Table 1 Definitions, assumptions and approximations for PAF when the exposure is binary, multi-category and logistic
From: Graphical comparisons of relative disease burden across multiple risk factors
Binary | Multicategory | Continuous | |
|---|---|---|---|
Counterfactual definition of PAF | \( \frac{P\left(Y=1\right)-P\left({Y}^{a=0}=1\right)}{P\left(Y=1\right)} \) | \( \frac{P\left(Y=1\right)-P\left({Y}^{a=0}=1\right)}{P\left(Y=1\right)} \) | \( \frac{P\left(Y=1\right)-P\left({Y}^{a={j}_0}=1\right)}{P\left(Y=1\right)} \) |
Assumptions: | 1. Standard causal inference assumptions • Conditional exchangeability (counterfactual outcome Ya = j and assigned risk factor A are independent random variables, within strata of observed confounders c • Consistency of counterfactuals: Ya = j = Y when A = j for all levels j of the risk factor A • Positivity 0 < P(Ya = j = 1| C = c) < 1 for all j and strata c 2. No interactions (P(Ya = j = 1| C = c)/P(Ya = k = 1| C = c) does not depend on c), for any possible values of exposure j and k 3. Rare disease assumption (P(Y = 1) small) | ||
Re-expression of PAF (given assumptions 1. and 2.) | P(A = 1| Y = 1)(RR − 1)/RR | \( \sum \limits_{j=1}^KP\left(A=j|Y=1\right)\left(R{R}_j-1\right)/R{R}_j \)** | \( {\int}_{-\infty}^{\infty }f\left(j|1\right)\frac{RR(j)-1}{RR(j)} dj \) ** |
aCorresponding logistic model (Given assumption 3.) | logit(P(Y = 1| A = j, C = c)) =μ + βj + γ(c) | logit(P(Y = 1| A = j, C = c)) = μ + βj + γ(c) | logit(P(Y = 1| A = j, C = c)) = μ + β(j) + γ(c) |
Logistic Approximation for PAF (Given assumptions 1,2 and 3) | \( \frac{\hat{P\left(A=1|Y=1\right)}\left({e}^{\hat{\beta_1}}-1\right)}{e^{\hat{\beta_1}}} \) | \( \sum \limits_{j=1}^K\hat{P}\left(A=j|Y=1\right)\left({e}^{\hat{\beta_j}}-1\right)/{e}^{\hat{\beta_j}} \) | \( {\int}_{-\infty}^{\infty}\hat{f}\left(j|1\right)\left({e}^{\hat{\beta (j)}}-1\right)/{e}^{\hat{\beta (j)}} dj \)*** |
Graphical Approximation | \( \hat{P\left(A=1|Y=0\right)}\times {\hat{\ \beta}}^{ave} \) | \( \hat{P}\left(A>0|Y=0\right)\times {\hat{\ \beta}}^{ave} \) | \( 1\times {\hat{\beta}}^{ave} \)**** |
“Average” estimated log-odds ratio: \( {\hat{\beta}}^{ave} \) | \( \hat{\beta_1} \) | \( \frac{\sum \limits_{j=1}^K\hat{P}\left(A=j|Y=0\right)\hat{\beta_j}}{1-\hat{P}\left(A=0|Y=0\right)} \) | \( {\int}_{-\infty}^{\infty}\hat{f}\left(j|0\right)\hat{\beta (j)} dj \) |