Identifying the odds ratio estimated by a two‐stage instrumental variable analysis with a logistic regression model

Abstract

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. An instrumental variable can be used 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 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 randomized controlled trial 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 which has arguably more relevance to an epidemiologist or a policy-maker, especially in the context of Mendelian randomization.

Network Mendelian randomization: using genetic variants as instrumental variables to investigate mediation in causal pathways

“…We consider the situation where the exposure, mediator and outcome are all continuous and assume that the effects of the exposure on the mediator ( X on Z ), and of the exposure and mediator on the outcome [( X , Z ) on Y ] are linear without interactions. Similar methods could be used in a case of a binary exposure, mediator and/or outcome, but we do not address the additional complications of non-collapsibility that arise in this paper 14 , 15 . We allow unmeasured confounding of the exposure–mediator, exposure–outcome and mediator–outcome relationships.…”

Modal-based estimation via heterogeneity-penalized weighting: model averaging for consistent and efficient estimation in Mendelian randomization when a plurality of candidate instruments are valid

“…The standard error of the association estimate with the outcome is $\text{se}({\widehat{\beta }}_{Yj})$. If any of the variables is binary, then these summarized association estimates may be replaced with association estimates from logistic regression; as has been shown previously, the interpretation of the causal estimate in this case is not clear due to non-collapsibility, but estimates still represent valid tests of the causal null hypothesis 12 , 13 . See Bowden et al 14 .…”

Blood lipids and colorectal polyps: testing an etiologic hypothesis using phenotypic measurements and Mendelian randomization

“…For each of LDL, HDL, TG, and TC, we estimated 3 primary associations: 1) lipid-polyp odds ratios (OR) with 95% confidence intervals (CI) comparing each case group to controls using polytomous logistic regression; 2) genotype-lipid associations using ordinary linear regression; and 3) genotype-polyp Mendelian randomization ORs using 2-stage linear-logistic regression [17]. We used trait-specific allele scores created by counting alleles associated with an increased mean in the GLGC GWAS, weighted by effect size from their analysis (for HDL, the score was based on alleles associated with decreased mean HDL) [18].…”