# Average treatment effect using relative risk difference?

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?

• From logistic regression with logit link, you can get odds ratio (OR=$[\frac {\pi_1}{1-\pi_1}]/[\frac{\pi_2}{1-\pi_2}]$), how can you get the risk ratio or rate ratio (RR = $\pi_1/\pi_2$)? Commented Nov 20, 2018 at 19:41
• Odds ratios provide a reasonable approximation of the relative risk but I could also get it through Invlog(odds) right? It doesn't really matter my question is whether or not the risk difference of the odds or the RR is the same as the average treatment effect. Commented Nov 20, 2018 at 19:45
• The approximation is OK when $\pi$ is small, say <0.05. If $\pi$ is large, the assumption that OR is constant (which is required for logit link) is equivalent to the assumption that RR is not constant. Commented Nov 20, 2018 at 19:49
• Its when it's <0.10 but ok, thank you for your contribution but that wasn't my question. Commented Nov 20, 2018 at 19:51
• If I want to get super anal about it I can just get the RR by just calculatin OR/(1-Pref) + (Pref * OR) And since I'm following them I can easily calculate the incidence of smoking cessation and if it by some miracle is above 10% since im studying adolescents then I'll do the required conversions. Commented Nov 20, 2018 at 19:54

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


Where 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).

• Thank you. Just to clarify the c proportion. Would that in my case be the prevalence of smoking cessation in my control arm or the incidence of smoking cessation in the control arm as in calculated with persontime in the denominator? Commented Nov 20, 2018 at 20:17