So I'm designing a RCT to evaluate the effectiveness of a smoking cessation intervention. For my analysis I've decided to look at three different things. 1) the point prevalence of smoking cessation at each follow-up time-point 2) The relative risks of smoking cessation in the intervention and control arm respectively (estimated through Odds ratios from logistic regression) 3) I want to estimate the average treatment effect and this is were my question comes in, am I right to assume that if I take the risk difference (from the RR I get from my logistic regression) than this is essentially the average treatment effect?
The average treatment effect can be expressed in logits, as an odds ratio, as relative risk, or as risk difference. Many people prefer risk difference, as in: "The treatment increased positive outcomes by 5 percentage points."
The function I use to convert between OR and risk difference (called "lift" in marketing and advertising contexts) is:
or_to_lift <- function(or, c) (c * or) / (c * (or - 1) + 1) - c
or is an odds ratio and
c is the baseline (i.e., control) proportion. So, if there was an odds ratio of 1.3 and a control treatment incidence of .5, I would say that the lift (a.k.a., risk difference) was .065 or 6.5 percentage points.
All of these represent the average treatment effect, though, when it is collapsed across all subgroups in the sample (i.e., you don't include any interactions).