Package 'Rsurrogate'

Calculates the augmented estimator of 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 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

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, 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.
basis.delta.one	either a vector of length $n_1$ or a matrix with $n_1$ rows; this is the basis transformation used for augmentation of $\hat{\Delta}(t)$ for treated observations only, all values must be numeric
basis.delta.zero	either a vector of length $n_0$ or a matrix with $n_0$ rows; this is the basis transformation used for augmentation of $\hat{\Delta}(t)$ for control observations only, all values must be numeric
basis.delta.s.one	either a vector of length $n_1$ or a matrix with $n_1$ rows; this is the basis transformation used for augmentation of $\hat{\Delta}_S(t,t_0)$ for treated observations only, all values must be numeric; default is to assume this is the same as basis.delta.one i.e. that the same basis transformation is used for both quantities
basis.delta.s.zero	either a vector of length $n_0$ or a matrix with $n_0$ rows; this is the basis transformation used for augmentation of $\hat{\Delta}_S(t,t_0)$ for control observations only, all values must be numeric; default is to assume this is the same as basis.delta.zero i.e. that the same basis transformation is used for both quantities
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

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 $\hat{\Delta}(t)$ is defined as:

$\hat{\Delta}(t)^{AUG} = \hat{\Delta}(t) + \gamma \{n_1^{-1}\sum_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}\sum_{i=1}^{n_0}h(Z_{0i}) \}$

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, $\{n_1^{-1}\sum_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}\sum_{i=1}^{n_0}h(Z_{0i}) \}$ 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 $\gamma$ is selected such that the variance of $\hat{\Delta}(t)^{AUG}$ is minimized. That is, $\gamma = (\Xi_{12}) ( \Xi_{22} ) ^{-1}$ where

$\Xi_{12} = \mbox{cov} \{ \hat{\Delta}(t), n_1^{-1}\sum_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}\sum_{i=1}^{n_0}h(Z_{0i}) \},$

$\Xi_{22} = \mbox{var} \{n_1^{-1}\sum_{i=1}^{n_1}h(Z_{1i})-n_0^{-1}\sum_{i=1}^{n_0}h(Z_{0i})\}$

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

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

When only a single $Z_{gi}$ is provided in the basis argument, the following basis is used in this function: $h(Z_{gi}) = (1, Z_{gi}, Z_{gi}^2)'.$

Value

A list is returned:

aug.delta	the estimate, $\hat{\Delta}(t)^{AUG}$.
aug.delta.s	the estimate, $\hat{\Delta}_S(t,t_0)^{AUG}$.
aug.R.s	the estimate, $\hat{R}_S(t,t_0)^{AUG}$.
aug.delta.var	the variance estimate of $\hat{\Delta}(t)^{AUG}$.
aug.delta.s.var	the variance estimate of $\hat{\Delta}_S(t,t_0)^{AUG}$.
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 $\hat{\Delta}(t)^{AUG}$ based on a normal approximation.
conf.int.quantile.aug.delta	a vector of size 2; the 95% confidence interval for $\hat{\Delta}(t)^{AUG}$ based on sample quantiles of the perturbed values.
conf.int.normal.aug.delta.s	a vector of size 2; the 95% confidence interval for $\hat{\Delta}_S(t,t_0)^{AUG}$ based on a normal approximation.
conf.int.quantile.aug.delta.s	a vector of size 2; the 95% confidence interval for $\hat{\Delta}_S(t,t_0)^{AUG}$ 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, $\hat{\Delta}_T(t,t_0)^{AUG}$; 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 $\hat{\Delta}_T(t,t_0)^{AUG}$; 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 $\hat{\Delta}_T(t,t_0)^{AUG}$ 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 $\hat{\Delta}_T(t,t_0)^{AUG}$ 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 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

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

Examples

#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)

---

Hypothetical data

Description

Hypothetical data to be used in examples.

Usage

data(d_example)

Format

A list with 8 elements representing 500 observations from a control group and 500 observations from a treatment group:

s1.a

First surrogate marker measurement for treated observations.

s1.b

Second surrogate marker measurement for treated observations.

s1.c

Third surrogate marker measurement for treated observations.

y1

Primary outcome for treated observations.

s0.a

First surrogate marker measurement for control observations.

s0.b

Second surrogate marker measurement for control observations.

s0.c

Third surrogate marker measurement for control observations.

y0

Primary outcome for control observations.

Examples

data(d_example)
names(d_example)

---

Hypothetical data with replicate measurements

Description

Hypothetical data to be used in measurement error example.

Usage

data(d_example_me)

Format

A list with 10 elements representing 500 observations from a control group and 500 observations from a treatment group:

y1

Primary outcome for treated observations.

s1

Surrogate marker for treated observations.

s1_rep1

Replicate measurement of the surrogate marker for treated observations.

s1_rep2

Replicate measurement of the surrogate marker for treated observations.

s1_rep3

Replicate measurement of the surrogate marker for treated observations.

y0

Primary outcome for control observations.

s0

Surrogate marker for control observations.

s0_rep1

Replicate measurement of the surrogate marker for control observations.

s0_rep2

Replicate measurement of the surrogate marker for control observations.

s0_rep3

Replicate measurement of the surrogate marker for control observations.

