R.multiple.surv {Rsurrogate} | R Documentation |
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 and primary outcome information up to
. 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 |
szero |
matrix of numeric values; surrogate marker measurements at |
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 |
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 be the binary treatment indicator and we assume that subjects are randomly assigned to either treatment group
or
at baseline. Let
denote the time to the occurrence of the primary outcome, death for example, and
denote the vector of
surrogate markers measured at a given time
. Let
and
denote the counterfactual event time and surrogate marker measurements under treatment
for
. In practice, we only observe
or
depending on whether
or
The treatment effect,
, is the treatment difference in survival rates at time
,
where
is the indicator function. For individuals who are censored or experience the primary outcome before
, we assume that their
information is not available.
The surrogate marker information at time is defined as a combination of the observed information on
and the observed
at
, denoted by
, where
. With information on
, the residual treatment effect is defined as:
where
,
The proportion of the treatment effect on the primary outcome that is explained by the treatment effect on
is
. This function provides 6 different estimators for
using censored data.
Due to censoring, the observed data consist of observations
from the two treatment groups, where
,
, and
denotes the censoring time for the
th subject. We assume independent censoring i.e.,
. For ease of notation, we also let
denote
observations from treatment group
where
and
Without loss of generality, we assume that
as
. Throughout, we estimate the treatment effect
as
where
is the Kaplan-Meier estimator of
.
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 . The idea is simply to summarize
into a univariate score
and then construct a nonparametric estimator for
treating
as
. To construct
, we approximate the conditional distribution of
by using a working semiparametric model such as the landmark proportional hazards model
where
is the unspecified baseline cumulative hazard function for
conditional on
and
is an unknown vector of coefficients. Let
be the maximizer of the corresponding log partial likelihood function and
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
would simply be:
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
as a univariate “pseudo-marker" to summarize the
surrogates. In the second stage, to estimate
, we apply a nonparametric approach with
represented by the univariate marker
. Specifically, we use a kernel Nelson-Aalen estimator to nonparametrically estimate
as
, where
,
,
,
is a smooth symmetric density function,
and
is a given monotone transformation function. We then estimate
as
and
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 . The weighting enables us to “adjust" the distribution of
before constructing the conditional survival estimate
This approach results in a double-robust estimator of
, which is consistent when either
captures all the information about the relationship between
and
or the propensity score model for
is correctly specified. While
depends on
, for simplicity, we drop
from our notation and simply use
.
Regression models can be imposed to obtain estimates for . For example, a simple logistic regression model can be imposed for
with
where
and
are estimated only among those with
to account for censoring. The propensity score of interest,
can be derived from
directly since
which follows from the assumption that
We then modify the above expression by weighting observations with the estimated
and obtain
,
where
Subsequently, we define
and
where
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:
and
, where
is the (unweighted) estimate of
used in
. We refer to this estimate as the double robust estimator (option 3 for type).
The weighted estimator (option type 5) is defined as:
and
This estimator completely relies on the correct specification of
. The double-robust model-based estimator (option 6 for type) is defined as
and
which are constructed parallel to the construction of
i.e., a combination of
and
.
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 alone up to
. 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, |
delta.s |
the residual treatment effect estimate, |
R.s |
the estimated proportion of treatment effect explained by the set of markers, |
delta.var |
the variance estimate of |
delta.s.var |
the variance estimate of |
R.s.var |
the variance estimate of |
conf.int.normal.delta |
a vector of size 2; the 95% confidence interval for |
conf.int.quantile.delta |
a vector of size 2; the 95% confidence interval for |
conf.int.normal.delta.s |
a vector of size 2; the 95% confidence interval for |
conf.int.quantile.delta.s |
a vector of size 2; the 95% confidence interval for |
conf.int.normal.R.s |
a vector of size 2; the 95% confidence interval for |
conf.int.quantile.R.s |
a vector of size 2; the 95% confidence interval for |
conf.int.fieller.R.s |
a vector of size 2; the 95% confidence interval for |
delta.t |
the estimate, |
R.t |
the estimated proportion of treatment effect explained by survival only, |
incremental.value |
the estimate of the incremental value of the surrogate markers, |
delta.t.var |
the variance estimate of |
R.t.var |
the variance estimate of |
incremental.value.var |
the variance estimate of |
conf.int.normal.delta.t |
a vector of size 2; the 95% confidence interval for |
conf.int.quantile.delta.t |
a vector of size 2; the 95% confidence interval for |
conf.int.normal.R.t |
a vector of size 2; the 95% confidence interval for |
conf.int.quantile.R.t |
a vector of size 2; the 95% confidence interval for |
conf.int.fieller.R.t |
a vector of size 2; the 95% confidence interval for |
conf.int.normal.iv |
a vector of size 2; the 95% confidence interval for |
conf.int.quantile.iv |
a vector of size 2; the 95% confidence interval for |
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)