Repeated measures designs contain several measurements for the same experimental unit. This generally occurs when we assign different doses of a treatment to a unit and want to evaluate the response through time.

We now have a fixed effect for time, which is typically coded as 1,2,3..,k that needs to be added. We would expect that this variable/effect will have an impact on the response.

These designs raise four major points, and we need to decide how to solve them:

• Do we think that time has the same impact for all of the experimental units? We have two options for this:
  • If yes, we will just add the time variable as a fixed effect.
  • If no, we will still add the time variable as a fixed effect, plus a random effect for time by subject. What this means is that the time effect can be obtained as the sum of the fixed effect (average for all units) plus the random effect. This is a way of adding heterogeneity in the effect, ensuring that the total effects are pulled toward the mean effect (fixed effect).
• Should the time effect be linear or nonlinear? Or should each timestep have a distinct effect? There are two options:
  • If we assume linearity, we will just treat the time variable as numeric 
  • If we assume that each time step has a distinct effect, we should add the time variable as a factor (it will have k levels) 
• Do we need a special covariance structure? By default, once we set up a random effect (typically by experimental unit), all of the observations for that unit will be correlated (will share a common shock). This is usually questionable, as it implies that the covariance between a pair of any two observations will be constant. Observations that are further away in time should be less correlated than observations for consecutive time steps.
• Since we are including the time effect, should we consider an interaction between the main effect and time? This implies that the response to the treatment changes according to the time step we are in. In general, this interaction is initially added, and discarded if not significative.