Examples

data(d_example_me)
names(d_example_me)

---

Hypothetical survival data with multiple surrogate markers

Description

Hypothetical survival data with multiple surrogate markers to be used in examples.

Usage

data(d_example_multiple)

Format

A list with 6 elements representing 1000 observations from a control group and 1000 observations from a treatment group:

s1

Surrogate marker measurements for treated observations; these markers are measured at time = 0.5. For observations that experience the primary outcome or are censored before 0.5, the surrogate values are NA.

x1

The observed event or censoring time for treated observations; X = min(T, C) where T is the time of the primary outcome and C is the censoring time.

delta1

The indicator identifying whether the treated observation was observed to have the event or was censored; D =1*(T<C) where T is the time of the primary outcome and C is the censoring time.

s0

Surrogate marker measurements for control observations; these markers are measured at time = 0.5. For observations that experience the primary outcome or are censored before 0.5, the surrogate values are NA.

x0

The observed event or censoring time for control observations; X = min(T, C) where T is the time of the primary outcome and C is the censoring time.

delta0

The indicator identifying whether the control observation was observed to have the event or was censored; D =1*(T<C) where T is the time of the primary outcome and C is the censoring time.

Examples

data(d_example_multiple)
names(d_example_multiple)

---

Hypothetical survival data

Description

Hypothetical survival data to be used in examples.

Usage

data(d_example_surv)

Format

A list with 8 elements representing 500 observations from a control group and 500 observations from a treatment group:

s1

Surrogate marker measurement for treated observations; this marker is measured at time = 0.5. For observations that experience the primary outcome or are censored before 0.5, this value is NA.

x1

The observed event or censoring time for treated observations; X = min(T, C) where T is the time of the primary outcome and C is the censoring time.

delta1

The indicator identifying whether the treated observation was observed to have the event or was censored; D =1*(T<C) where T is the time of the primary outcome and C is the censoring time.

s0

Surrogate marker measurement for control observations; this marker is measured at time = 0.5. For observations that experience the primary outcome or are censored before 0.5, this value is NA.

x0

The observed event or censoring time for control observations; X = min(T, C) where T is the time of the primary outcome and C is the censoring time.

delta0

The indicator identifying whether the control observation was observed to have the event or was censored; D =1*(T<C) where T is the time of the primary outcome and C is the censoring time.

z1

A baseline covariate value for treated observations.

z0

A baseline covariate value for control observations.

Examples

data(d_example_surv)
names(d_example_surv)

---

Calculates treatment effect

Description

This function calculates the treatment effect estimate, the difference in the average outcome in the treatment group minus the control group. This function is intended to be used for a fully observed continuous outcome. The user can also request a variance estimate, estimated using perturbating-resampling, and a 95% confidence interval. If a confidence interval is requested two versions are provided: a normal approximation based interval and a quantile based interval, both use perturbation-resampling.

Usage

delta.estimate(yone,yzero, var = FALSE, conf.int = FALSE, weight = NULL, 
weight.perturb = NULL)

Arguments

yone	numeric vector; primary outcome for treated observations.
yzero	numeric vector; primary outcome for control observations.
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.
weight	a n1+n0 by x matrix of weights where n1 = length of yone and n0 = length of yzero, default is null; generally not supplied by use but only used by other functions.
weight.perturb	a n1+n0 by x matrix of weights where n1 = length of yone and n0 = length of yzero, default is null; generally used for confidence interval construction and may be supplied by user.

Details

Let $Y^{(1)}$ and $Y^{(0)}$ denote the primary outcome under the treatment and primary outcome under the control,respectively. The treatment effect, $\Delta$, is the expected difference in $Y^{(1)}$ compared to $Y^{(0)}$, $\Delta=E(Y^{(1)}-Y^{(0)}).$ We estimate $\Delta$ as

$\hat{\Delta} = n_1^{-1} \sum_{i=1}^{n_1} Y_{1i} - n_0^{-1} \sum_{i=1}^{n_0} Y_{0i}$

where $Y_{1i}$ is the observed primary outcome for person $i$ in the treated group, $Y_{0i}$ is the observed primary outcome for person $i$ in the control group, and $n_1$ and $n_0$ are the number of individuals in the treatment and control group, respectively. Randomized treatment assignment is assumed throughout this package.

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

$\hat{\Delta}^{(b)} = \frac{ \sum_{i=1}^{n_1} V_{1i}^{(b)} Y_{1i}}{ \sum_{i=1}^{n_1} V_{1i}^{(b)}} - \frac{ \sum_{i=1}^{n_0} V_{0i}^{(b)} Y_{0i}}{ \sum_{i=1}^{n_0} V_{0i}^{(b)}}.$

The variance of $\hat{\Delta}$ is obtained as the empirical variance of $\{\hat{\Delta}^{(b)}, b = 1,...,D\}.$ In this package, we use weights generated from an Exponential(1) distribution and use $D=500$. We construct two versions of the $95\%$ confidence interval for $\hat{\Delta}$: one based on a normal approximation confidence interval using the estimated variance and another taking the 2.5th and 97.5th empirical percentiles of $\hat{\Delta}^{(b)}$.

