On Instrumental Variables Estimation of Causal Odds Ratios

Abstract

Inference for causal effects can benefit from the availability of an instrumental variable (IV) which, by definition, is associated with the given exposure, but not with the outcome of interest other than through a causal exposure effect. Estimation methods for instrumental variables are now well established for continuous outcomes, but much less so for dichotomous outcomes. In this article we review IV estimation of so-called conditional causal odds ratios which express the effect of an arbitrary exposure on a dichotomous outcome conditional on the exposure level, instrumental variable and measured covariates. In addition, we propose IV estimators of so-called marginal causal odds ratios which express the effect of an arbitrary exposure on a dichotomous outcome at the population level, and are therefore of greater public health relevance. We explore interconnections between the different estimators and support the results with extensive simulation studies and three applications.Comment: Published in at http://dx.doi.org/10.1214/11-STS360 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

“…We found that the two-stage method using the whole sample in the first-stage regression gave estimates close to the PLOR. The adjusted two-stage method gave biased results in this case, even with no covariate effect, similar to the bias of the adjusted two-stage method observed by Vansteelandt et al under the null [3].…”

“…For a binary outcome Y = 0, 1, the conditional odds ratio (COR) is defined as the odds ratio for unit increase in the risk factor from x to x + 1 for a given value of v: (1) where and Y|(σ X = x, V = v) is the outcome random variable, conditional on covariate level v, where the risk factor level is set to x using the σ X notation for intervention Pearl [21]. Under the assumptions that Y equals Y|(σ X = x, V = v) if X = x and V = v (consistency) and that Y(x) ⫫ X|V (ignorability) [22], this can be expressed as: (2) In general, the COR may be a function of x and v, although in a logistic-linear model of association, where the logit of the probability of outcome (π) is a linear function in X and V with no interaction term: (3) the COR is exp(β 1 ) independently of x and v. This is the odds ratio estimated by a logistic regression of Y on X and V.…”

“…Secondly, the two-stage and adjusted two-stage estimands are of interest as they are similar to the odds ratios estimated by other IV methods. For example, in the case that the risk factor is normally distributed, the logistic structural mean model estimate is equivalent to that from an adjusted two-stage approach [3]. Thirdly, the IV estimand is of interest when it is close to the population odds ratio, a parameter of intrinsic natural interest.…”

“…If the IV explained a large proportion of the variation in the risk factor, such as treatment assignment as an IV for treatment received in a RCT, the finding that the IV estimand and the population odds ratio are similar may not be valid. An alternative approach in this case is to use an alternative method which specifically targets a specific marginal or a conditional odds ratio [3].…”

Improving the accuracy of two-sample summary data Mendelian randomization: moving beyond the NOME assumption

“…For example, the modelling assumptions are very often knowingly violated in applied MR studies when the outcome is binary and the SNP-outcome association is measured an odds ratio [5]. Indeed, the MR study we analyse in this paper relates to a binary outcome.…”