Summary Pilot phases of a randomized clinical trial often suggest that a parametric model may be an accurate description of the trial’s longitudinal trajectories. However, parametric models are often not used for fear that they may invalidate tests of null hypotheses of equality between the experimental groups. Existing work has shown that when, for some types of data, certain parametric models are used, the validity for testing the null is preserved even if the parametric models are incorrect. Here, we provide a broader and easier to check characterization of parametric models that can be used to (i) preserve nonparametric validity of testing the null hypothesis, i.e., even when the models are incorrect, and (ii) increase power compared to the non- or semiparametric bounds when the models are close to correct. We demonstrate our results in a clinical trial of depression in Alzheimer’s patients.