Value

A list is returned:

delta	the estimate, $\hat{\Delta}$, described above.
var	the variance estimate of $\hat{\Delta}$; if var = TRUE or conf.int = TRUE.
conf.int.normal	a vector of size 2; the 95% confidence interval for $\hat{\Delta}$ based on a normal approximation; if conf.int = TRUE.
conf.int.quantile	a vector of size 2; the 95% confidence interval for $\hat{\Delta}$ based on sample quantiles of the perturbed values, described above; if conf.int = TRUE.

Author(s)

Layla Parast

Examples

data(d_example)
names(d_example)
delta.estimate(yone=d_example$y1, yzero=d_example$y0)

---

Calculates robust residual treatment effect accounting for multiple surrogate markers at a specified time and primary outcome information up to that specified time

Description

This function calculates the robust estimate of the residual treatment effect accounting for multiple surrogate markers measured at $t_0$ and primary outcome information up to $t_0$ i.e. the hypothetical treatment effect if both the surrogate marker distributions at $t_0$ and survival up to $t_0$ in the treatment group look like the surrogate marker distributions and survival up to $t_0$ in the control group. Ideally this function is only used as a helper function and is not directly called.

Usage

delta.multiple.surv(xone, xzero, deltaone, deltazero, sone, szero, type =1, t, 
weight.perturb = NULL, landmark, extrapolate = FALSE, transform = 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	matrix of numeric values; surrogate marker measurements at $t_0$ for treated observations. If $X_{1i}<t_0$, then the surrogate marker measurements should be NA.
szero	matrix of numeric values; surrogate marker measurements at $t_0$ for control observations. If $X_{0i}<t_0$, then the surrogate marker measurements should be NA.
type	type of estimate; options are 1 = two-stage robust estimator, 2 = weighted two-stage robust estimator, 3 = double-robust estimator, 4 = two-stage model-based estimator, 5 = weighted estimator, 6 = double-robust model-bsed estimator; default is 1.
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 psuedo-score.
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

Details are included in the documentation for R.multiple.surv.

Value

$\hat{\Delta}_S(t,t_0)$, the residual treatment effect estimate accounting for multiple surrogarte markers measured at $t_0$ and primary outcome information up to $t_0$.

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

References

Parast, L., Cai, T., & Tian, L. (2021). Evaluating multiple surrogate markers with censored data. Biometrics, 77(4), 1315-1327.

Examples

data(d_example_multiple)
names(d_example_multiple)
## Not run: 
delta.multiple.surv(xone = d_example_multiple$x1, xzero = d_example_multiple$x0, deltaone =
 d_example_multiple$delta1, deltazero = d_example_multiple$delta0, sone = 
 as.matrix(d_example_multiple$s1), szero = as.matrix(d_example_multiple$s0), 
 type =1, t = 1, landmark=0.5)

## End(Not run)

---

Calculates model-based or robust residual treatment effect

Description

This function calculates the model-based or robust estimate of the residual treatment effect i.e. the hypothetical treatment effect if the distribution of the surrogate in the treatment group looks like the distribution of the surrogate in the control group. Ideally, this function is only used as a helper function and is not directly called.

Usage

delta.s.estimate(sone, szero, yone, yzero, weight.perturb = NULL, number="single",
type="robust", warn.te = FALSE, warn.support = FALSE, extrapolate = FALSE, 
transform = FALSE)

Arguments

sone	numeric vector or matrix; surrogate marker for treated observations, assumed to be continuous. If there are multiple surrogates then this should be a matrix with $n_1$ (number of treated observations) rows and n.s (number of surrogate markers) columns.
szero	numeric vector or matrix; surrogate marker for control observations, assumed to be continuous. If there are multiple surrogates then this should be a matrix with $n_0$ (number of control observations) rows and n.s (number of surrogate markers) columns.
yone	numeric vector; primary outcome for treated observations.
yzero	numeric vector; primary outcome for control observations.
weight.perturb	a $n_1+n_0$ by x matrix of weights where $n_1 =$ length of yone and $n_0 =$ length of yzero; generally used for variance estimation and confidence interval construction, default is null.
number	specifies the number of surrogate markers; choices are "multiple" or "single", default is "single".
type	specifies the type of estimation; choices are "robust" or "model", default is "robust".
warn.te	value passed from R.s.estimate function to control warnings; user does not need to specify.
warn.support	value passed from R.s.estimate function to control warnings; user does not need to specify.
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.

Details

Details are included in the documentation for R.s.estimate.

Value

$\hat{\Delta}_S$, the model-based or robust residual treatment effect estimate.

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". In the single marker case with the robust estimation approach, 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., McDermott, M., Tian, L. (2015). Robust estimation of the proportion of treatment effect explained by surrogate marker information. Statistics in Medicine, 35(10):1637-1653.

Wang, Y., & Taylor, J. M. (2002). A measure of the proportion of treatment effect explained by a surrogate marker. Biometrics, 58(4), 803-812.

