# Aug.R.s.surv.estimate: Calculates the augmented estimator of the proportion of... In Rsurrogate: Robust Estimation of the Proportion of Treatment Effect Explained by Surrogate Marker Information

## Description

This function calculates the augmented version of the proportion of treatment effect on the primary outcome explained by the surrogate marker information measured at t_0 and primary outcome information up to t_0. Variance estimates and 95 % confidence intervals for the augmented estimates are provided automatically; three versions of the confidence interval are provided: a normal approximation based interval, a quantile based interval and Fieller's confidence interval, all using perturbation-resampling. The user can also request an estimate of the incremental value of surrogate marker information.

## Usage

 1 2 3 4 Aug.R.s.surv.estimate(xone, xzero, deltaone, deltazero, sone, szero, t, weight.perturb = NULL, landmark, extrapolate = FALSE, transform = FALSE, basis.delta.one, basis.delta.zero, basis.delta.s.one = NULL, basis.delta.s.zero = NULL, incremental.value = FALSE) 

## Arguments

 xone numeric vector, the observed event times in the treatment group, X = min(T,C) where T is the time of the primary outcome and C is the censoring time. xzero numeric vector, the observed event times in the control group, X = min(T,C) where T is the time of the primary outcome and C is the censoring time. deltaone numeric vector, the event indicators for the treatment group, D = I(T

## Details

Please see R.s.surv.estimate documention for details about the estimates before augmentation is performed. Recent work has shown that augmentation can lead to improvements in efficiency by taking advantage of the association between baseline information, denoted here as Z, and the primary outcome. This function calculates the augmented estimates of the quantities of interest. For example, the augmented version of \hat{Δ}(t) is defined as:

\hat{Δ}(t)^{AUG} = \hat{Δ}(t) + γ \{n_1^{-1}∑_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}∑_{i=1}^{n_0}h(Z_{0i}) \}

where Z_{gi}, i=1, 2, \cdots, n_g are i.i.d. random vectors of baseline covariates from treatment group g and h(\cdot) is a basis transformation given a priori. Due to treatment randomization, \{n_1^{-1}∑_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}∑_{i=1}^{n_0}h(Z_{0i}) \} converges to zero in probability as the sample size goes to infinity and thus the augmented estimator converges to the same limit as the original counterparts. The quantity γ is selected such that the variance of \hat{Δ}(t)^{AUG} is minimized. That is, γ = (Ξ_{12}) ( Ξ_{22} ) ^{-1} where

Ξ_{12} = \mbox{cov} \{ \hat{Δ}(t), n_1^{-1}∑_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}∑_{i=1}^{n_0}h(Z_{0i}) \},

Ξ_{22} = \mbox{var} \{n_1^{-1}∑_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}∑_{i=1}^{n_0}h(Z_{0i})\}

and thus we can obtain \hat{Δ}(t)^{AUG} by replacing γ with a consistent estimator, \hat{γ} obtained using perturbation-resampling. A similar approach is used to obtain \hat{Δ}_S(t)^{AUG} and thus construct

\hat{R}_S(t,t_0)^{AUG}=1-\frac{\hat{Δ}_S(t,t_0)^{AUG}}{\hat{Δ}(t)^{AUG}}.

## Value

A list is returned:

