are shown (see figure legend). BMI, body mass index.

Figure 3.. Simulated relative bias in the exposure risk estimate from the mixture model as a function of the mixture parameter. Based on cohort size N = 20,000, baseline prevalence p₀ = 0.05, and control:case ratio m = 2. Simulated relative biases for the excess relative risk (ERR) and logistic regression (log RR, where RR is relative risk) models are shown next to their equivalents in the mixture model for comparison. BMI, body mass index.