Examples

data(d_example)
names(d_example)
delta.s.estimate(yone=d_example$y1, yzero=d_example$y0, sone=d_example$s1.a, szero=
d_example$s0.a, number = "single", type = "robust")
delta.s.estimate(yone=d_example$y1, yzero=d_example$y0, sone=d_example$s1.a, szero=
d_example$s0.a, number = "single", type = "model")
delta.s.estimate(yone=d_example$y1, yzero=d_example$y0, sone=cbind(d_example$s1.a, 
d_example$s1.b, d_example$s1.c), szero=cbind(d_example$s0.a, d_example$s0.b, d_example$s0.c), 
number = "multiple", type = "robust")
delta.s.estimate(yone=d_example$y1, yzero=d_example$y0, sone=cbind(d_example$s1.a, 
d_example$s1.b, d_example$s1.c), szero=cbind(d_example$s0.a, d_example$s0.b, d_example$s0.c),
number = "multiple", type = "model")

---

Calculates robust residual treatment effect accounting for surrogate marker information measured at a specified time and primary outcome information up to that specified time

Description

This function calculates the robust estimate of the residual treatment effect accounting for surrogate marker information measured at $t_0$ and primary outcome information up to $t_0$ i.e. the hypothetical treatment effect if both the surrogate marker distribution at $t_0$ and survival up to $t_0$ in the treatment group look like the surrogate marker distribution and survival up to $t_0$ in the control group. Ideally this function is only used as a helper function and is not directly called.

Usage

delta.s.surv.estimate(xone, xzero, deltaone, deltazero, sone, szero, t, 
weight.perturb = NULL, landmark, extrapolate = FALSE, transform = FALSE,
approx = T, warn.te = FALSE, warn.support = 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<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.
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.
warn.te	value passed from R.s.estimate function to control warnings; user does not need to specify.
warn.support	value passed from R.s.estimate function to control warnings; user does not need to specify.

Details

Details are included in the documentation for R.s.surv.estimate.

Value

$\hat{\Delta}_S(t,t_0)$, the robust residual treatment effect estimate accounting for surrogate marker information measured at $t_0$ and primary outcome information up to $t_0$.

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)

---

Calculates treatment effect in a survival setting

Description

This function calculates the treatment effect in the survival setting i.e. the difference in survival at time t between the treatment group and the control group. The user can also request a variance estimate, estimated using perturbating-resampling, and a 95% confidence interval. If a confidence interval is requested two versions are provided: a normal approximation based interval and a quantile based interval, both use perturbation-resampling.

Usage

delta.surv.estimate(xone, xzero, deltaone, deltazero, t, var = FALSE, conf.int 
= FALSE, weight = NULL, weight.perturb = NULL, 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.
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.
weight	a $n_1+n_0$ by $x$ matrix of weights where $n_1 =$ sample size in treatment group and $n_0 =$ sample size in the control group, default is null; generally not supplied by use but only used by other functions.
weight.perturb	a $n_1+n_0$ by $x$ matrix of weights where $n_1 =$sample size in treatment group and $n_0 =$ sample size in the control group, default is null; generally used for confidence interval construction and may be supplied by user.
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 $G = g$. We define the treatment effect, $\Delta(t)$, as the difference in survival rates by time $t$ under treatment versus control,

$\Delta(t)=E\{ I(T^{(1)}>t)\} - E\{I(T^{(0)}>t)\} = P(T^{(1)}>t) - P(T^{(0)}>t)$

where $t>t_0$

Due to censoring, our data consist of $n_1$ observations $\{(X_{1i}, \delta_{1i}), i=1,...,n_1\}$ from the treatment group $G=1$ and $n_0$ observations $\{(X_{0i}, \delta_{0i}), i=1,...,n_0\}$ from the control group $G=0$ where $X_{gi} = \min(T_{gi}, C_{ gi})$, $\delta_{gi} = I(T_{gi} < C_{gi})$, and $C_{gi}$ denotes the censoring time for $g= 1,0$, for individual $i$. Throughout, we estimate the treatment effect $\Delta(t)$ as

$\hat{\Delta}(t) = n_1^{-1} \sum_{i=1}^{n_1} \frac{I(X_{1i}>t)}{\hat{W}^C_1(t)} - n_0^{-1} \sum_{i=1}^{n_0} \frac{I(X_{0i}>t)}{\hat{W}^C_0(t)}$

where $\hat{W}^C_g(\cdot)$ is the Kaplan-Meier estimator of survival for censoring for $g=1,0.$

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 $\{\hat{\Delta}(t)^{(b)}, b = 1,...,D\}.$ We construct two versions of the $95\%$confidence interval for $\hat{\Delta}(t)$: one based on a normal approximation confidence interval using the estimated variance and another taking the 2.5th and 97.5th empirical percentiles of $\hat{\Delta}(t)^{(b)}$.

Value

A list is returned:

delta	the estimate, $\hat{\Delta}(t)$, described above.
var	the variance estimate of $\hat{\Delta}(t)$; if var = TRUE or conf.int = TRUE.
conf.int.normal	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	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.

