# Attributable fraction function based on a Cox Proportional Hazard regression model as a `coxph` object (commonly used for cohort sampling designs with time-to-event outcomes). `AFcoxph` estimates the model-based adjusted attributable fraction function from a Cox Proportional Hazard regression model in form of a `coxph` object. This model is commonly used for data from cohort sampling designs with time-to-event outcomes. ```r AFcoxph(object, data, exposure, times, clusterid) ``` ## Arguments - `object`: a fitted Cox Proportional Hazard regression model object of class "`coxph`". Method for handling ties must be breslow since this is assumed in the calculation of the standard errors. No special terms such as `cluster`, `strata` and `tt` is allowed in the formula for the fitted object. - `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 event times will be used. - `clusterid`: the name of the cluster identifier variable as a string, if data are clustered. Cluster robust standard errors will be calculated. ## 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 `AFcoxph` 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 Cox proportional hazards model (`coxph`). 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` is sufficient for confounding control, then $S0(t)$ can be expressed as $E_z{S(t|X=0,Z)}$. The function uses a fitted Cox proportional hazards regression to estimate $S(t|X=0,Z)$, and the marginal sample distribution of `Z` to approximate the outer expectation ( and Vansteelandt, 2014). If `clusterid` is supplied, then a clustered sandwich formula is used in all variance calculations. ## Examples ```r # Simulate a sample from a cohort sampling design with time-to-event outcome expit <- function(x) 1 / (1 + exp( - x)) n <- 500 time <- c(seq(from = 0.2, to = 1, by = 0.2)) Z <- rnorm(n = n) X <- rbinom(n = n, size = 1, prob = expit(Z)) Tim <- rexp(n = n, rate = exp(X + Z)) C <- rexp(n = n, rate = exp(X + Z)) Tobs <- pmin(Tim, C) D <- as.numeric(Tobs < C) #Ties created by rounding Tobs <- round(Tobs, digits = 2) # Example 1: non clustered data from a cohort sampling design with time-to-event outcomes data <- data.frame(Tobs, D, X, Z) # Fit a Cox PH regression model fit <- coxph(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data, ties="breslow") # Estimate the attributable fraction from the fitted Cox PH regression model AFcoxph_est <- AFcoxph(fit, data=data, exposure ="X", times = time) summary(AFcoxph_est) # Example 2: clustered data from a cohort sampling design with time-to-event outcomes # Duplicate observations in order to create clustered data id <- rep(1:n, 2) data <- data.frame(Tobs = c(Tobs, Tobs), D = c(D, D), X = c(X, X), Z = c(Z, Z), id = id) # Fit a Cox PH regression model fit <- coxph(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data, ties="breslow") # Estimate the attributable fraction from the fitted Cox PH regression model AFcoxph_clust <- AFcoxph(object = fit, data = data, exposure = "X", times = time, clusterid = "id") summary(AFcoxph_clust) plot(AFcoxph_clust, CI = TRUE) # Estimate the attributable fraction from the fitted Cox PH regression model, time unspecified AFcoxph_clust_no_time <- AFcoxph(object = fit, data = data, exposure = "X", clusterid = "id") summary(AFcoxph_clust_no_time) plot(AFcoxph_clust, CI = TRUE) ``` ## References Chen, L., Lin, D. Y., and Zeng, D. (2010). Attributable fraction functions for censored event times. **Biometrika** 97 , 713-726. , A. and Vansteelandt, S. (2014). Doubly robust estimation of attributable fractions in survival analysis. **Statistical Methods in Medical Research**. doi: 10.1177/0962280214564003. ## See Also `coxph` and `Surv` used for fitting the Cox proportional hazards model. ## Author(s) Elisabeth Dahlqwist, Arvid

AFcoxph function