First and second authors contributed equally.


The conditional relative risk is an important measure in medical and epidemiological studies when the outcome of interest is binary (i.e. disease vs. no disease). When the outcome is common, estimation of conditional relative risk and related parameters can be problematic, especially when the exposure or covariates are continuous. We propose a new estimation procedure based on targeted maximum likelihood methodology that targets the parameters relating to the conditional relative risk for common outcomes under a log-linear, or multiplicative, semi-parametric model. In this paper, we present three possible targeted maximum likelihood estimators for relative risk parameters implied by such a model: log-binomial based, Poisson-based, and a general semi-parametric approach. We present the properties and trade-offs of each of these estimators, focusing in particular on the Poisson-based estimator, which is most practical for implementation. We show that the resulting estimator is double robust and asymptotically linear, and inference can be obtained using the corresponding influence curve. The robustness of our estimator is compared to alternative methods (e.g. log-linear, Poisson regression) through simulation under model misspecification and increasing violations of the positivity assumption. The estimation procedure is then applied to a study of HIV genetic susceptibility scores, which aims to determine the effects of different genetic susceptibility scores on viral response. Effect modification by other covariates in the model is also explored.



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