Author(s)

Layla Parast

Examples

data(d_example_surv)
names(d_example_surv)
delta.surv.estimate(xone = d_example_surv$x1, xzero = d_example_surv$x0,  
deltaone = d_example_surv$delta1, deltazero = d_example_surv$delta0, t = 3)

---

Calculates robust residual treatment effect accounting only for primary outcome information up to a specified time

Description

This function calculates the robust estimate of the residual treatment effect accounting only for primary outcome information up to $t_0$ i.e. the hypothetical treatment effect if survival up to $t_0$ in the treatment group looks like survival up to $t_0$ in the control group. Ideally this function is only used as a helper function and is not directly called.

Usage

delta.t.surv.estimate(xone, xzero, deltaone, deltazero, t, weight.perturb = NULL,
landmark, 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.
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

Details are included in the documentation for R.t.surv.estimate.

Value

$\hat{\Delta}_T(t,t_0)$, the robust residual treatment effect estimate accounting only for survival up to $t_0$.

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

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)

---

Constructs Fieller's confidence interval.

Description

Constructs Fieller's confidence interval.

Usage

fieller.ci(perturb.delta.s, perturb.delta, delta.s, delta)

Arguments

perturb.delta.s	numeric vector; the perturbed values for $\hat{\Delta}_S$, the residual treatment effect estimate, used in variance estimation and confidence interval construction.
perturb.delta	numeric vector; the perturbed values for $\hat{\Delta}$, the treatment effect estimate, used in variance estimation and confidence interval construction.
delta.s	the residual treatment effect, $\Delta_S$, estimate, $\hat{\Delta}_S$.
delta	the treatment effect, $\Delta$, estimate, $\hat{\Delta}$.

Details

See documention for R.s.estimate for more detail.

Value

Returns a vector of length 2, lower bound of the 95% confidence interval and upper bound of the 95% confidence interval.

Author(s)

Layla Parast

References

Fieller, Edgar C. (1954). Some problems in interval estimation. Journal of the Royal Statistical Society. Series B (Methodological), 175-185.

Fieller, E. C. (1940). The biological standardization of insulin. Supplement to the Journal of the Royal Statistical Society, 1-64.

Parast, L., McDermott, M., Tian, L. (2016). Robust estimation of the proportion of treatment effect explained by surrogate marker information. Statistics in Medicine, 35(10):1637-1653.

---

Estimates measurement error variance given replicate data.

Description

Estimates measurement error variance given replicate data using a simple components of variance analysis.

Usage

me.variance.estimate(replicates)

Arguments

replicates

matrix of data where each row indicates a subject and each column is a replicated measurement; columns can have NAs when subjects have different numbers of measurements.

Details

Estimates measurement error variance given replicate data using a simple components of variance analysis.

Value

estimate of measurement error variance

Author(s)

Layla Parast

References

Carroll, R. J., Ruppert, D., Crainiceanu, C. M., and Stefanski, L. A. (2006). Measurement error in nonlinear models: a modern perspective. Chapman and Hall/CRC.

Parast, L., Garcia, T. P., Prentice, R. L., & Carroll, R. J. (2022). Robust methods to correct for measurement error when evaluating a surrogate marker. Biometrics, 78(1), 9-23.

---

Calculates the proportion of treatment effect explained by multiple surrogate markers 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 multiple surrogate markers 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 the surrogate marker information.

Usage

R.multiple.surv(xone, xzero, deltaone, deltazero, sone, szero, type =1, 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	matrix of numeric values; surrogate marker measurements at $t_0$ for treated observations. If $X_{1i}<t_0$, then the surrogate marker measurements should be NA.
szero	matrix of numeric values; surrogate marker measurements at $t_0$ for control observations. If $X_{0i}<t_0$, then the surrogate marker measurements should be NA.
type	type of estimate; options are 1 = two-stage robust estimator, 2 = weighted two-stage robust estimator, 3 = double-robust estimator, 4 = two-stage model-based estimator, 5 = weighted estimator, 6 = double-robust model-based estimator; default is 1.
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 pseudo-score.
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 $G \in \{A, B\}$ be the binary treatment indicator and we assume that subjects are randomly assigned to either treatment group $A$ or $B$ at baseline. Let $T$ denote the time to the occurrence of the primary outcome, death for example, and $S = (S_1,S_2,...,S_k)^{T}$ denote the vector of $k$ surrogate markers measured at a given time $t_0$. Let $T^{(g)}$ and $S^{(g)}$ denote the counterfactual event time and surrogate marker measurements under treatment $G = g$ for $g \in \{A, B\}$. In practice, we only observe $(T, S)=(T^{(A)}, S^{(A)})$ or $(T^{(B)}, S^{(B)})$ depending on whether $G=A$ or $B.$ The treatment effect, $\Delta(t)$, is the treatment difference in survival rates at time $t > t_0$, $\Delta(t)=E\{ I(T^{(A)}>t)\} - E\{I(T^{(B)}>t)\} = P(T^{(A)}>t) - P(T^{(B)}>t)$ where $I(\cdot)$ is the indicator function. For individuals who are censored or experience the primary outcome before $t_0$, we assume that their $S$ information is not available.

