Structural mean models for instrumented difference-in-differences

Abstract

In the standard difference-in-differences research design, the parallel trend assumption can be violated when the effect of some unmeasured confounders on the outcome trend is different between the treated and untreated populations. Progress can be made if there is an exogenous variable that (i) does not directly influence the change in outcome (i.e. the outcome trend) except through influencing the change in exposure (i.e. the exposure trend), and (ii) is not related to the unmeasured exposure - outcome confounders on the trend scale. Such exogenous variable is called an instrument for difference-in-differences. For continuous outcomes that lend themselves to linear modelling, so-called instrumented difference-in-differences methods have been proposed. In this paper, we will suggest novel multiplicative structural mean models for instrumented difference-in-differences, which allow one to identify and estimate the average treatment effect that is stable over time on the multiplicative scale, in the whole population or among the treated, when (i) a valid instrument for difference-in-differences is available and (ii) there is no carry-over effect across periods. We discuss the identifiability of these models, then develop efficient semi-parametric estimation approaches that allow the use of flexible, data-adaptive or machine learning methods to estimate the nuisance parameters. We apply our proposal on health care data to investigate the risk of moderate to severe weight gain under sulfonylurea treatment compared to metformin treatment, among new users of antihyperglycemic drugs.