R.s.surv.estimate {Rsurrogate}

Calculates the proportion of treatment effect explained by the surrogate marker information measured at a specified time and primary outcome information up to that specified time

Description

This function calculates 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. 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. The user can also request an estimate of the incremental value of surrogate marker information.

Usage

R.s.surv.estimate(xone, xzero, deltaone, deltazero, sone, szero, t, 
weight.perturb = NULL, landmark, extrapolate = FALSE, transform = FALSE, 
conf.int = FALSE, var = FALSE, incremental.value = 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.

sone

numeric vector; surrogate marker measurement at t_0 for treated observations, assumed to be continuous. If X_{1i}<t_0, then the surrogate marker measurement should be NA.

szero

numeric vector; surrogate marker measurement at t_0 for control observations, assumed to be continuous. If X_{1i}<t_0, then the surrogate marker measurement should be NA.

t

the time of interest.

weight.perturb

weights used for perturbation resampling.

landmark

the landmark time t_0 or time of surrogate marker measurement.

extrapolate

TRUE or FALSE; indicates whether the user wants to use extrapolation.

transform

TRUE or FALSE; indicates whether the user wants to use a transformation for the surrogate marker.

conf.int

TRUE or FALSE; indicates whether a 95% confidence interval for delta is requested, default is FALSE.

var

TRUE or FALSE; indicates whether a variance estimate is requested, default is FALSE.

incremental.value

TRUE or FALSE; indicates whether the user would like to see the incremental value of the surrogate marker information, 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^{(1)} and T^{(0)} denote the time of the primary outcome of interest, death for example, under the treatment and under the control, respectively. Let S^{(1)} and S^{(0)} denote the surrogate marker measured at time t_0 under the treatment and the control, respectively.

The residual treatment effect is defined as

\Delta_S(t,t_0) = P(T^{(0)} > t_0) \left \{\int \psi_1(t | s, t_0) dF_0(s | t_0)-P(T^{(0)}> t| T^{(0)}> t_0) \right \}

where F_0(\cdot | t_0) is the cumulative distribution function of S^{(0)} conditional on T^{(0)}> t_0 and \psi_1(t | s,t_0) = P(T^{(1)}> t | S^{(1)}=s, T ^{(1)} > t_0). The proportion of treatment effect explained by the surrogate marker information measured at t_0 and primary outcome information up to t_0, which we denote by R_S(t,t_0), can be expressed using a contrast between \Delta_S(t,t_0) and \Delta(t):

R_S(t,t_0)=\{\Delta(t)-\Delta_S(t,t_0)\}/\Delta(t)=1-\Delta_S(t,t_0)/\Delta(t).

The definition and estimation of \Delta(t) is described in the delta.surv.estimate documentation.

Due to censoring, our data consist of n_1 observations \{(X_{1i}, \delta_{1i}, S_{1i}), i=1,...,n_1\} from the treatment group G=1 and n_0 observations \{(X_{0i}, \delta_{0i}, S_{0i}), i=1,...,n_0\} from the control group G=0 where X_{gi} = \min(T_{gi}, C_{ gi}), \delta_{gi} = I(T_{gi} < C_{gi}), C_{gi} denotes the censoring time, and S_{gi} denotes the surrogate marker information measured at time t_0, for g= 1,0, for individual i. Note that if X_{gi} < t_0, then S_{gi} should be NA (not available).

To estimate \Delta_S(t,t_0), we use a nonparametric kernel Nelson-Aalen estimator to estimate \psi_1(t | s,t_0) as \hat{\psi}_1(t| s,t_0) = \exp\{-\hat{\Lambda}_1(t| s,t_0) \}, where

