Modelling treatment-effect heterogeneity in randomized controlled trials of complex interventions (psychological treatments).

We describe instrumental variable (IV) methods for the estimation of the 'dose'-response effects of psychological interventions in randomized controlled trials in which there is variability in the patients' adherence to the allocated therapy (that is, variability in the actual number of sessions of therapy attended) and also variability in the strength of the therapeutic alliance between patients and their therapists. The effect of the therapy on outcome is assumed to be a function of both the number of sessions attended and the strength of the therapeutic alliance, with no intervention effects in the absence of any sessions attended (an exclusion restriction) and the effect of the strength of the alliance being represented by a multiplicative term (interaction) in the treatment-effect model. The IV methods that are described allow for: (a) hidden confounding between sessions, alliance and outcome; (b) measurement errors in the alliance; and (c) that alliance is only measured in those receiving treatment. Three two-stage estimation procedures are illustrated, and their equivalence demonstrated, through Monte Carlo simulation and analysis of the results of an actual trial.