# Attributable fraction for cross-sectional sampling designs. NOTE! Deprecated function. Use `AFglm`. `AF.cs` estimates the model-based adjusted attributable fraction for data from cross-sectional sampling designs. ```r AF.cs(formula, data, exposure, clusterid) ``` ## Arguments - `formula`: an object of class "`formula`" (or one that can be coerced to that class): a symbolic description of the model used for adjusting for confounders. The exposure and confounders should be specified as independent (right-hand side) variables. The outcome should be specified as dependent (left-hand side) variable. The formula is used to object a logistic regression by `glm`. - `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. - `clusterid`: the name of the cluster identifier variable as a string, if data are clustered. ## Returns - **AF.est**: estimated attributable fraction. - **AF.var**: estimated variance of `AF.est`. The variance is obtained by combining the delta method with the sandwich formula. - **P.est**: estimated factual proportion of cases; $Pr(Y=1)$. - **P.var**: estimated variance of `P.est`. The variance is obtained by the sandwich formula. - **P0.est**: estimated counterfactual proportion of cases if exposure would be eliminated; $Pr(Y0=1)$. - **P0.var**: estimated variance of `P0.est`. The variance is obtained by the sandwich formula. - **object**: the fitted model. Fitted using logistic regression, `glm`. ## Details `Af.cs` estimates the attributable fraction for a binary outcome `Y` under the hypothetical scenario where a binary exposure `X` is eliminated from the population. The estimate is adjusted for confounders `Z` by logistic regression (`glm`). Let the AF be defined as $$ AF=1-\frac{Pr(Y_0=1)}{Pr(Y=1)}AF = 1 - Pr(Y0 = 1) / Pr(Y = 1) $$ where $Pr(Y0 = 1)$ denotes the counterfactual probability of the outcome if the exposure would have been eliminated from the population and $Pr(Y = 1)$ denotes the factual probability of the outcome. If `Z` is sufficient for confounding control, then $Pr(Y0 = 1)$ can be expressed as $E_z{Pr(Y = 1 |X = 0,Z)}.$ The function uses logistic regression to estimate $Pr(Y=1|X=0,Z)$, and the marginal sample distribution of `Z` to approximate the outer expectation ( and Vansteelandt, 2012). If `clusterid` is supplied, then a clustered sandwich formula is used in all variance calculations. ## Examples ```r # Simulate a cross-sectional sample expit <- function(x) 1 / (1 + exp( - x)) n <- 1000 Z <- rnorm(n = n) X <- rbinom(n = n, size = 1, prob = expit(Z)) Y <- rbinom(n = n, size = 1, prob = expit(Z + X)) # Example 1: non clustered data from a cross-sectional sampling design data <- data.frame(Y, X, Z) # Estimation of the attributable fraction AF.cs_est <- AF.cs(formula = Y ~ X + Z + X * Z, data = data, exposure = "X") summary(AF.cs_est) # Example 2: clustered data from a cross-sectional sampling design # Duplicate observations in order to create clustered data id <- rep(1:n, 2) data <- data.frame(id = id, Y = c(Y, Y), X = c(X, X), Z = c(Z, Z)) # Estimation of the attributable fraction AF.cs_clust <- AF.cs(formula = Y ~ X + Z + X * Z, data = data, exposure = "X", clusterid = "id") summary(AF.cs_clust) ``` ## References Greenland, S. and Drescher, K. (1993). Maximum Likelihood Estimation of the Attributable Fraction from logistic Models. **Biometrics** 49 , 865-872. , A. and Vansteelandt, S. (2011). Doubly robust estimation of attributable fractions. **Biostatistics** 12 , 112-121. ## See Also The new and more general version of the function: `AFglm`. ## Author(s) Elisabeth Dahlqwist, Arvid

AF.cs function