# Attributable fraction function based on a Weibull gamma-frailty model as a `parfrailty` object (commonly used for cohort sampling family designs with time-to-event outcomes). `AFparfrailty` estimates the model-based adjusted attributable fraction function from a shared Weibull gamma-frailty model in form of a `parfrailty` object. This model is commonly used for data from cohort sampling familty designs with time-to-event outcomes. ```r AFparfrailty(object, data, exposure, times, clusterid) ``` ## Arguments - `object`: a fitted Weibull gamma-parfrailty object of class "`parfrailty`". - `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. - `exposure`: the name of the exposure variable as a string. The exposure must be binary (0/1) where unexposed is coded as 0. - `times`: a scalar or vector of time points specified by the user for which the attributable fraction function is estimated. If not specified the observed death times will be used. - `clusterid`: the name of the cluster identifier variable as a string, if data are clustered. ## Returns - **AF.est**: estimated attributable fraction function for every time point specified by `times`. - **AF.var**: estimated variance of `AF.est`. The variance is obtained by combining the delta methods with the sandwich formula. - **S.est**: estimated factual survival function; $S(t)$. - **S.var**: estimated variance of `S.est`. The variance is obtained by the sandwich formula. - **S0.est**: estimated counterfactual survival function if exposure would be eliminated; $S0(t)$. - **S0.var**: estimated variance of `S0.est`. The variance is obtained by the sandwich formula. ## Details `AFparfrailty` estimates the attributable fraction for a time-to-event outcome under the hypothetical scenario where a binary exposure `X` is eliminated from the population. The estimate is adjusted for confounders `Z` by the shared frailty model (`parfrailty`). The baseline hazard is assumed to follow a Weibull distribution and the unobserved shared frailty effects `U` are assumed to be gamma distributed. Let the AF function be defined as $$ AF=1-\frac{\{1-S_0(t)\}}{\{1-S(t)\}}AF = 1 - {1 - S0(t)} / {1 - S(t)} $$ where $S0(t)$ denotes the counterfactual survival function for the event if the exposure would have been eliminated from the population at baseline and $S(t)$ denotes the factual survival function. If `Z` and `U` are sufficient for confounding control, then $S0(t)$ can be expressed as $E_z{S(t|X=0,Z)}$. The function uses a fitted Weibull gamma-frailty model to estimate $S(t|X=0,Z)$, and the marginal sample distribution of `Z` to approximate the outer expectation. A clustered sandwich formula is used in all variance calculations. ## Examples ```r # Example 1: clustered data with frailty U expit <- function(x) 1 / (1 + exp( - x)) n <- 100 m <- 2 alpha <- 1.5 eta <- 1 phi <- 0.5 beta <- 1 id <- rep(1:n,each=m) U <- rep(rgamma(n, shape = 1 / phi, scale = phi), each = m) Z <- rnorm(n * m) X <- rbinom(n * m, size = 1, prob = expit(Z)) # Reparametrize scale as in rweibull function weibull.scale <- alpha / (U * exp(beta * X)) ^ (1 / eta) t <- rweibull(n * m, shape = eta, scale = weibull.scale) # Right censoring c <- runif(n * m, 0, 10) delta <- as.numeric(t < c) t <- pmin(t, c) data <- data.frame(t, delta, X, Z, id) # Fit a parfrailty object library(stdReg) fit <- parfrailty(formula = Surv(t, delta) ~ X + Z + X * Z, data = data, clusterid = "id") summary(fit) # Estimate the attributable fraction from the fitted frailty model time <- c(seq(from = 0.2, to = 1, by = 0.2)) AFparfrailty_est <- AFparfrailty(object = fit, data = data, exposure = "X", times = time, clusterid = "id") summary(AFparfrailty_est) plot(AFparfrailty_est, CI = TRUE, ylim=c(0.1,0.7)) ``` ## See Also `parfrailty` used for fitting the Weibull gamma-frailty and `stdParfrailty` used for standardization of a `parfrailty` object. ## Author(s) Elisabeth Dahlqwist, Arvid

AFparfrailty function