The surrogate marker information at time $t_0$ is defined as a combination of the observed information on $I(T >t_0)$ and the observed $S$ at $t_0$, denoted by $Q_{t_0} = \{Q_{t_0}^{(g)}, g = A, B\}$, where $Q_{t_0} ^{(g)} = \{ T ^{(g)} \wedge t_0, S ^{(g)} I(T ^{(g)} > t_0)\}$. With information on $Q_{t_0}$, the residual treatment effect is defined as: $\Delta_{S}(t,t_0) = E\{ I(T ^{(A)} > t) - I(T ^{(B)}>t) \mid Q_{t_0}^{(A)} = Q_{t_0}^{(B)} \} =P(T^{(B)} > t_0) \int_{S} \psi_A(t \mid S, t_0) dF_{B}(S \mid t_0) -P(T^{(B)}> t)$ where $S = (s_1, ..., s_k)^{T}$, $F_{g}(S \mid t_0) = P(S^{(g)}\le S \mid T^{(g)}>t_0),$ $\psi_g(t\mid S,t_0) = P(T^{(g)}> t\mid S^{(g)}=S, T^{(g)}> t_0).$ The proportion of the treatment effect on the primary outcome that is explained by the treatment effect on $Q_{t_0}$ is $R_{S}(t,t_0)=1-\Delta_{S}(t,t_0)/\Delta$. This function provides 6 different estimators for $R_{S}(t,t_0)$ using censored data.

Due to censoring, the observed data consist of $n$ observations $\{(G_i, X_{i}, \delta_{i}, S_{i}I(X_{i} > t_0)), i=1,...,n\}$ from the two treatment groups, where $X_{i} = \min(T_{i}, C_{i})$, $\delta_{i} = I(T_{i} < C_{i})$, and $C_{i}$ denotes the censoring time for the $i$th subject. We assume independent censoring i.e., $(T_i, S_i) \perp C_i \mid G_i$. For ease of notation, we also let $\{(X_{gi}, \delta_{gi}, S_{gi}I(X_{gi} > t_0)), i=1,...,n_g\}$ denote $n_g = \sum_{i=1}^n I(G_i = g)$ observations from treatment group $g \in \{A,B\},$ where $X_{gi}=\min(T_{gi}, C_{gi})$ and $\delta_{gi}=I(T_{gi}<C_{gi}).$ Without loss of generality, we assume that $\bar{\pi}_g = n_g/n \to \pi_g \in (0,1)$ as $n\to \infty$. Throughout, we estimate the treatment effect $\Delta(t) =P(T^{(A)}>t) - P(T^{(B)}>t)$ as $\hat{\Delta}(t) = n_A^{-1} \sum_{i=1}^{n_A} \frac{I(X_{Ai}>t)}{\hat{W}^C_A(t)} - n_B^{-1} \sum_{i=1}^{n_B} \frac{I(X_{Bi}>t)}{\hat{W}^C_B(t)}$ where $\hat{W}^C_g(t)$ is the Kaplan-Meier estimator of $W^C_g(t) = P(C_{g} > t)$.

We first describe the two-stage robust estimator which involves a two-stage procedure combining the use of a working model and a nonparametric estimation procedure for $\Delta_{S}(t,t_0)$. The idea is simply to summarize $S$ into a univariate score $U$ and then construct a nonparametric estimator for $R_S(t,t_0)$ treating $U$ as $S$. To construct $U$, we approximate the conditional distribution of $T^{(A)} \mid S^{(A)}, T^{(A)} > t_0$ by using a working semiparametric model such as the landmark proportional hazards model $q_A(S) = P(T^{(A)} >t \mid T^{(A)} >t_0, S^{(A)}) = \exp\{-\Lambda_0(t|t_0) \exp(\beta ^{T} S^{(A)}) \}, t>t_0,$ where $\Lambda_0(t|t_0)$ is the unspecified baseline cumulative hazard function for $T^{(A)}$ conditional on $\{T^{(A)} > t_0\}$ and $\beta$ is an unknown vector of coefficients. Let $\hat{\beta}$ be the maximizer of the corresponding log partial likelihood function and $\hat{\Lambda}_0(t|t_0)$ be the Breslow-type estimate of baseline hazard. If one were to assume that this working model is correctly specified, then a consistent estimate of $\Delta_{S}(t,t_0)$ would simply be: $\hat{\Delta}_{S}^M=n_B^{-1} \sum_{i=1}^{n_B} [ \exp \{ -\hat{\Lambda}_0(t|t_0)\exp(\hat{\beta} ^{T} S_{Bi}) \} \frac{I(X_{Bi} > t_0)}{\hat{W}^C_B(t_0)} - \frac{I(X_{Bi} > t)}{\hat{W}^C_B(t)}].$ We refer to this estimate as the two-stage model-based estimator (option 4 for type). Instead of relying on correct specification of this model, we use the resulting score $U = \beta_0^{T}S$ as a univariate “pseudo-marker" to summarize the $k$ surrogates. In the second stage, to estimate $\Delta_{S}(t,t_0)$, we apply a nonparametric approach with $S$ represented by the univariate marker $U$. Specifically, we use a kernel Nelson-Aalen estimator to nonparametrically estimate $\phi_A(t|u, t_0)=P(T^{(A)}> t\mid U^{(A)}=u, T^{(A)}> t_0) = \exp\{-\Lambda_A(t|u, t_0 )\}$ as $\hat \phi_A(t|u, t_0) = \exp\{-\hat{\Lambda}_A(t|u, t_0) \}$, where $\hat{\Lambda}_A(t|u, t_0) = \int_{t_0}^t \frac{\sum_{i=1}^{n_A} I(X_{Ai}>t_0) K_h\{\gamma(\hat{U}_{Ai}) - \gamma(u)\}dN_{Ai}(z)}{\sum_{i=1}^{n_A} I(X_{Ai}>t_0) K_h\{\gamma(\hat{U}_{Ai}) - \gamma(u)\} Y_{Ai}(z)},$ $\hat{U}_{Ai} = \hat{\beta} ^{T} S_{Ai}$, $\hat{U}_{Bi} = \hat{\beta} ^{T} S_{Bi}$, $Y_{Ai} = I(X_{Ai} \geq t)$, $N_{Ai}(t) = I(X_{Ai} \leq t) \delta_{Ai}, K(\cdot)$ is a smooth symmetric density function, $K_h(x) = K(x/h)/h,$ and $\gamma(\cdot)$ is a given monotone transformation function. We then estimate $\Delta_{S}(t,t_0)$ as $\hat{\Delta}_{S}(t,t_0) = n_B^{-1} \sum_{i=1}^{n_B} [\hat{\phi}_A(t|\hat{U}_{Bi},t_0) \frac{I(X_{Bi} > t_0)}{\hat{W}^C_B(t_0)} - \frac{I(X_{Bi} > t)}{\hat{W}^C_B(t)}]$ and $\hat{R}_{S}(t,t_0) =1- \hat{\Delta}_{S}(t,t_0)/\hat{\Delta}(t).$ We refer to this estimate as the two-stage robust estimator (option 1 for type).