\hat{\Lambda}_1(t| s,t_0) = \int_{t_0}^t \frac{\sum_{i=1}^{n_1} I(X_{1i}>t_0) K_h\{\gamma(S_{1i}) - \gamma(s)\}dN_{1i}(z)}{\sum_{i=1}^{n_1} I(X_{1i}>t_0) K_h\{\gamma(S_{1i}) - \gamma(s)\} Y_{1i}(z)},

is a consistent estimate of \Lambda_1(t| s,t_0 ) = -\log [\psi_1(t| s,t_0)], Y_{1i}(t) = I(X_{1i} \geq t), N_{1i}(t) = I(X_{1i} \leq t) \delta_i, K(\cdot) is a smooth symmetric density function, K_h(x) = K(x/h)/h, \gamma(\cdot) is a given monotone transformation function, and h is a specified bandwidth. To obtain an appropriate h we first use bw.nrd to obtain h_{opt}; and then we let h = h_{opt}n_1^{-c_0} with c_0 = 0.11.

Since F_0(s | t_0) = P(S_{0i} \le s | X_{0i} > t_0), we empirically estimate F_0(s | t_0) using all subjects with X_{0i} > t_0 as

\hat{F}_0(s | t_0) = \frac{\sum_{i=1}^{n_0} I(S_{0i} \le s, X_{0i} > t_0)}{\sum_{i=1}^{n_0} I(X_{0i} > t_0)}.

Subsequently, we construct an estimator for \Delta_{S}(t,t_0) as

\hat{\Delta}_S(t,t_0) = n_0^{-1} \sum_{i=1}^{n_0} \left[\hat{\psi}_1(t| S_{0i},t_0) \frac{I(X_{0i} > t_0)}{\hat{W}^C_0(t_0)} - \frac{I(X_{0i} > t)}{\hat{W}^C_0(t)}\right]

where \hat{W}^C_g(\cdot) is the Kaplan-Meier estimator of survival for censoring for g=1,0. Finally, we estimate R_S(t,t_0) as \hat{R}_S(t,t_0) =1- \hat{\Delta}_S(t,t_0)/\hat{\Delta}(t).

Variance estimation and confidence interval construction are performed using perturbation-resampling. Specifically, let \left \{ V^{(b)} = (V_{11}^{(b)}, ...V_{1n_1}^{(b)}, V_{01}^{(b)}, ...V_{0n_0}^{(b)})^T, b=1,....,D \right \} be n \times D independent copies of a positive random variables V from a known distribution with unit mean and unit variance. Let

\hat{\Delta}^{(b)}(t) = \frac{ \sum_{i=1}^{n_1} V_{1i}^{(b)} I(X_{1i}>t)}{ \sum_{i=1}^{n_1} V_{1i}^{(b)} \hat{W}_1^{C(b)}(t)} -\frac{ \sum_{i=1}^{n_0} V_{0i}^{(b)} I(X_{0i}>t)}{ \sum_{i=1}^{n_0} V_{0i}^{(b)} \hat{W}_0^{C(b)}(t)}.

In this package, we use weights generated from an Exponential(1) distribution and use D=500. The variance of \hat{\Delta}(t) is obtained as the empirical variance of \{\hat{\Delta}(t)^{(b)}, b = 1,...,D\}. Variance estimates for \hat{\Delta}_S(t,t_0) and \hat{R}_S(t,t_0) are calculated similarly. We construct two versions of the 95\% confidence interval for each estimate: one based on a normal approximation confidence interval using the estimated variance and another taking the 2.5th and 97.5th empirical percentile of the perturbed quantities. In addition, we use Fieller's method to obtain a third confidence interval for R_S(t,t_0) as

\left\{1-r: \frac{(\hat{\Delta}_S(t,t_0)-r\hat{\Delta}(t))^2}{\hat{\sigma}_{11}-2r\hat\sigma_{12}+r^2\hat\sigma_{22}} \le c_{\alpha}\right\},

where \hat{\Sigma}=(\hat\sigma_{ij})_{1\le i,j\le 2} and c_\alpha is the (1-\alpha)th percentile of

\left\{\frac{\{\hat{\Delta}^{(b)}_S(t)-(1-\hat R_S(t,t_0))\hat{\Delta}(t)^{(b)}\}^2}{\hat{\sigma}_{11}-2(1-\hat R_S(t,t_0))\hat\sigma_{12}+(1-\hat R_S(t,t_0))^2\hat\sigma_{22}}, b=1, \cdots, C\right\}

where \alpha=0.05.

