Vignette for the ltrmst2adapt function in
the survRM2adapt package
Miki Horiguchi, Hajime Uno
March 23, 2023
1 Introduction
Cancer immunotherapy is an important treatment option for cancer
patients, and clinical trials that evaluate immunotherapy will
increasingly be conducted toward optimizing cancer management. Unlike
cytotoxic agents, immunotherapies do not attack the cancer cell directly
but utilize the immune system to do so. This unique approach leads to
the possibility of delayed treatment effects, but when successful can
deliver long-lasting benefits for treated patients.
A 2020 study\(^{1}\) showed that, in
almost all cancer phase III randomized controlled trials, the log-rank
test and Cox’s hazard ratio (HR) are used for testing the equality of
the survival functions and estimating the magnitude of the treatment
effect, respectively. However, the log-rank test may not be powerful
enough to capture the delayed treatment effect we often see in
immunotherapy studies. In addition, the basis of the traditional Cox’s
HR method is the proportional hazards (PH) assumption which assumes that
the hazard functions from two groups are constant over the study time.
It has been shown that the PH assumption does not hold in most
immunotherapy studies. Therefore, the conventional analysis methods
using the log-rank test and HR estimation may be suboptimal, and
development of alternative approaches are critical for the successful
development of immunotherapy.
The long-term restricted mean survival time (LT-RMST) and the
adaptive (or flexible) version of the LT-RMST have been proposed for
possible application to immunotherapy studies.\(^{2-4}\) In this vignette, we briefly
review these methods and illustrate how to implement them using the
ltrmst2adapt function in the survRM2adapt package. The
package was made and tested on R version 4.2.3.
2 Installation
Open the R or RStudio applications. Then, copy and paste either of
the following scripts to the command line.
To install the package from the CRAN:
install.packages("survRM2adapt")
To install the development version:
install.packages("devtools") #-- if the devtools package has not been installed
devtools::install_github("uno1lab/survRM2adapt")
2 Sample Reconstructed Data
Throughout this vignette, we use sample data which we reconstructed
from the CheckMate214 study reported by Motzer et al.\(^{5}\) The data consists of 847 patients
with previously untreated clear-cell advanced renal-cell carcinoma; 425
for the nivolumab plus ipilimumab group (treatment) and 422 for the
sunitinib group (control).
The sample reconstructed data from the CheckMate214 study is
available on survRM2adapt package as cm214_pfs. To
load the data, copy and paste the following scripts to the command
line.
library(survRM2adapt)
nrow(cm214_pfs)
#> [1] 847
cm214_pfs[1:10,]
#> time status arm
#> 1 0.451 1 1
#> 2 0.451 1 1
#> 3 0.451 1 1
#> 4 0.451 1 1
#> 5 0.451 1 1
#> 6 0.710 1 1
#> 7 0.710 1 1
#> 8 0.710 1 1
#> 9 0.919 1 1
#> 10 0.919 1 1
Here, time is months from the registration to
progression-free survival (PFS), status is the
indicator of the event (1: event, 0: censor), and arm
is the treatment assignment indicator (1: Treatment group, 0: Control
group).
Below are the Kaplan-Meier estimates for the PFS for each treatment
group.
3 Long-Term Restricted Mean Survival Time (LT-RMST)
Let \(S\) be the survival function
for the event time. The LT-RMST is defined as \[R(\eta, \tau) = \int_\eta^\tau S(s)ds,\]
where \([\eta, \tau]\) is a fixed range
within the study time \((0<\eta<\tau)\). \(R(\eta, \tau)\) can be estimated
nonparametrically using the Kaplan-Meier estimator \(\hat{S}\). It can be interpretated as
average survival time during the time range \([\eta, \tau]\) (i.e., the time treatment
effect appears until the specific time \(\tau\)) and provide a clinically
interpretable treatment effect associated with a long-lasting survival
benefit for treated patients. Standardized by the width of the time
window \([\eta, \tau]\), \[\frac{R(\eta, \tau)}{\tau-\eta}\] can be
interpreted as an average survival probability on \([\eta, \tau]\) or average proportion of the
time range spent alive.
