AF.ch {AF}

Attributable fraction function for cohort sampling designs with time-to-event outcomes. NOTE! Deprecated function. Use AFcoxph.

Description

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

Usage

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

Arguments

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.

Details

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-\frac{\{1-S_0(t)\}}{\{1-S(t)\}}

where S_0(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 S_0(t) can be expressed as E_Z\{S(t\mid{X=0,Z })\}. The function uses Cox proportional hazards regression to estimate S(t\mid{X=0,Z}), and the marginal sample distribution of Z to approximate the outer expectation (Sjölander and Vansteelandt, 2014). If clusterid is supplied, then a clustered sandwich formula is used in all variance calculations.

Value

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; S_0(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.

Author(s)

Elisabeth Dahlqwist, Arvid Sjölander

References

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

Sjölander, 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

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

Examples

# 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)