A model for a proportional treatment effect on disease progression

Treatments intended to slow the progression of chronic diseases are often h ypothesized to reduce the rate of further injury to a biological system wit hout improving the current level of functioning. In this situation. the tre atment effect may be negligible for patients whose disease would have been stable without the treatment hut would be expected to be an increasing func tion of the progression rate in patients with worsening disease. This artic le 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 di sease. Inference based on maximum likelihood and a generalized estimating e quations procedure is considered. Under the proportional effect assumption, the precision of the estimated treatment effect carl be increased by incor porating the functional relationship between the model parameters and the v ariance of the outcome variable. particularly when the magnitude of the mea n 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 illust rate insights provided by the proportional effect model that may be overloo ked with models assuming additive treatment effects.