For simplicity, assume we have a linear model which looks like this:
Outcome = beta0 + beta1*Treatment + beta2*Time + beta3*Treatment*Time + error
where Treatment is a dummy variable (0 = Control, 1 = Novel Treatment) and Time is also a dummy variable (0 = Occasion 1, 1 = Occasion 2).
If we wanted to report the effect of Treatment averaged across both time occasions, am I right in thinking that we would obtain it by computing a simple average of the simple effects of Treatment at each time occasion?
Since the simple effect of Treatment when Time = 0 is beta1 and the simple effect of Treatment when Time = 1 is beta1 + beta3, the simple average of these two simple effects would be [beta1 + (beta1 + beta3)]/2 = beta1 + 0.5*beta3. In other words, the contrast needed for testing whether or not this effect is 0 would be c = (0, 1, 0, 0.5).
Since I haven't worked with this type of contrasts before, here's what I am wondering:
Does it actually make sense to compute such a contrast in a linear regression model which includes an interaction term between Treatment and Time as a convenient way to "summarize" the treatment effect across time (rather than report the treatment effect separately for each time occasion)?
Is a simple average the right way to compute the contrast in item 1.? Or should the contrast reflect the fact that the underlying data contains more/less records for one temporal occasion than for the other? If the contrast should reflect a weighted average of beta1 and beta1 + beta3, how should the weights be set up and what would the weighted contrast look like? Would it look like w1*beta1 + w2*0.5*beta3, where w1 is the proportion of data records for which Time = 0 and w2 is the proportion of data records for which Time = 1?
Assuming that Time were treated as a continuous variable (and would span plenty of occasions) rather than a dummy variable, what would the equivalent contrast be for reporting an average effect of Treatment across the continuous values of Time? What would that contrast look like?
Thank you in advance for any insights you can provide.