convert the ATE Average Treatment Effect from a doubly robust model into Odds Ratio How would one convert the ATE Average Treatment Effect from a doubly robust model into Odds Ratio? Or is it better to discuss this by converting into risk ratio? If so, how is this done.


If your outcome is binary, this is straightforward by noting that that the estimator is the difference between two counterfactual probabilities, and the odds ratio is just a ratio of the odds instead of of the probabilities. So, if
$$\hat{P}(Y^1=1) = n^{-1}\sum_{i=1}^n \left\{ \frac{A_i}{\hat{\pi}_i} Y_i - \frac{A_i - \hat{\pi}_i}{\hat{\pi}_i}\hat{E}(Y|A=1,X_i) \right\}$$
and
$$\hat{P}(Y^0=1) = n^{-1}\sum_{i=1}^n \left\{ \frac{1-A_i}{1-\hat{\pi}_i} Y_i - \frac{A_i - \hat{\pi}_i}{1-\hat{\pi}_i}\hat{E}(Y|A=0,X_i) \right\}$$
and we know $\text{odds}(Y=1)=\frac{P(Y=1)}{1-P(Y=1)}$, then the DR estimator for the marginal causal odds ratio is simply
$$\frac{\hat{\text{odds}}(Y^1=1)}{ \hat{\text{odds}}(Y^0=1)} = \frac{\frac{\hat{P}(Y^1=1)}{1-\hat{P}(Y^1=1)}}{\frac{\hat{P}(Y^0=1)}{1-\hat{P}(Y^0=1)}}$$
To get confidence interval you can bootstrap or use the delta method or M-estimation to estimate the marginal log odds ratio from the marginal probabilities.
