Instrumental variables regression with latent variables: Accounting for treatment-based differential item functioning as item-level heterogeneity or item parameter moderation

Recent research has highlighted that modeling heterogeneous treatment effects across the individual items on an outcome measure can yield additional insights into interventions' effects beyond what is available from comparisons of average scores. So far, this work has generally focused on randomized controlled trials. However, in social science research, randomization often does not hold perfectly, or may be impossible or impractical. This can lead to treatment endogeneity---correlation between the predictor and the error term of the outcome. Instrumental variables regression (IVR) is a popular method for correcting the resulting bias by introducing an exogenous variable that drives change in the predictor but is otherwise uncorrelated with the outcome. Though typically estimated via two-stage least squares, IVR can also be estimated in a latent variable framework. We develop two models for estimating treatment effects that are heterogeneous across the items used to measure a latent outcome in an IVR, supporting valid causal inferences about impacts of nonrandom treatments/dosages on test items. We present methods for modeling items as fixed when most items are equally sensitive to treatment, or random when item-level variation in treatment sensitivity is pervasive. We outline the affordances and tradeoffs of the two approaches. We illustrate the use of these models via an empirical example analyzing potential item-level variability in sensitivity to time spent on homework using data a large-scale international assessment of high school mathematics.