AF.ch: Attributable fraction function for cohort sampling designs...
In AF: Model-Based Estimation of Confounder-Adjusted Attributable Fractions

Description Usage Arguments Details Value Author(s) References See Also Examples

AF.ch estimates the model-based adjusted attributable fraction function for data from cohort sampling designs with time-to-event outcomes.

1	AF.ch(formula, data, exposure, ties = "breslow", times, clusterid)

`formula`	a formula object, with the response on the left of a ~ operator, and the terms on the right. The response must be a survival object as returned by the `Surv` function (`Surv`). The exposure and confounders should be specified as independent (right-hand side) variables. The time-to-event outcome should be specified by the survival object. The formula is used to fit a Cox proportional hazards model.
`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.
`ties`	a character string specifying the method for tie handling. If there are no tied death times all the methods are equivalent. Uses the Breslow method by default.
`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.

Af.ch 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 - {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 Cox proportional hazards regression to estimate S(t|X=0,Z), and the marginal sample distribution of Z to approximate the outer expectation (Sj<c3><b6>lander and Vansteelandt, 2014). If clusterid is supplied, then a clustered sandwich formula is used in all variance calculations.

`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.
`object`	the fitted model. Fitted using Cox proportional hazard, `coxph`.

Elisabeth Dahlqwist, Arvid Sj<c3><b6>lander

Chen, L., Lin, D. Y., and Zeng, D. (2010). Attributable fraction functions for censored event times. Biometrika 97, 713-726.

Sj<c3><b6>lander, A. and Vansteelandt, S. (2014). Doubly robust estimation of attributable fractions in survival analysis. Statistical Methods in Medical Research. doi: 10.1177/0962280214564003.

The new and more general version of the function: AFcoxph. coxph and Surv used for fitting the Cox proportional hazards model.

# 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)

# Estimation of the attributable fraction
AF.ch_est <- AF.ch(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data,
                   exposure = "X", times = time)
summary(AF.ch_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)

# Estimation of the attributable fraction
AF.ch_clust <- AF.ch(formula = Surv(Tobs, D) ~ X + Z + X * Z, data = data,
                         exposure = "X", times = time, clusterid = "id")
summary(AF.ch_clust)
plot(AF.ch_clust, CI = TRUE)