Identifying the Odds Ratio Estimated by a Two-stage Instrumental Variable Analysis with a Logistic Regression Model

An adjustment for an uncorrelated covariate in a logistic regression changes the true value of an odds ratio for a unit increase in a risk factor. Even when there is no variation due to covariates, the odds ratio for a unit increase in a risk factor also depends on the distribution of the risk factor. We can use an instrumental variable to consistently estimate a causal effect in the presence of arbitrary confounding. With a logistic outcome model, we show that the simple ratio or two-stage instrumental variable estimate is consistent for the odds ratio of an increase in the population distribution of the risk factor equal to the change due to a unit increase in the instrument divided by the average change in the risk factor due to the increase in the instrument. This odds ratio is conditional within the strata of the instrumental variable, but marginal across all other covariates, and is averaged across the population distribution of the risk factor. Where the proportion of variance in the risk factor explained by the instrument is small, this is similar to the odds ratio from a RCT without adjustment for any covariates, where the intervention corresponds to the effect of a change in the population distribution of the risk factor. This implies that the ratio or two-stage instrumental variable method is not biased, as has been suggested, but estimates a different quantity to the conditional odds ratio from an adjusted multiple regression, a quantity that has arguably more relevance to an epidemiologist or a policy maker, especially in the context of Mendelian randomization.