In progressive diseases such as Alzheimer's, treatments that slow progression should start early to preserve higher levels of functioning for a longer period. In corresponding clinical trials, treatment effects are usually expressed as mean differences on a clinical scale at fixed time points. Early in the disease course, however, these mean differences may appear small but may nonetheless correspond to an important slowing of disease progression. This complicates the appreciation of the relevance of observed treatment effects. We introduce a class of target parameters that quantify treatment effects on the time scale in longitudinal studies; for instance, in terms of time saved or percentage slowing of progression. We focus on data from randomized trials where the target parameters are identified under regularity assumptions. These target parameters remain well defined if treatment was not randomized, but additional untestable assumptions are required for identification. We propose general two-step estimators. In the first step, the data can be analyzed with standard methods for longitudinal data and standard software can thus be used. In the second step, summary statistics from the first step are used for inferences about the target parameters. The second step has been implemented in the TCT R package. We study the asymptotic properties and efficiency of these two-step estimators, and evaluate them in an extensive simulation study. These estimators are used in a phase 2/3 clinical trial for Alzheimer's disease, leading to important additional insights into the treatment effect.
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