Instrumental variable analysis with a nonlinear exposure-outcome relationship

Abstract

Background: Instrumental variable methods can estimate the causal effect of an exposure on an outcome using observational data. Many instrumental variable methods assume that the exposure-outcome relation is linear, but in practice this assumption is often in doubt, or perhaps the shape of the relation is a target for investigation. We investigate this issue in the context of Mendelian randomization, the use of genetic variants as instrumental variables.

Methods: Using simulations, we demonstrate the performance of a simple linear instrumental variable method when the true shape of the exposure-outcome relation is not linear. We also present a novel method for estimating the effect of the exposure on the outcome within strata of the exposure distribution. This enables the estimation of localized average causal effects within quantile groups of the exposure or as a continuous function of the exposure using a sliding window approach.

Results: Our simulations suggest that linear instrumental variable estimates approximate a population-averaged causal effect. This is the average difference in the outcome if the exposure for every individual in the population is increased by a fixed amount. Estimates of localized average causal effects reveal the shape of the exposure-outcome relation for a variety of models. These methods are used to investigate the relations between body mass index and a range of cardiovascular risk factors.

Conclusions: Nonlinear exposure-outcome relations should not be a barrier to instrumental variable analyses. When the exposure-outcome relation is not linear, either a population-averaged causal effect or the shape of the exposure-outcome relation can be estimated.

FIGURE 1.

Mean level of cardiovascular risk factors stratified by quintile of body mass index against mean value of body mass index in quintile (lines are ±1.96 standard errors).

FIGURE 2.

Nonlinear relationships between exposure and outcome for quadratic, J-shaped, U-shaped, and threshold relationship models.

FIGURE 3.

Distribution of body mass index in subgroups defined by genetic variant rs1421085: solid line, major homozygotes; dashed line, heterozygotes; dotted line, minor homozygotes. Densities are smoothed using a kernel-density method with a common bandwidth of 0.8.

FIGURE 4.

Directed acyclic graph of relationships among instrumental variable (IV) G, exposure X, IV-free exposure X₀, confounder U, and outcome Y.

FIGURE 5.

Localized average causal effect estimates of body mass index on glycated hemoglobin at various levels of body mass index from EPIC-InterAct data set: sliding window approach with window sizes 500, 1,000, 1,500, 2,000, 3,000, and 4,000 (top-left to bottom-right). Gray lines represent point wise 95% CIs.

FIGURE 6.

Localized average causal effect estimates of body mass index on cardiovascular risk factors at various levels of body mass index from EPIC-InterAct data set: sliding window approach with window size 2,000. Gray lines represent point wise 95% CIs.