A model for a proportional treatment effect on disease progression

Treatments intended to slow the progression of chronic diseases are often hypothesized to reduce the rate of further injury to a biological system without improving the current level of functioning. In this situation, the treatment effect may be negligible for patients whose disease would have been stable without the treatment but would be expected to be an increasing function of the progression rate in patients with worsening disease. This article considers a variation of the Laird Ware mixed effects model in which the effect of the treatment on the slope of a longitudinal outcome is assumed to be proportional to the progression rate for patients with progressive disease. Inference based on maximum likelihood and a generalized estimating equations procedure is considered. Under the proportional effect assumption, the precision of the estimated treatment effect can be increased by incorporating the functional relationship between the model parameters and the variance of the outcome variable, particularly when the magnitude of the mean slope of the outcome is small compared with the standard deviation of the slopes. An example from a study of chronic renal disease is used to illustrate insights provided by the proportional effect model that may be overlooked with models assuming additive treatment effects.