Impact of wrongly assuming a linear trajectory, when the treatment effect is truly proportional to time
The empirical power for the random slopes model was close to nominal when the observational data and the planned trial had the same duration and visit schedule, regardless of whether the mean trajectories are linear or not, provided that the treatment effect is correctly specified.This result agrees with the simulation study in reference [Using an observational study with a different schedule of visits to that in the proposed trial can impact power when there is non-linearity, even for a proportional treatment effect. When using a 5-year observational study to specify sample sizes for a 3-year trial, empirical powers for the non-linear trajectories are higher than nominal. If we had used a 3-year observational study to specify the sample sizes for a 3-year trial, we might have again seen that a non-linear mean trajectory has little effect on the empirical power.To understand why powers are high for shorter trials in the presence of non-linearity, we first consider the effect of shortening trials when trajectories are linear. When mean trajectories are truly linear, the change in sample size for a shorter trial is governed by the size of the residual variance relative to the between person variance. When the residual variance is larger, the effect of shortening follow-up is greater than for a smaller residual variance. Intuitively, there is a greater gain in efficiency (in terms of sample size reduction) from increasing the trial’s length when the residual within-person error is greater. When mean trajectories are non-linear, the residual error in the random slopes model is over-estimated whereas the intercept and slope variances are generally under-estimated (Figures A8.1.2 and A8.1.3, The incorrect powering for a 3-year trial is less of an issue for a greater residual error variance, since the large residual error variance is dominating the sample size calculation, and the residual variance is generally over-estimated by only a small amount for the non-linear trajectories (less than 5% over-estimated, compared to up to around 60% overestimated for the lower residual error variance).While we have only looked at a 3-year trial versus a 5-year observational study in our simulation study, these findings are likely to apply to any trial with a different length or pattern of visits to the observational study. For example, the bottom panel of Fig. 