For between-group comparison, we consider \[D(\eta, \tau) = R_{1}(\eta, \tau)-R_{0}(\eta,
\tau)\] as a summary measure of the magnitude of the treatment
effect. The asymptotic properties of \(\hat{D}(\eta, \tau)\) can be seen in Zhao
et al.\(^{2}\)
4 Adaptive Long-Term Restricted Mean Survival Time (adaptive
LT-RMST)
A challenge of the LT-RMST estimation would be specifying \(\eta\) for the approximate time point where
the treatment effect will begin to appear. We have proposed the adaptive
LT-RMST-based test/estimation methods to get around this
limitation.\(^{4}\) This method is a
modified version of the LT-RMST test/estimation methods with improved
flexibility in choosing \(\eta\).
Specifically, instead of choosing one specific value of \(\eta\), we specify a time range or a set of
time values where \(\eta\) value can be
included. For example, if we expect the treatment effect to appear
sometime between 6 to 12 months, a time range \(B = \{b|b \in [6,12]\}\) (months) can be
prespecified. The procedure selects one time value as \(\eta\) among \(B\) that can detect the treatment effect
with the most statistical significance, which is not always the time
point of separation of the two survival curves.
For testing the null hypothesis using the adaptive LT-RMST (i.e.,
\(H_{0}: D(\eta, \tau)=0\) for all
\(\eta \in B\)), the following test
statistic is used:
\[\sup_{b \in B} \left| D(b,\tau) /
\sqrt{V(W_n(b,\tau))/n}\right|.\] The procedure derives the
reference distribution of the test statistic using a zero-mean Gaussian
process with the covariance function of the process \(W_n(\eta,\tau) \in B\) to obtain a p-value
and a corresponding \((1-\alpha)\)
simultaneous confidence bands for \(D(b,\tau)\). The detailed procedure is
given in the full paper.\(^{4}\)
During the procedure, the largest value of \(\eta\), say, \(\eta^{*}\), that satisfies \[\left|\hat{D}(\eta^{*},
\tau)/\sqrt{\hat{V}(W_n(\eta^{*},\tau))/n}\right|=\sup_{b \in
B}\left|\hat{D}(b, \tau)/\sqrt{\hat{V}(W_n(b,\tau))/n}\right|\]
is chosen. A \((1-\alpha)\) confidence
interval (CI) for \(D(\eta^{*},\tau)\)
is then given by \[\hat{D}(\eta^{*}, \tau)
\pm c_{\alpha}\sqrt{\hat{V}(W_n(\eta^{*},\tau))/n},\] where \(c_{\alpha}\) is derived using the reference
distribution.
5 Implementation of Long-Term RMST Test/Estimation Methods
The procedure below shows how to use the function,
ltrmst2adapt, to implement the LT-RMST method with a
pre-specified \(\eta\).
ltrmst2adapt(indata=cm214_pfs, tau1=3, tau2=10, seed=123)
#>
#> RMST over the time window [3,10]
#>
#> RMST(arm1) RMST(arm0)
#> 4.415 4.191
#>
#> RMST(arm1-arm0) lower 0.95 upper 0.95 p_value
#> 0.224 -0.209 0.657 0.312
The first argument (indata) is the data frame you
want to analyze. The 1st column is the time-to-event variable, the 2nd
column is the event indicator (1=event, 0=censor), and the 3rd column is
the treatment indicator (1=treatment, 0=control). No missing values are
allowed in this data matrix. The second argument (tau1)
is an integer value indicating the lower end of the time window (\(\eta\)). The third argument
(tau2) is an integer value indicating the upper end of
the time window (\(\tau\)). The fourth
argument (seed) is an integer value for random number
generation in the resampling procedures.
When a fixed value is specified as tau1, the
ltrmst2adapt function returns RMST over the time window
[\(\tau_{1}\), \(\tau_{2}\)] on each group and the results
of the between-group contrast measure (difference in the LT-RMST between
the two groups). The estimated LT-RMSTs were 4.415 months and 4.191
months for the treatment group and the control group, respectively. The
difference between the LT-RMSTs was 0.224 months (95% CI: -0.209 to
0.657, p-value=0.312).
