# How to calculate the observed absolute treatment effect?

I'm reading this article. The authors indicated that for a random sample of 12 characteristics and 3600 patients affected in tow arms (control and treatment arm), they fitted some treatment effect models consisting of all 12 characteristics and treatment interactions. They calculated the predicted absolute treatment benefit, the mean predicted absolute treatment benefit and the observed absolute treatment benefit.

But, they don't give any mathematical formula.

Below, the paragraph in the article on the calculation of the predicted absolute treatment benefit, the mean predicted absolute treatment benefit and the observed absolute treatment benefit:

For each effect model that was fitted in each consecutive sample, the predicted absolute treatment benefit (i.e., predicted risk difference) in the population was calculated as the model's outcome prediction conditional on assignment to the control arm minus the outcome prediction conditional on assignment to the treatment arm. The population was stratified into quartiles of predicted absolute treatment benefit. In each quartile of the population, we calculated the mean predicted absolute treatment benefit and the observed absolute treatment benefit, that is, the event rate of control patients minus the event rate of treated patients.

I have summarized in the following figure the treatment effect models used:

I would be very grateful if someone could explain to me more (with a mathematical formula) how to calculate these measures?

• See fharrell.com/post/varyor and make sure in the article you cited that the authors provided evidence for the existence of interactions, e.g. the AIC for a model with interactions is better than one without. You'll see in my example the the optimum penalty is infinity meaning no evidence for any interactions improving prediction. – Frank Harrell Jan 15 at 15:36