Bias and Variance of Adjusting for Instruments

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Abstract
Propensity score adjustment is commonly used in observational research to address confounding. Controversy persists about how to select covariates as possible confounders to generate the propensity model. A desire to include all possible confounders is offset by a concern that more covariates will augment bias or increase variance. Much of concern is over instruments, which are variables that affect the treatment but not the outcome. Adjusting for an instrument has been shown to increase bias due to unadjusted confounding and to increase the variance of the effect estimate. Large-scale propensity score (LSPS) adjustment includes most available pre-treatment covariates in its propensity model. It addresses instruments with a pair of diagnostics, ceasing the analysis if any covariate exceeds a correlation coefficient of 0.5 with the treatment and checking for an aggregation of instruments with equipoise reported as a preference score. Our simulation assesses the impact of adjusting for instruments in the context of LSPS’s diagnostics. In our simulation, even when the variance of the treatment contributed by the adjusted instrument(s) exceeds an unadjusted confounder by over twenty-fold, when the correlation between the instrument(s) and the treatment was less than 0.5 and the equipoise was greater than 0.5, the additional shift in the effect estimate due to adjusting for the instrument(s) was less than the shift due to confounding by itself. Therefore, we find in this simulation that adjusting for instruments contributed a minor amount of bias to the effect estimate. This simulation aligns well with a previous assessment of the impact of adjusting for instruments and with separate empirical evidence that adjusting for many covariates surpasses attempts to identify a limited set of confounders.

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