R.t.surv.estimate function

Calculates the proportion of treatment effect explained by the primary outcome information up to a specified time

This function calculates the proportion of treatment effect on the primary outcome explained by the treatment effect on the primary outcome up to $t_0$. The user can also request a variance estimate, estimated using perturbating-resampling, and a 95% confidence interval. If a confidence interval is requested three versions are provided: a normal approximation based interval, a quantile based interval and Fieller's confidence interval, all using perturbation-resampling.

R.t.surv.estimate(xone, xzero, deltaone, deltazero, t, weight.perturb = NULL, 
landmark, var = FALSE, conf.int = FALSE, approx = T)

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<C) where T is the time of the primary outcome and C is the censoring time.
• deltazero: numeric vector, the event indicators for the control group, D = I(T<C) where T is the time of the primary outcome and C is the censoring time.
• t: the time of interest.
• weight.perturb: weights used for perturbation resampling.
• landmark: the landmark time $t_0$ or time of surrogate marker measurement.
• var: TRUE or FALSE; indicates whether a variance estimate for delta is requested, default is FALSE.
• conf.int: TRUE or FALSE; indicates whether a 95% confidence interval for delta is requested, default is FALSE.
• approx: TRUE or FALSE indicating whether an approximation should be used when calculating the probability of censoring; most relevant in settings where the survival time of interest for the primary outcome is greater than the last observed event but before the last censored case, default is TRUE.

Details

Let $G$ be the binary treatment indicator with $G=1$ for treatment and $G=0$ for control and we assume throughout that subjects are randomly assigned to a treatment group at baseline. Let $T$ denote the time of the primary outcome of interest, death for example. We use potential outcomes notation such that $T^{(g)}$ denotes the time of the primary outcome under treatment $G = g$. The proportion of treatment effect explained by T observed up to $t_0$ only is $R_T(t,t_0) = 1-\Delta_T(t,t_0)/\Delta(t)$ where

To estimate $R_T(t,t_0)$, we use the estimator $\hat{R}_T(t,t_0) = 1-\hat{\Delta}_T(t,t_0)/\hat{\Delta}(t)$ where $\hat{\Delta}_T(t,t_0) = \hat{\phi}_0(t_0)\hat{\phi}_1(t)/\hat{\phi}_1(t_0) - \hat{\phi}_0(t)$ and $\hat{\phi}_g(u) = n_g^{-1} \sum_{i=1}^{n_g} \frac{I(X_{gi}>u)}{\hat{W}^C_g(u)}$ for $g=1,0$ where $\widehat{W}^C_g(\cdot)$ is the Kaplan-Meier estimator of survival for censoring for $g=1,0.$

Returns

A list is returned: - delta: the estimate, $\hat{\Delta}(t)$, described in delta.estimate documentation.

• delta.t: the estimate, $\hat{\Delta}_T(t,t_0)$, described above.

• R.t: the estimate, $\hat{R}_T(t,t_0)$, described above.

• delta.var: the variance estimate of $\hat{\Delta}(t)$; if var = TRUE or conf.int = TRUE.

• delta.t.var: the variance estimate of $\hat{\Delta}_T(t,t_0)$; if var = TRUE or conf.int = TRUE.

• R.t.var: the variance estimate of $\hat{R}_T(t,t_0)$; if var = TRUE or conf.int = TRUE.

• conf.int.normal.delta: a vector of size 2; the 95% confidence interval for $\hat{\Delta}(t)$ based on a normal approximation; if conf.int = TRUE.

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

• conf.int.normal.delta.t: a vector of size 2; the 95% confidence interval for $\hat{\Delta}_T(t,t_0)$ based on a normal approximation; if conf.int = TRUE.

• conf.int.quantile.delta.t: a vector of size 2; the 95% confidence interval for $\hat{\Delta}_T(t,t_0)$ based on sample quantiles of the perturbed values, described above; if conf.int = TRUE.

• conf.int.normal.R.t: a vector of size 2; the 95% confidence interval for $\hat{R}_T(t,t_0)$ based on a normal approximation; if conf.int = TRUE.

• conf.int.quantile.R.t: a vector of size 2; the 95% confidence interval for $\hat{R}_T(t,t_0)$ based on sample quantiles of the perturbed values, described above; if conf.int = TRUE.

• conf.int.fieller.R.t: a vector of size 2; the 95% confidence interval for $\hat{R}_T(t,t_0)$ based on Fieller's approach, described above; if conf.int = TRUE.

References

Parast, L., Cai, T., & Tian, L. (2017). Evaluating surrogate marker information using censored data. Statistics in Medicine, 36(11), 1767-1782.

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".

Author(s)

Layla Parast

Examples

data(d_example_surv)
names(d_example_surv)