A general approach to evaluating the bias of 2‐stage instrumental variable estimators

Abstract

Unmeasured confounding is a common concern when researchers attempt to estimate a treatment effect using observational data or randomized studies with nonperfect compliance. To address this concern, instrumental variable methods, such as 2-stage predictor substitution (2SPS) and 2-stage residual inclusion (2SRI), have been widely adopted. In many clinical studies of binary and survival outcomes, 2SRI has been accepted as the method of choice over 2SPS, but a compelling theoretical rationale has not been postulated. We evaluate the bias and consistency in estimating the conditional treatment effect for both 2SPS and 2SRI when the outcome is binary, count, or time to event. We demonstrate analytically that the bias in 2SPS and 2SRI estimators can be reframed to mirror the problem of omitted variables in nonlinear models and that there is a direct relationship with the collapsibility of effect measures. In contrast to conclusions made by previous studies (Terza et al, 2008), we demonstrate that the consistency of 2SRI estimators only holds under the following conditions: (1) when the null hypothesis is true; (2) when the outcome model is collapsible; or (3) when estimating the nonnull causal effect from Cox or logistic regression models, the strong and unrealistic assumption that the effect of the unmeasured covariates on the treatment is proportional to their effect on the outcome needs to hold. We propose a novel dissimilarity metric to provide an intuitive explanation of the bias of 2SRI estimators in noncollapsible models and demonstrate that with increasing dissimilarity between the effects of the unmeasured covariates on the treatment versus outcome, the bias of 2SRI increases in magnitude.

Matched or unmatched analyses with propensity‐score–matched data?

“…As shown in Figure , sin( θ ) is a distance metric measuring the dissimilarity between β 2 and α 1 . When β 2 and α 1 are perpendicular to each other ( ), the sine distance between them is the maximum (sin( θ ) = 1).…”

“…As shown in Figure 1, sin( ) is a distance metric measuring the dissimilarity between 2 and 1 . 13,14 When 2 and 1 are perpendicular to each other ( = 2 ), the sine distance between them is the maximum (sin( ) = 1). When 2 and 1 are parallel to each other in either direction ( = 0 or = ), the sine distance is the minimum (sin( ) = 0).…”

A note on a naive regression‐based test on the validity of an instrumental variable

“…Second, V is independent of U. Third, the fact that V is included in the treatment model (2) not in the outcome model (1) suggests the exclusion restriction assumption holds.…”

“…Proposition 1. Let L( |D, V) be the least squares projection of the error in the treatment model (2) onto the space spanned by (1, D, V ) and…”

“…Instrumental variable (IV) methods have been commonly used to address the bias from unmeasured confounding in observational studies. As depicted in Figure , a valid IV needs to satisfy three assumptions: (i) IV has a nonzero effect on treatment, (ii) IV is independent of unmeasured confounders, and (iii) IV cannot have a direct effect on outcome and all of its effects on outcome has to go through treatment. This assumption is also known as exclusion restriction .…”