I fitted a Logistic regression with a number of variables, and was asked about the absolute risk (AR) and number needed to treat (NNT) for one of the variable with the outcome. I understand how to calculate them in a 2x2 table, but I am wondering if it can be calculated from the logistic regression?

From my understanding, in a Logistic regression we have (say I'm interested in $X_1$):
$$\newcommand{\exposed}{{\rm exposed}}\newcommand{\unexposed}{{\rm unexposed}} \ln\bigg[\frac{\pi(\exposed_1)}{\pi(\unexposed_1)}\bigg] = \ln\bigg[\frac{(\beta_0 + \beta_1X_1 + \beta_2X_2 + ...)}{(\beta_0 + \beta_2X_2 + ...)}\bigg] $$

and since $AR=\pi(\unexposed1)-\pi(\exposed1)$, can we calculate AR like this? \begin{align} \pi(\unexposed) &= \frac{\exp(\beta_0+\beta_2X_2+...)}{(1+\exp(\beta_0+\beta_2X_2+...)} \\[5pt] &\qquad{\rm and} \\[5pt] \pi(\exposed) &= \frac{\exp(\beta_0+\beta_1X_1+\beta_2X_2+...)}{(1+\exp(\beta_0+\beta_1X_1+\beta_2X_2+...)} \end{align} And the NNT is:
$$ \frac 1 {\pi(\unexposed)-\pi(\exposed))} $$

But since it varies with different $X_2$, $X_3$, etc., is it possible to get a single number for AR and NNT? Would it be appropriate to calculate the $\pi$'s for all possible $X_n$, and then just take the average?

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    $\begingroup$ Regardless of the method of calculation, please make sure that the study design "allows" you to calculate an NNT and AR. Remember that these are risk-based measures. A case-control study, for example, cannot provide you with estimates of the baseline risk unless fulfill specific assumptions. $\endgroup$ – user140401 Dec 5 '16 at 2:43
  • $\begingroup$ @Elmer Thank you! That is a very good point. (sorry if this side-tracks the question) May I ask under which specific assumptions/circumstances does a Case-control study "allows" for NNT and AR calculation? I understand that we cannot estimate the incidence from such studies, but in this analysis I know the prevalence from population data, and the proportion of cases in the study is similar to the prevalence. $\endgroup$ – Inji Dec 5 '16 at 4:37
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    $\begingroup$ Thanks, @lnji. I suggest that you refer to King and Zeng's comprehensive exposition on the estimation of risks and rates from case-control studies: gking.harvard.edu/files/1s.pdf. Good luck! $\endgroup$ – user140401 Dec 5 '16 at 7:09

With great thanks to @Elmer, I think I have an answer to my question in King and Zeng's reference. There are more details on their website (and an R package called zelig).

They also expanded more on the risk difference (RD) [which is essentially the opposite of AR], risk ratio (RR), and NNT in this reference:

Gary King, Langche Zeng, and Shein-Chung Chow. 2010. “Inference in Case Control Studies.” In Encyclopedia of Biopharmaceutical Statistics, 3rd ed. New York: Marcel Dekker. http://gking.harvard.edu/files/gking/files/1s-enc.pdf

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  • $\begingroup$ Brilliant. I wasn't aware of the website or the R package. $\endgroup$ – user140401 Dec 6 '16 at 4:50

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