Since the definition of R_S(t,t_0) considers the surrogate information as a combination of both S information and T information up to t_0, a logical inquiry would be how to assess the incremental value of the S information in terms of the proportion of treatment effect explained, when added to T information up to t_0. The proportion of treatment effect explained by T information up to t_0 only is denoted as R_T(t,t_0) and is described in the documentation for R.t.surv.estimate. The incremental value of S information is defined as:

IV_S(t,t_0) = R_S(t,t_0) - R_T(t,t_0) = \frac{\Delta_T(t,t_0) - \Delta_S(t,t_0)}{\Delta (t)}.

For estimation of R_T(t_0), see documentation for R.t.surv.estimate. The quantity IV_S(t,t_0) is then estimated by \hat{IV}_S(t,t_0) = \hat{R}_S(t,t_0) - \hat{R}_T(t,t_0). Perturbation-resampling is used for variance estimation and confidence interval construction for this quantity, similar to the other quantities in this package.

Note that if the observed supports for S are not the same, then \hat{\Lambda}_1(t| s,t_0) for S_{0i} = s outside the support of S_{1i} may return NA (depending on the bandwidth). If extrapolation = TRUE, then the \hat{\Lambda}_1(t| s,t_0) values for these surrogate values are set to the closest non-NA value. If transform = TRUE, then S_{1i} and S_{0i} are transformed such that the new transformed values, S^{tr}_{1i} and S^{tr}_{0i} are defined as: S^{tr}_{gi} = F([S_{gi} - \mu]/\sigma) for g=0,1 where F(\cdot) is the cumulative distribution function for a standard normal random variable, and \mu and \sigma are the sample mean and standard deviation, respectively, of \{(S_{1i}, S_{0i})^T, i \quad s.t. X_{gi} > t_0\}.

Value

A list is returned:

delta

the estimate, \hat{\Delta}(t), described in delta.estimate documentation.

delta.s

the estimate, \hat{\Delta}_S(t,t_0), described above.

R.s

the estimate, \hat{R}_S(t,t_0), described above.

delta.var

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

delta.s.var

the variance estimate of \hat{\Delta}_S(t,t_0); if var = TRUE or conf.int = TRUE.

R.s.var

the variance estimate of \hat{R}_S(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.s

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

conf.int.quantile.delta.s

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

conf.int.normal.R.s

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

conf.int.quantile.R.s

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

conf.int.fieller.R.s

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

delta.t

the estimate, \hat{\Delta}_T(t,t_0), described above; if incremental.vaue = TRUE.

R.t

the estimate, \hat{R}_T(t,t_0), described above; if incremental.vaue = TRUE.

incremental.value

the estimate, \hat{IV}_S(t,t_0), described above; if incremental.vaue = TRUE.

delta.t.var

the variance estimate of \hat{\Delta}_T(t,t_0); if var = TRUE or conf.int = TRUE and incremental.vaue = TRUE.

R.t.var

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

incremental.value.var

the variance estimate of \hat{IV}_S(t,t_0); if var = TRUE or conf.int = TRUE and incremental.vaue = 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 and incremental.vaue = 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 and incremental.vaue = 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 and incremental.vaue = 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 and incremental.vaue = 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 and incremental.vaue = TRUE.

conf.int.normal.iv

a vector of size 2; the 95% confidence interval for \hat{IV}_S(t,t_0) based on a normal approximation; if conf.int = TRUE and incremental.vaue = TRUE.

conf.int.quantile.iv

a vector of size 2; the 95% confidence interval for \hat{IV}_S(t,t_0) based on sample quantiles of the perturbed values, described above; if conf.int = TRUE and 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 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".

Author(s)

Layla Parast

References

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

Examples

data(d_example_surv)
names(d_example_surv)