Conclusion
When the treatment effect in the trial is proportional to time, having non-linear trajectories has little or no effect on the empirical power, unless the proposed trial has a different visit schedule to the observational study. The extent of the effect on power depends on the underlying data generation parameters, for example the size of the residual error variance. If investigators anticipate that the treatment effect in the proposed trial will be proportional to time (or close to this), then powering the trial assuming a random slopes model should yield empirical powers close to those expected. However, if there are marked non-linearities in the trajectories, then observational data with the same visit schedule as the proposed trial should be used where possible, especially in settings where the random intercepts and slopes variances are comparable in size to the residual variance. Alternatively, investigators could switch the analysis model to one with a less restrictive structure, for example, the free trajectories, free covariance model, and power accordingly.When treatment effects are not proportional to time, this has a large impact on the empirical power, even for a trial with the same visits as the observational study. If investigators want to guard against the possibility of a non-proportional treatment effect, a free trajectory, free covariance model could be specified as a second line analysis method, bearing in mind that power will be lost when using a less restrictive model if the linearity assumptions are in fact true. Alternatively, if investigators have strong reasons to suspect any treatment effect in the proposed trial will not be proportional to time, then such a model should be used at the sample size calculation stage.