AFparfrailty: Attributable fraction function based on a Weibull...
In AF: Model-Based Estimation of Confounder-Adjusted Attributable Fractions

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

AFparfrailty estimates the model-based adjusted attributable fraction function from a shared Weibull gamma-frailty model in form of a parfrailty object. This model is commonly used for data from cohort sampling familty designs with time-to-event outcomes.

1	AFparfrailty(object, data, exposure, times, clusterid)

`object`	a fitted Weibull gamma-parfrailty object of class "`parfrailty`".
`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 death times will be used.
`clusterid`	the name of the cluster identifier variable as a string, if data are clustered.

AFparfrailty 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 shared frailty model (parfrailty). The baseline hazard is assumed to follow a Weibull distribution and the unobserved shared frailty effects U are assumed to be gamma distributed. 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 and U are sufficient for confounding control, then S0(t) can be expressed as E_z{S(t|X=0,Z)}. The function uses a fitted Weibull gamma-frailty model to estimate S(t|X=0,Z), and the marginal sample distribution of Z to approximate the outer expectation. 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.

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

parfrailty used for fitting the Weibull gamma-frailty and stdParfrailty used for standardization of a parfrailty object.

# Example 1: clustered data with frailty U
expit <- function(x) 1 / (1 + exp( - x))
n <- 100
m <- 2
alpha <- 1.5
eta <- 1
phi <- 0.5
beta <- 1
id <- rep(1:n,each=m)
U <- rep(rgamma(n, shape = 1 / phi, scale = phi), each = m)
Z <- rnorm(n * m)
X <- rbinom(n * m, size = 1, prob = expit(Z))
# Reparametrize scale as in rweibull function
weibull.scale <- alpha / (U * exp(beta * X)) ^ (1 / eta)
t <- rweibull(n * m, shape = eta, scale = weibull.scale)

# Right censoring
c <- runif(n * m, 0, 10)
delta <- as.numeric(t < c)
t <- pmin(t, c)

data <- data.frame(t, delta, X, Z, id)

# Fit a parfrailty object
library(stdReg)
fit <- parfrailty(formula = Surv(t, delta) ~ X + Z + X * Z, data = data, clusterid = "id")
summary(fit)

# Estimate the attributable fraction from the fitted frailty model

time <- c(seq(from = 0.2, to = 1, by = 0.2))

AFparfrailty_est <- AFparfrailty(object = fit, data = data, exposure = "X",
                                  times = time, clusterid = "id")
summary(AFparfrailty_est)
plot(AFparfrailty_est, CI = TRUE, ylim=c(0.1,0.7))