delta.estimate {SurrogateOutcome}

Estimates the treatment effect at time t, defined as the difference in the restricted mean survival time

Description

Estimates the treatment effect at time t, defined as the difference in the restricted mean survival time.

Usage

delta.estimate(xone, xzero, deltaone, deltazero, t, std = FALSE, conf.int = FALSE, 
weight.perturb = NULL)

Arguments

xone

numeric vector, observed event times for the primary outcome in the treatment group.

xzero

numeric vector, observed event times for the primary outcome in the control group.

deltaone

numeric vector, event/censoring indicators for the primary outcome in the treatment group.

deltazero

numeric vector, event/censoring indicators for the primary outcome in the control group.

t

time of interest for treatment effect.

std

TRUE or FALSE; indicates whether standard error estimates should be provided, default is FALSE. Estimates are calculated using perturbation-resampling. Two versions are provided: one that takes the standard deviation of the perturbed estimates (denoted as "sd") and one that takes the median absolute deviation (denoted as "mad").

conf.int

TRUE or FALSE; indicates whether 95% confidence intervals should be provided. Confidence intervals are calculated using the percentiles of perturbed estimates, default is FALSE. If this is TRUE, standard error estimates are automatically provided.

weight.perturb

weights used for perturbation resampling.

Details

Let G \in \{1,0\} be the randomized treatment indicator and T denote the time of the primary outcome of interest. We use potential outcomes notation such that T^{(G)} denotes the time of the primary outcome under treatment G, for G \in \{1, 0\}. We define the treatment effect as the difference in restricted mean survival time up to a fixed time t under treatment 1 versus under treatment 0,

\Delta(t)=E\{T^{(1)}\wedge t\} - E\{T^{(0)}\wedge t \}

where \wedge indicates the minimum. Due to censoring, data consist of n = n_1 + n_0 independent observations \{X_{gi}, \delta_{gi}, i=1,...,n_g, g = 1,0\}, where X_{gi} = T_{gi}\wedge C_{ gi}, \delta_{gi} = I(T_{gi} < C_{gi}), C_{gi} denotes the censoring time, T_{gi} denotes the time of the primary outcome, and \{(T_{gi}, C_{gi}), i = 1, ..., n_g\} are identically distributed within treatment group. The quantity \Delta(t) is estimated using inverse probability of censoring weights:

\hat{\Delta}(t) = n_1^{-1} \sum_{i=1}^{n_1} \hat{M}_{1i}(t)- n_0^{-1} \sum_{i=1}^{n_0} \hat{M}_{0i}(t)

where \hat{M}_{gi}(t) = I(X_{gi} > t)t/\hat{W}^C_g(t) + I(X_{gi} < t)X_{gi}\delta_{gi}/\hat{W}^C_g(X_{gi}) and \hat{W}^C_g(t) is the Kaplan-Meier estimator of P(C_{gi} \ge t).

Value

A list is returned:

delta

the estimate, \hat{\Delta}(t), described above.

rmst.1

the estimated restricted mean survival time in group 1, described above.

rmst.0

the estimated restricted mean survival time in group 0, described above.

delta.sd

the standard error estimate of \hat{\Delta}(t); if std = TRUE or conf.int = TRUE.

delta.mad

the standard error estimate of \hat{\Delta}(t) using the median absolute deviation; if std = TRUE or conf.int = TRUE.

conf.int.delta

a vector of size 2; the 95% confidence interval for \hat{\Delta}(t) based on sample quantiles of the perturbed values; if conf.int = TRUE.

Author(s)

Layla Parast

References

Parast L, Tian L, and Cai T (2020). Assessing the Value of a Censored Surrogate Outcome. Lifetime Data Analysis, 26(2):245-265.

Tian, L, Zhao, L, & Wei, LJ (2013). Predicting the restricted mean event time with the subject's baseline covariates in survival analysis. Biostatistics, 15(2), 222-233.

Royston, P, & Parmar, MK (2011). The use of restricted mean survival time to estimate the treatment effect in randomized clinical trials when the proportional hazards assumption is in doubt. Statistics in Medicine, 30(19), 2409-2421.

Examples

data(ExampleData)
names(ExampleData)

delta.estimate(xone = ExampleData$x1, xzero = ExampleData$x0, deltaone = ExampleData$delta1, 
deltazero = ExampleData$delta0, t = 5)


delta.estimate(xone = ExampleData$x1, xzero = ExampleData$x0, deltaone = ExampleData$delta1, 
deltazero = ExampleData$delta0, t = 5, std = TRUE, conf.int = TRUE)