Reconsidering the denominator of the attributable proportion for interaction

Appendices

Appendix 1: Invariance to recoding the outcome

Here we show that the alternative definition of the attributable proportion, $AP^{\ast}=\frac{RERI}{RR_{11}-1},$ is invariant to the recoding of the outcome, whereas Rothman’s primary definition, $AP^{\ast }=\frac{RERI}{RR_{11}},$ is not. The definition of the alternative attributable proportion is [6]:

$$ AP^{\ast }=\frac{RERI}{RR_{11}-1}=\frac{RR_{11}-RR_{10}-RR_{01}+1}{RR_{11}-1} =\frac{\frac{p_{11}}{p_{00}}-\frac{p_{10}}{p_{00}}-\frac{p_{01}}{p_{00}}+1}{ \frac{p_{11}}{p_{00}}-1}=\frac{(p_{11}-p_{10}-p_{01}+p_{00})}{p_{11}-p_{00}}. $$

If we reverse the coding of the outcome, the relative risks for the categories (X ₁ = 1, X ₂ = 1),  (X ₁ = 1, X ₂ = 0),  (X ₁ = 0, X ₂ = 1) become respectively $\frac{1-p_{11}}{1-p_{00}},\,\frac{1-p_{10}}{1-p_{00}}$, and $\frac{1-p_{01}}{1-p_{00}},$ and the alternative attributable proportion measure is then

$$ \begin{aligned} AP^{\ast}&=\frac{\frac{1-p_{11}}{1-p_{00}}-\frac{1-p_{10}}{1-p_{00}}- \frac{1-p_{01}}{1-p_{00}}+1}{\frac{1-p_{11}}{1-p_{00}}-1}=\frac{ (1-p_{11})-(1-p_{10})-(1-p_{01})+(1-p_{00})}{(1-p_{11})-(1-p_{00})}\\ &=\frac{(p_{11}-p_{10}-p_{01}+p_{00})}{p_{11}-p_{00}} \\ \end{aligned} $$

which is the same measure we obtained under the original coding of the outcome.

For Rothman’s primary definition of the attributable proportion, $AP=\frac{RERI}{RR_{11}}$, under the original coding we have that this is

$$ AP=\frac{RERI}{RR_{11}}=\frac{RR_{11}-RR_{10}-RR_{01}+1}{RR_{11}}=\frac{ \frac{p_{11}}{p_{00}}-\frac{p_{10}}{p_{00}}-\frac{p_{01}}{p_{00}}+1}{\frac{ p_{11}}{p_{00}}}=\frac{(p_{11}-p_{10}-p_{01}+p_{00})}{p_{11}}. $$

If we reverse the coding of the outcome, the relative risks for the categories (X ₁ = 1, X ₂ = 1),  (X ₁ = 1, X ₂ = 0),  (X ₁ = 0, X ₂ = 1) become respectively $\frac{1-p_{11}}{1-p_{00}},$ $\frac{1-p_{10}}{1-p_{00}}$, and $\frac{1-p_{01}}{1-p_{00}},$ and Rothman’s primary attributable proportion measure is then

$$ \begin{aligned} AP&=\frac{\frac{1-p_{11}}{1-p_{00}}-\frac{1-p_{10}}{1-p_{00}}-\frac{ 1-p_{01}}{1-p_{00}}+1}{\frac{1-p_{11}}{1-p_{00}}}=\frac{ (1-p_{11})-(1-p_{10})-(1-p_{01})+(1-p_{00})}{(1-p_{11})} \\ &=\frac{-(p_{11}-p_{10}-p_{01}+p_{00})}{1-p_{11}} \end{aligned} $$

so that the measure reverses sign and is of a very different magnitude.

Appendix 2: SAS and Stata code to implement the alternative attributable proportion measure

Suppose the outcome is in variable ‘d’, the first exposure in variable ‘g’ and the second exposure in variable ‘e’ with three covariates ‘c1 c2 c3’. The following SAS code will estimate the alternative attributable proportion measure, $AP^{\ast }=\frac{RERI}{RR_{11}-1},$ and its confidence interval using the delta method (see Online Appendix):