# Attributable fraction estimation based on a logistic regression model from a `glm` object (commonly used for cross-sectional or case-control sampling designs). `AFglm` estimates the model-based adjusted attributable fraction for data from a logistic regression model in the form of a `glm` object. This model is commonly used for data from a cross-sectional or non-matched case-control sampling design. ```r AFglm(object, data, exposure, clusterid, case.control = FALSE) ``` ## Arguments - `object`: a fitted logistic regression model object of class "`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. Cluster robust standard errors will be calculated. - `case.control`: can be set to `TRUE` if the data is from a non-matched case control study. By default `case.control` is set to `FALSE` which is used for cross-sectional sampling designs. ## 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; $Pr(Y=1)$. Returned by default when `case.control = FALSE`. - **P.var**: estimated variance of `P.est`. The variance is obtained by the sandwich formula. Returned by default when `case.control = FALSE`. - **P0.est**: estimated counterfactual proportion of cases if exposure would be eliminated; $Pr(Y0=1)$. Returned by default when `case.control = FALSE`. - **P0.var**: estimated variance of `P0.est`. The variance is obtained by the sandwich formula. Returned by default when `case.control = FALSE`. - **log.or**: a vector of the estimated log odds ratio for every individual. `log.or` contains the estimated coefficient for the exposure variable `X` for every level of the confounder `Z` as specified by the user in the formula. If the model to be estimated is $$ logit\{Pr(Y=1|X,Z)\} = \alpha+\beta{X}+\gamma{Z}logit {Pr(Y=1|X,Z)} = \alpha + \beta X + \gamma Z $$ then `log.or` is the estimate of $\beta$. If the model to be estimated is $$ logit\{Pr(Y=1|X,Z)\}=\alpha+\beta{X}+\gamma{Z}+\psi{XZ}logit{Pr(Y=1|X,Z)} = \alpha + \beta X +\gamma Z +\psi XZ $$ then `log.odds` is the estimate of $\beta + \psi Z$. Only returned if argument `case.control` is set to `TRUE`. ## Details `AFglm` 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 using the (`glm`) function. The estimation strategy is different for cross-sectional and case-control sampling designs even if the underlying logististic regression model is the same. For cross-sectional sampling designs the AF can 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). For case-control sampling designs the outcome prevalence is fixed by sampling design and absolute probabilities (`P.est` and `P0.est`) can not be estimated. Instead adjusted log odds ratios (`log.or`) are estimated for each individual. This is done by setting `case.control` to `TRUE`. It is then assumed that the outcome is rare so that the risk ratio can be approximated by the odds ratio. For case-control sampling designs the AF be defined as (Bruzzi et. al) $$ AF = 1 - \frac{Pr(Y_0=1)}{Pr(Y = 1)}AF = 1 - Pr(Y0 = 1) / Pr(Y = 1) $$ where $Pr(Y0 = 1)$ denotes the counterfactual probability of the outcome if the exposure would have been eliminated from the population. If `Z` is sufficient for confounding control then the probability $Pr(Y0 = 1)$ can be expressed as $$ Pr(Y_0=1)=E_Z\{Pr(Y=1\mid{X}=0,Z)\}.Pr(Y0=1) = E_z{Pr(Y = 1 | X = 0, Z)}. $$ Using Bayes' theorem this implies that the AF can be expressed as $$ AF = 1-\frac{E_Z\{Pr(Y=1\mid X=0,Z)\}}{Pr(Y=1)}=1-E_Z\{RR^{-X}(Z)\mid{Y = 1}\}AF = 1 - E_z{Pr( Y = 1 | X = 0, Z)} / Pr(Y = 1) = 1 - E_z{RR^{-X} (Z) | Y = 1} $$ where $RR(Z)$ is the risk ratio $$ \frac{Pr(Y=1\mid{X=1,Z})}{Pr(Y=1\mid{X=0,Z})}.Pr(Y = 1 | X = 1,Z)/Pr(Y=1 | X = 0, Z). $$ Moreover, the risk ratio can be approximated by the odds ratio if the outcome is rare. Thus, $$ AF \approx 1 - E_Z\{OR^{-X}(Z)\mid{Y = 1}\}.AF is approximately 1 - E_z{OR^{-X}(Z) | Y = 1}. $$ 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) # Fit a glm object fit <- glm(formula = Y ~ X + Z + X * Z, family = binomial, data = data) # Estimate the attributable fraction from the fitted logistic regression AFglm_est <- AFglm(object = fit, data = data, exposure = "X") summary(AFglm_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)) # Fit a glm object fit <- glm(formula = Y ~ X + Z + X * Z, family = binomial, data = data) # Estimate the attributable fraction from the fitted logistic regression AFglm_clust <- AFglm(object = fit, data = data, exposure = "X", clusterid = "id") summary(AFglm_clust) # Example 3: non matched case-control # Simulate a sample from a non matched case-control sampling design # Make the outcome a rare event by setting the intercept to -6 expit <- function(x) 1 / (1 + exp( - x)) NN <- 1000000 n <- 500 intercept <- -6 Z <- rnorm(n = NN) X <- rbinom(n = NN, size = 1, prob = expit(Z)) Y <- rbinom(n = NN, size = 1, prob = expit(intercept + X + Z)) population <- data.frame(Z, X, Y) Case <- which(population$Y == 1) Control <- which(population$Y == 0) # Sample cases and controls from the population case <- sample(Case, n) control <- sample(Control, n) data <- population[c(case, control), ] # Fit a glm object fit <- glm(formula = Y ~ X + Z + X * Z, family = binomial, data = data) # Estimate the attributable fraction from the fitted logistic regression AFglm_est_cc <- AFglm(object = fit, data = data, exposure = "X", case.control = TRUE) summary(AFglm_est_cc) ``` ## References Bruzzi, P., Green, S. B., Byar, D., Brinton, L. A., and Schairer, C. (1985). Estimating the population attributable risk for multiple risk factors using case-control data. **American Journal of Epidemiology** 122 , 904-914. 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 `glm` used for fitting the logistic regression model. For conditional logistic regression (commonly for data from a matched case-control sampling design) see `AFclogit`. ## Author(s) Elisabeth Dahlqwist, Arvid

AFglm function