How can I convert relative risk into absolute risk?
How can I estimate absolute risk for an individual from the available sources of RR and OR?
Solved – How to convert relative risk into absolute risk
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Well, from what you've already said, I think you've got most of it covered but just need to put it in her language: One is a difference of risks, one is a ratio. So one hypothesis test asks if $p_2 - p_1 = 0$ while the other asks if $\frac{p_2}{p_1} = 1$. Sometimes these are "close" sometimes not. (Close in quotes because clearly they aren't close in the usual arithmetic sense). If the risk is rare, these are typically "far apart". e.g. $.002/.001 = 2$ (far from 1) while $.002-.001 = .001$ (close to 0); but if the risk is high, then these are "close": $.2/.1 = 2$ (far from 0) and $.2 - .1 = .1$ (also far from 0, at least compared to the rare case.
In the formula for the doubly robust estimator provided (augmented inverse probability weighting), it is shown for the ATE. The key piece to note is that
$$\hat{\theta}_{1} = n^{-1} \sum_{i=1}^n \frac{I(A_i=1)Y_i}{\hat{\pi}(W_i)} - \frac{A_i-\hat{\pi}(W_i)}{\hat{\pi}(W_i)} \hat{E}[Y|A=1,X_i]$$
is the estimator for the risk had everyone been given $A=1$. The second part similar follows for the risk had everyone been given $A=0$, which I indicated as $\hat{\theta}_0$. This estimator consists of 3 pieces, the inverse probability weighting part ($\frac{I(A_i=1)Y_i}{\hat{\pi}(W_i)}$), the predicted outcome ($\hat{E}[Y|A=1,X_i]$), and the 'glue' ($\frac{A_i-\hat{\pi}(W_i)}{\hat{\pi}(W_i)}$). Together these come together to estimate the risk. This form has the nice property of double robustness, but are still estimating the risk.
Once you have the two $\hat{\theta}$ or risks, you can calculate whatever contrast you prefer by manipulating those risks. For example the risk difference is just the difference in those risks (this is written out above without my $\theta$ shorthand) $$ \hat{\theta}_1 - \hat{\theta}_0$$ If you want the risk ratio, $$ \hat{\theta}_1 / \hat{\theta}_0$$ This second contrast for the risk ratio is equivalent to replacing the minus sign in the provided formula with a division.
The key piece is that the doubly robust estimator estimates each risk individually. The ATE is often frame as the estimand, so that's why it is usually presented in that form. However, the estimated risks from the doubly robust estimator can be transformed as desired.
For the estimator of the variance, influence curves are often used for the above doubly robust estimator. It is important to note that the form of the influence curve changes between the risk difference and risk ratio. Therefore, the variance estimator is a little more complicated. A simple solution is to bootstrap the variance.
Best Answer
Absolute risk is the risk (or prevalence within the group) that a group has X. For example, if 10% of men have cancer, the absolute risk of cancer in men is 10%.
Relative risk is the ratio of this risk between two groups (e.g., 2 to 1). So if men are twice as likely to have cancer compared to women, the relative risk is 2 to 1.
To calculate absolute risk from relative risk, you need to know the absolute risk for at least one of the groups. So if the relative risk for men of having X compared to women having X is 3, and you know the absolute risk of X in women is 1/100, then you know the absolute risk of having X in men is 3/100. If you do not know the absolute risk in either group you cannot calculate absolute risk.