The next estimator borrows ideas from the extensive causal inference literature focusing on double robust estimators two-stage weighted estimator with a propensity score weight explicitly balancing the two treatment groups with respect to the distribution of $S$. The weighting enables us to “adjust" the distribution of $S^{(A)}$ before constructing the conditional survival estimate $\hat \phi_A(t|u, t_0).$ This approach results in a double-robust estimator of $\Delta_{S}(t, t_0)$, which is consistent when either $U^{(A)}$ captures all the information about the relationship between $I(T^{(A)}\ge t)$ and $S^{(A)}$ or the propensity score model for $\pi(S,t_0)=P(G_i=B|S_i=S, T_{i} > t_0)$ is correctly specified. While $\pi(S,t_0)$ depends on $t_0$, for simplicity, we drop $t_0$ from our notation and simply use $\pi(S)$.

Regression models can be imposed to obtain estimates for $\pi(S)$. For example, a simple logistic regression model can be imposed for $\tilde{\pi}(S)=P(G_i=B|S_i=S, X_{i}>t_0)$ with $logit\{\tilde{\pi}(S)\} = \alpha_0 + \alpha_1 ^{T} S,$ where $\alpha_0$ and $\alpha_1$ are estimated only among those with $X_{gi} > t_0$ to account for censoring. The propensity score of interest, $\pi(S),$ can be derived from $\tilde{\pi}(S)$ directly since $logit\{\pi(S)\}=logit\{\tilde{\pi}(S)\} + \log\{W_A^C(t_0)/W_B^C(t_0)\},$ which follows from the assumption that $(T_{gi}, S_{gi}) \perp C_{gi}.$ We then modify the above expression by weighting observations with the estimated $L(S_{Ai})=\pi(S_{Ai})/\{1-\pi(S_{Ai})\}$ and obtain

$\hat{\Lambda}^w_A(t|u, t_0) = \int_{t_0}^t \frac{\sum_{i=1}^{n_A} \hat{L}(S_{Ai}) I(X_{Ai}>t_0) K_h\{\gamma(\hat{U}_{Ai}) - \gamma(u)\}dN_{Ai}(z)}{\sum_{i=1}^{n_A} \hat{L}(S_{Ai}) I(X_{Ai}>t_0) K_h\{\gamma(\hat{U}_{Ai}) - \gamma(u)\} Y_{Ai}(z)},$

, where $\hat{L}(S_{gi}) = \exp(\hat{\alpha}_0+\hat{\alpha}_1^{T} S_{gi})\hat{W}^C_B(t_0)/\hat{W}^C_A(t_0).$

Subsequently, we define $\hat{\Delta}^w_{S}(t,t_0) = n_B^{-1} \sum_{i=1}^{n_B} [\hat{\phi}^w_A(t|\hat{U}_{Bi},t_0) \frac{I(X_{Bi} > t_0)}{\hat{W}^C_B(t_0)} - \frac{I(X_{Bi} > t)}{\hat{W}^C_B(t)}]$ and $\hat{R}^w_{S}(t,t_0) =1- \hat{\Delta}^w_{S}(t,t_0)/\hat{\Delta}(t)$ where $\hat \phi^w_A(t|t_0,u) = \exp\{-\hat{\Lambda}^w_A(t|t_0,u) \}.$ We refer to this estimate as the weighted two-stage robust estimator (option 2 for type).

