# Attributable fraction function based on Instrumental Variables (IV) regression as an `ivglm` object in the `ivtools` package. `AFivglm` estimates the model-based adjusted attributable fraction from a Instrumental Variable regression from a `ivglm` object. The IV regression can be estimated by either G-estimation or Two Stage estimation for a binary exposure and outcome. ```r AFivglm(object, data) ``` ## Arguments - `object`: a fitted Instrumental Variable regression of class "`ivglm`". - `data`: an optional data frame, list or environment (or object coercible by `as.data.frame` to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment (`formula`), typically the environment from which the function is called. ## Returns - **AF.est**: estimated attributable fraction. - **AF.var**: estimated variance of `AF.est`. The variance is obtained by combining the delta methods with the sandwich formula. ## Details `AFivglm` estimates the attributable fraction for an IV regression under the hypothetical scenario where a binary exposure `X` is eliminated from the population. The estimate can be adjusted for IV-outcome confounders `L` in the `ivglm` function. Let the AF function be defined as $$ AF=1-\frac{Pr(Y_0=1)}{Pr(Y=1)}AF = 1 - \frac{Pr(Y0=1)}{Pr(Y=1)} $$ where $Pr(Y0=1)$ denotes the counterfactual outcome prevalence had everyone been unexposed and $Pr(Y=1)$ denotes the factual outcome prevalence. If the instrument `Z` is valid, conditional on covariates `L`, i.e. fulfills the IV assumptions 1) the IV should have a (preferably strong) association with the exposure, 2) the effect of the IV on the outcome should only go through the exposure and 3) the IV-outcome association should be unconfounded (Imbens and Angrist, 1994) then $Pr(Y0=1)$ can be estimated. ## Examples ```r # Example 1 set.seed(2) n <- 5000 ## parameter a0 determines the outcome prevalence a0 <- -4 psi.true <- 1 l <- rbinom(n, 1, 0.5) u <- rbinom(n, 1, 0.5) z <- rbinom(n, 1, plogis(a0)) x <- rbinom(n, 1, plogis(a0+3*z+ u)) y <- rbinom(n, 1, exp(a0+psi.true*x+u)) d <- data.frame(z,u,x,y,l) ## Outcome prevalence mean(d$y) ####### G-estimation ## log CRR fitz.l <- glm(z~1, family=binomial, data=d) gest_log <- ivglm(estmethod="g", X="x", Y="y", fitZ.L=fitz.l, data=d, link="log") AFgestlog <- AFivglm(gest_log, data=d) summary(AFgestlog) ## log COR ## Associational model, saturated fit_y <- glm(y~x+z+x*z, family="binomial", data=d) ## Estimations of COR and AF gest_logit <- ivglm(estmethod="g", X="x", Y="y", fitZ.L=fitz.l, fitY.LZX=fit_y, data=d, link="logit") AFgestlogit <- AFivglm(gest_logit, data = d) summary(AFgestlogit) ####### TS estimation ## log CRR # First stage fitx <- glm(x ~ z, family=binomial, data=d) # Second stage fity <- glm(y ~ x, family=poisson, data=d) ## Estimations of CRR and AF TSlog <- ivglm(estmethod="ts", X="x", Y="y", fitY.LX=fity, fitX.LZ=fitx, data=d, link="log") AFtslog <- AFivglm(TSlog, data=d) summary(AFtslog) ## log COR # First stage fitx_logit <- glm(x ~ z, family=binomial, data=d) # Second stage fity_logit <- glm(y ~ x, family=binomial, data=d) ## Estimations of COR and AF TSlogit <- ivglm(estmethod="ts", X="x", Y="y", fitY.LX=fity_logit, fitX.LZ=fitx_logit, data=d, link="logit") AFtslogit <- AFivglm(TSlogit, data=d) summary(AFtslogit) ## Example 2: IV-outcome confounding by L ####### G-estimation ## log CRR fitz.l <- glm(z~l, family=binomial, data=d) gest_log <- ivglm(estmethod="g", X="x", Y="y", fitZ.L=fitz.l, data=d, link="log") AFgestlog <- AFivglm(gest_log, data=d) summary(AFgestlog) ## log COR ## Associational model fit_y <- glm(y~x+z+l+x*z+x*l+z*l, family="binomial", data=d) ## Estimations of COR and AF gest_logit <- ivglm(estmethod="g", X="x", Y="y", fitZ.L=fitz.l, fitY.LZX=fit_y, data=d, link="logit") AFgestlogit <- AFivglm(gest_logit, data = d) summary(AFgestlogit) ####### TS estimation ## log CRR # First stage fitx <- glm(x ~ z+l, family=binomial, data=d) # Second stage fity <- glm(y ~ x+l, family=poisson, data=d) ## Estimations of CRR and AF TSlog <- ivglm(estmethod="ts", X="x", Y="y", fitY.LX=fity, fitX.LZ=fitx, data=d, link="log") AFtslog <- AFivglm(TSlog, data=d) summary(AFtslog) ## log COR # First stage fitx_logit <- glm(x ~ z+l, family=binomial, data=d) # Second stage fity_logit <- glm(y ~ x+l, family=binomial, data=d) ## Estimations of COR and AF TSlogit <- ivglm(estmethod="ts", X="x", Y="y", fitY.LX=fity_logit, fitX.LZ=fitx_logit, data=d, link="logit") AFtslogit <- AFivglm(TSlogit, data=d) summary(AFtslogit) ``` ## References Dahlqwist E., Kutalik Z., , A. (2019). Using Instrumental Variables to estimate the attributable fraction. **Manuscript**. ## See Also `ivglm` used for fitting the causal risk ratio or odds ratio using the G-estimator or Two stage estimator. ## Author(s) Elisabeth Dahlqwist, Arvid

AFivglm function