In a study of a dose-response relationship, flexibility in modelling is essential to capturing the treatment effect when the mean effect of other covariates is not fully understood, so that observed treatment effect is not due to the imposition of a rigid model for the relationship between response, treatment, and other variables. A semiparametric additive linear mixed (SPALM) model (Ruppert et al. 2003) provides a tractable and flexible approach to modelling the influence of potentially confounding variables. In this paper, we present pure likelihood and Bayesian versions of the SPALM model. Both methods of inference are readily implementable, but the Bayesian approach allows coherent propagation of uncertainty in the model, and, more importantly, allows prediction of future experimental results for as yet untreated individuals, thus allowing an assessment of the merits of different dosing strategies. We motivate the use of the methodology with the Monitored Occlusion Treatment of Amblyopia Study (MOTAS), which investigated the relationship between duration of occlusion and improvement in visual acuity.



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