The impact of unmeasured within- and between-cluster confounding on the bias of effect estimatorsof a continuous exposure

Unmeasured confounding almost always exists in observational studies and can bias estimates of exposure effects. Instrumental variable methods are popular choices in combating unmeasured confounding to obtain less biased effect estimates. However, we demonstrate that alternative methods may give less biased estimates depending on the nature of unmeasured confounding. Treatment preferences of clusters (e.g. physician practices) are the most frequently used instruments in instrumental variable analyses. These preference-based instrumental variable analyses are usually conducted on data clustered by region, hospital/facility, or physician, where unmeasured confounding often occurs within or between clusters. We aim to quantify the impact of unmeasured confounding on the bias of effect estimators in instrumental variable analysis, as well as several common alternative methods including ordinary least squares regression, linear mixed models, and fixed-effect models to study the effect of a continuous exposure (e.g. treatment dose) on a continuous outcome. We derive closed-form expressions of asymptotic bias of estimators from these four methods in the presence of unmeasured within- and/or between-cluster confounders. Simulations demonstrate that the asymptotic bias formulae well approximate bias in finite samples for all methods. The bias formulae show that instrumental variable analyses can provide consistent estimates when unmeasured within-cluster confounding exists, but not when between-cluster confounding exists. On the other hand, fixed-effect models and linear mixed models can provide consistent estimates when unmeasured between-cluster confounding exits, but not for within-cluster confounding. Whether instrumental variable analyses are advantageous in reducing bias over fixed-effect models and linear mixed models depends on the extent of unmeasured within-cluster confounding relative to between-cluster confounding. Furthermore, the impact of unmeasured between-cluster confounding on instrumental variable analysis estimates is larger than the impact of unmeasured within-cluster confounding on fixed-effect model and linear mixed model estimates. We illustrate the use of these methods in estimating the effect of erythropoiesis stimulating agents on hemoglobin levels. Our findings provide guidance for choosing appropriate methods to combat the dominant types of unmeasured confounders and help interpret statistical results in the context of unmeasured confounding.