While the two-stage weighted estimator reflects one way to enhance the robustness of an initial estimator, the idea of combining a propensity-score type model and a regression-type model has certainly been extensively studied in the causal inference literature and a more familiar double-robust estimator can be constructed as: $\hat{\Delta}_{S}^{DR}(t,t_0) = n^{-1} [\sum_{i=1}^{n_A}\frac{I(X_{Ai}>t)}{\hat{W}_{A}^C(t)\bar{\pi}_B} \hat{L}(S_{Ai}) - \sum_{i=1}^{n_B} \frac{I(X_{Bi}>t)}{\hat{W}_{B}^C(t)\bar{\pi}_B} ] - n^{-1} [ \sum_{i=1}^{n_A} \frac{ \hat{\phi}_A(t\mid\hat{U}_{Ai},t_0) I(X_{Ai} > t_0) }{\hat{W}_{A}^C(t_0)\bar{\pi}_B} \hat{L}(S_{Ai}) - \sum_{i=1}^{n_B}\frac{ \hat{\phi}_A(t\mid \hat{U}_{Bi},t_0) I(X_{Bi} > t_0) }{\hat{W}_{B}^C(t_0)\bar{\pi}_B} ]$ and $\hat{R}_{S}^{DR}(t,t_0) =1- \hat{\Delta}_{S}^{DR}(t,t_0)/\hat{\Delta}(t)$, where $\hat{\phi}_A(t|\hat{U}_{gi},t_0)$ is the (unweighted) estimate of $\phi_A(t\mid u,t_0)$ used in $\hat{\Delta}_{S}(t,t_0)$. We refer to this estimate as the double robust estimator (option 3 for type).

The weighted estimator (option type 5) is defined as: $\hat{\Delta}^{PS}_{S}(t,t_0) = n^{-1} \sum_{i=1}^n \{\frac{I(X_{i}>t)}{\hat{W}_{G_i}^C(t)\bar{\pi}_B} [ I(G_i = A)\hat{L}(S_{Ai}) - I(G_i = B ) ] \}$ and $\hat{R}^{PS}_{S}(t,t_0) =1- \hat{\Delta}^{PS}_{S}(t,t_0)/\hat{\Delta}(t).$ This estimator completely relies on the correct specification of $\pi(S)$. The double-robust model-based estimator (option 6 for type) is defined as $\hat{\Delta}_{S}^{DR2}(t,t_0)$ and $\hat{R}_{S}^{DR2}(t,t_0) =1- \hat{\Delta}_{S}^{DR2}(t,t_0)/\hat{\Delta}(t)$ which are constructed parallel to the construction of $\hat{\Delta}_{S}^{DR}(t,t_0)$ i.e., a combination of $\hat{\Delta}_{S}^M(t,t_0)$ and $\hat{R}^{PS}_{S}(t,t_0)$.

Variance estimates are obtained using perturbation resampling. 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. An estimate of the incremental value of the surrogate marker information can also be requested; this essentially compared the proportion explained by the surrogate information vs. the proportion explained by $T$ alone up to $t_0$. Details can be found in Parast, L., Cai, T., & Tian, L. (2021). Evaluating multiple surrogate markers with censored data. Biometrics, 77(4), 1315-1327.

Value

A list is returned:

delta	the estimate, $\hat{\Delta}(t)$, described in delta.estimate documentation.
delta.s	the residual treatment effect estimate, $\hat{\Delta}_S(t,t_0)$.
R.s	the estimated proportion of treatment effect explained by the set of markers, $\hat{R}_S(t,t_0)$.
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; 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; 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; 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; if conf.int = TRUE.
delta.t	the estimate, $\hat{\Delta}_T(t,t_0)$; if incremental.vaue = TRUE.
R.t	the estimated proportion of treatment effect explained by survival only, $\hat{R}_T(t,t_0)$; if incremental.vaue = TRUE.
incremental.value	the estimate of the incremental value of the surrogate markers, $\hat{IV}_S(t,t_0)$; 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; 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; 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; 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; 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".

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.

Parast, L., Cai, T., & Tian, L. (2021). Evaluating multiple surrogate markers with censored data. Biometrics, 77(4), 1315-1327.

Examples

data(d_example_multiple)
names(d_example_multiple)
## Not run: 
R.multiple.surv(xone = d_example_multiple$x1, xzero = d_example_multiple$x0, deltaone = 
d_example_multiple$delta1, deltazero = d_example_multiple$delta0, sone = 
as.matrix(d_example_multiple$s1), szero = as.matrix(d_example_multiple$s0), 
type =1, t = 1, landmark=0.5)
R.multiple.surv(xone = d_example_multiple$x1, xzero = d_example_multiple$x0, deltaone = 
d_example_multiple$delta1, deltazero = d_example_multiple$delta0, sone = 
as.matrix(d_example_multiple$s1), szero = as.matrix(d_example_multiple$s0), 
type =1, t = 1, landmark=0.5, conf.int = T)	
R.multiple.surv(xone = d_example_multiple$x1, xzero = d_example_multiple$x0, deltaone = 
d_example_multiple$delta1, deltazero = d_example_multiple$delta0, sone = 
as.matrix(d_example_multiple$s1), szero = as.matrix(d_example_multiple$s0), 
type =3, t = 1, landmark=0.5)

## End(Not run)

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