 aug.delta  the estimate, \hat{Δ}(t)^{AUG}. aug.delta.s  the estimate, \hat{Δ}_S(t,t_0)^{AUG}. aug.R.s  the estimate, \hat{R}_S(t,t_0)^{AUG}. aug.delta.var  the variance estimate of \hat{Δ}(t)^{AUG}. aug.delta.s.var  the variance estimate of \hat{Δ}_S(t,t_0)^{AUG}. aug.R.s.var  the variance estimate of \hat{R}_S(t,t_0)^{AUG}. conf.int.normal.aug.delta a vector of size 2; the 95% confidence interval for \hat{Δ}(t)^{AUG} based on a normal approximation. conf.int.quantile.aug.delta a vector of size 2; the 95% confidence interval for \hat{Δ}(t)^{AUG} based on sample quantiles of the perturbed values. conf.int.normal.aug.delta.s a vector of size 2; the 95% confidence interval for \hat{Δ}_S(t,t_0)^{AUG} based on a normal approximation. conf.int.quantile.aug.delta.s a vector of size 2; the 95% confidence interval for \hat{Δ}_S(t,t_0)^{AUG} based on sample quantiles of the perturbed values. conf.int.normal.R.s a vector of size 2; the 95% confidence interval for \hat{R}_S(t,t_0)^{AUG} based on a normal approximation. conf.int.quantile.aug.R.s a vector of size 2; the 95% confidence interval for \hat{R}_S(t,t_0)^{AUG} based on sample quantiles of the perturbed values.. conf.int.fieller.aug.R.s a vector of size 2; the 95% confidence interval for \hat{R}_S(t,t_0)^{AUG} based on Fieller's approach. aug.delta.t  the estimate, \hat{Δ}_T(t,t_0)^{AUG}; if incremental.vaue = TRUE. aug.R.t  the estimate, \hat{R}_T(t,t_0)^{AUG}; if incremental.vaue = TRUE. aug.incremental.value the estimate, \hat{IV}_S(t,t_0)^{AUG}; if incremental.vaue = TRUE. aug.delta.t.var  the variance estimate of \hat{Δ}_T(t,t_0)^{AUG}; if incremental.vaue = TRUE. aug.R.t.var  the variance estimate of \hat{R}_T(t,t_0)^{AUG}; if incremental.vaue = TRUE. aug.incremental.value.var  the variance estimate of \hat{IV}_S(t,t_0)^{AUG}; if incremental.vaue = TRUE. aug.conf.int.normal.delta.t a vector of size 2; the 95% confidence interval for \hat{Δ}_T(t,t_0)^{AUG} based on a normal approximation; if incremental.vaue = TRUE. aug.conf.int.quantile.delta.t a vector of size 2; the 95% confidence interval for \hat{Δ}_T(t,t_0)^{AUG} based on sample quantiles of the perturbed values; if incremental.vaue = TRUE. aug.conf.int.normal.R.t a vector of size 2; the 95% confidence interval for \hat{R}_T(t,t_0)^{AUG} based on a normal approximation; if incremental.vaue = TRUE. aug.conf.int.quantile.R.t a vector of size 2; the 95% confidence interval for \hat{R}_T(t,t_0)^{AUG} based on sample quantiles of the perturbed values; if incremental.vaue = TRUE. aug.conf.int.fieller.R.t a vector of size 2; the 95% confidence interval for \hat{R}_T(t,t_0)^{AUG} based on Fieller's approach, described above; if incremental.vaue = TRUE. aug.conf.int.normal.iv a vector of size 2; the 95% confidence interval for \hat{IV}_S(t,t_0)^{AUG} based on a normal approximation; if incremental.vaue = TRUE. aug.conf.int.quantile.iv a vector of size 2; the 95% confidence interval for \hat{IV}_S(t,t_0)^{AUG} based on sample quantiles of the perturbed values; if incremental.vaue = TRUE.

## Note

If the treatment effect is not significant, the user will receive the following message: "Warning: it looks like the treatment effect is not significant; may be difficult to interpret the residual treatment effect in this setting". If the treatment effect is negative, the user will receive the following message: "Warning: it looks like you need to switch the treatment groups" as this package assumes throughout that larger values of the event time are better. If the observed support of the surrogate marker for the control group is outside the observed support of the surrogate marker for the treatment group, the user will receive the following message: "Warning: observed supports do not appear equal, may need to consider a transformation or extrapolation".

Layla Parast

## References

Tian L, Cai T, Zhao L,Wei L. On the covariate-adjusted estimation for an overall treatment difference with data from a randomized comparative clinical trial. Biostatistics 2012; 13(2): 256-273.

Garcia TP, Ma Y, Yin G. Efficiency improvement in a class of survival models through model-free covariate incorporation. Lifetime Data Analysis 2011; 17(4): 552-565.

Zhang M, Tsiatis AA, Davidian M. Improving efficiency of inferences in randomized clinical trials using auxiliary covariates. Biometrics 2008; 64(3): 707-715.

Parast L, Cai T and Tian L. Evaluating Surrogate Marker Information using Censored Data. Under Review.

## Examples

 1 2 3 4 5 #computationally intensive #Aug.R.s.surv.estimate(xone = d_example_surv$x1, xzero = d_example_surv$x0, #deltaone = d_example_surv$delta1, deltazero = d_example_surv$delta0, #sone = d_example_surv$s1, szero = d_example_surv$s0, t=3, landmark = 1, #basis.delta.one = d_example_surv$z1 , basis.delta.zero = d_example_surv$z0) 

Rsurrogate documentation built on May 29, 2017, 6:16 p.m.