# Table 1 Definitions, assumptions and approximations for PAF when the exposure is binary, multi-category and logistic

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: $${\hat{\beta}}^{ave}$$ $$\hat{\beta_1}$$ $$\frac{\sum \limits_{j=1}^K\hat{P}\left(A=j|Y=0\right)\hat{\beta_j}}{1-\hat{P}\left(A=0|Y=0\right)}$$ $${\int}_{-\infty}^{\infty}\hat{f}\left(j|0\right)\hat{\beta (j)} dj$$
1. *Here β0 = 0 by definition for the Binary and Multicategory exposures and β(j0) = 0 for continuous exposures. Estimates $$\hat{\beta}(j)/{\hat{\beta}}_j\ \mathsf{and}$$ $$\hat{\gamma}(c)$$ could be found via generalized additive models with a logistic link, where the confounders and possibly the exposure are modelled non-parametrically
2. **Note that RRj = P(Y = 1| A = j, C = c)/P(Y = 1| A = 0, C = c) and RR(j) = P(Y = 1| A = j, C = c)/P(Y = 1| A = j0, C = c)
3. ***f(j| 1) is the conditional density of A when Y = 1; similarly f(j| 0) is the conditional density of A when Y = 0
4. ****Note that when A is continuous, the probability of a non-reference level of the exposure: $$\hat{P}\left(A\ne {j}_0|Y=0\right)$$ is 1