6 Implementation of adaptive Long-Term RMST Test/Estimation
Methods
Instead of specifying a fixed value of \(\eta\), the procedure below allows us to
specify a set of values for \(\eta\),
say, 0, 1, 2, and 3.
ltrmst2adapt(indata=cm214_pfs, tau1=c(0,1,2,3), tau2=10, seed=123)
#>
#> RMST over the time window [3,10]
#>
#> RMST(arm1) RMST(arm0)
#> 4.415 4.191
#>
#> RMST(arm1-arm0) lower 0.95 upper 0.95 p_value
#> 0.224 -0.218 0.666 0.332
Here, the second argument (tau1) is a vector
indicating the set of values for the lower end of the time window (\(\eta\)). The first, third, and fourth
arguments are the same as specified in Section 5.
The ltrmst2adapt function returns RMST over the time
window [\(\tau_{1}^{*}\), \(\tau_{2}\)] on each group and the results
of the between-group contrast measure (difference in the LT-RMSTs),
where \(\tau_{1}^{*}\) is the
adaptively selected value among the specified values as
tau1. With this example, 3 was selected as \(\tau_{1}^{*}\) among the candidates (0, 1,
2, and 3). The estimated LT-RMSTs for both groups and the difference
between them were the same as the ones in the LT-RMSTs with fixed \(\eta\) in Section 5. The 95% CI for the
difference was slightly wider (-0.218 to 0.666) than the LT-RMST with
fixed \(\eta\), and the p-value was
also slightly larger (0.332). These results suggest that the efficiency
loss by adopting the flexibility in the adaptive LT-RMST method is not
remarkable, compared to the LT-RMST with fixed \(\eta\).
7 Conclusions
The function, ltrmst2adapt(), in the survRM2adapt
package can easily implement the LT-RMST-based statistical comparison
with a pre-specified \(\eta\) and also
with a data-dependent choice of \(\eta\), which we have called the adaptive
LT-RMST approach in this vignette.
The adaptive LT-RMST approach does not require users to pre-specify a
single number for \(\eta\) but rather a
set of candidates for \(\eta\), making
the adaptive LT-RMST approach more flexible than the LT-RMST approach
with a fixed \(\eta\).
Both approaches provide estimates of the magnitude of the treatment
effect that do not depend on the study-specific censoring distribution.
They also offer greater power than the traditional Cox regression
approach to detect the treatment effect when the delayed survival
benefit is expected, and thus will be particularly useful for clinical
trials evaluating the efficacy and safety of cancer immunotherapies.
References
Uno H, Horiguchi M, Hassett MJ (2020). Statistical
test/estimation methods used in contemporary phase III cancer randomized
controlled trials with time-to-event outcomes. Oncologist,
25(2):91-93. https://doi.org/10.1634/theoncologist.2019-0464
Zhao L, Tian L, Uno H, et al (2012). Utilizing the integrated
difference of two survival functions to quantify the treatment contrast
for designing, monitoring, and analyzing a comparative clinical study.
Clinical Trials, 9(5):570-7. https://doi.org/10.1177/1740774512455464
Horiguchi M, Tian L, Uno H et al (2018). Quantification of
long-term survival benefit in a comparative oncology clinical study.
JAMA Oncology, 4(6):881-882. https://doi.org/10.1001/jamaoncol.2018.0518
Horiguchi M, Tian L, Uno H. On assessing survival benefit of
immunotherapy using long-term restricted mean survival time.
Statistics in Medicine, 42(8):1139-1155. https://doi.org/10.1002/sim.9662
Motzer RJ, Tannir NM, McDermott DF, et al. (2018). Nivolumab plus
ipilimumab versus sunitinib in advanced renal-cell carcinoma. New
England Journal of Medicine, 378(14):1277-1290. https://doi.org/10.1056/NEJMoa1712126