title

Summary

The restricted mean survival time (RMST) has recently been widely used due to its straightforward interpretation as a mean measure. Until now, there has not been a package that derives the distribution of RMST using parametric Bayesian methods. The rmstBayepara package enables parametric Bayesian analysis considering covariates and heterogeneity. It provides posterior samples of RMST, shrinkage-adjusted RMST for each cluster, and RMST under specific covariate conditions. The summarized results include the mean, median, mode, and 95% credible interval for the mean. The analysis considering heterogeneity includes methods that add random effects to the linear combination of covariates and apply frailty to the hazard. The rmstBayepara package can analyze exponential, Weibull, log-normal, and log-logistic models. It also implements model selection using the Widely Applicable Information Criterion (WAIC).

Statement of need

The restricted mean survival time (RMST) has recently been widely used due to its straightforward interpretation as a mean measure [@uno2014moving;@kim2017restricted]. The RMST is obtained by integrating the survival function up to a specified time limit. In non-parametric analysis methods, such as the Kaplan-Meier method, the survival function is estimated, and the RMST can be obtained by calculating the sum of the areas of each segment. In frequentist statistics, methods have been proposed for analyzing the influence of covariates and the heterogeneity between clusters[@andersen2003generalised;@tian2014predicting]. In Bayesian methods, non-parametric approaches for calculating RMST have been suggested by Zhang and Yin (2023) [@zhang2023bayesian,], but because they are non-parametric, it is not possible to perform analyses that adjust for covariates or consider heterogeneity.

For the RMST relating packages, survRM2 provides RMST, the difference in RMST, the confidence interval for the difference, the ratio, and the confidence interval for the ratio. It also allows for covariate adjustment. The rmst function in the RISCA package can also calculate RMST. There is no a package of non-parametric Bayesian analysis.

The rmstBayepara package enables parametric Bayesian analysis considering covariates and heterogeneity. It provides posterior samples of RMST, shrinkage-adjusted RMST for each cluster, and RMST under specific covariate conditions. The summarized results include the mean, median, mode, and 95% credible interval for the mean. The analysis considering heterogeneity includes methods that add random effects to the linear combination of covariates and apply frailty to the hazard. The rmstBayepara package can analyze exponential, Weibull, log-normal, and log-logistic models. It also implements model selection using the Widely Applicable Information Criterion (WAIC) [@watanabe2010asymptotic].

Illustration of rmstBayespara

We introduce how to use rmstBayespara. The package rmstBayespara can be installed and loaded using the following code.

library(rmstBayespara)
library(spBayesSurv)

The following main functions are demonstrated in this article:

brm_surv: Inference on the Bayesian regression models for parametric survival time.

rmstpara: Restricted mean survival time via parametric models.

The function provides the inference of Bayesian regression models using Stan for parametric survival time.

Usage: The brm_surv function is called using following syntax.

brm_surv(time, cnsr, var, rvar, family = "exponential", random = "fixed", data,
  iter = 2000, warmup = 1000, seed = NA, chains = 4)

Arguments: The main arguments of brm_surv function is tabulated as below:

time: name of time variable in data. Need to set character.
cnsr: name of censor variable in data. Need to set character.
var: vector of covariate names in data. Need to set character.
rvar: name of random effect in data. Need to set character.
family: A description of the response distribution and link function to be used in the model. 'exponential', 'Weibull', 'log-normal', and 'log-logistic' can be selected.
random: A description of random effect. 'fixed', 'normal', and 'frailty' are available.
data: An object of class data.frame (or one that can be coerced to that class) containing data of all variables used in the model.

Value: The brm_surv function returns a list of list of an object of class brmsfit or stanfit (see \CRANpkg{rstan} and \CRANpkg{brms}), sampling values from posterior distribution, leave-one-out cross-validation, and widely applicable information criterions.

fit: An object of class brmsfit or stanfit.
post_sample: Extract posterior samples of specified parameters.
waic: Widely applicable information criterion.
loo: Efficient approximate leave-one-out cross-validation.

The function provides the value of RMST via parametric models, such as exponential, Weibull, log-normal, and log-logistic models.

Usage: The rmstpara function is called using following syntax.

rmstpara(tau, var, rvar = NA, shape = NA, sigma = NA, 
         family = "exponential", random = "fixed")

Arguments: The main arguments of rmstpara function is tabulated as below:

tau: A value of pre-specified evaluation time point.
var: A vector of covariate values.
rvar: A vector of frailty effects. It is necessary when log-normal frailty and log-logistic frailty models.
shape: A vector of shape parameters. It is necessary when Weibull and log-logistic models.
sigma: A vector of standard error parameters. It is necessary when log-normal model.
family: A description of the response distribution and link function to be used in the model. 'exponential', 'Weibull', 'log-normal', and 'log-logistic' can be selected.
random: A description of random effect. 'fixed', 'normal', and 'frailty' are available.

Value: The rmstpara function returns a value or vector of RMST via the specific parametric model.

Example data

We illustrate a real example of the Leukemia Survival Data in spBayesSurv, which include 1,043 patients of survival of acute myeloid leukemia. We focus on the "time", "cens", "age" and "district" variables. "District" can be regarded as a cluster, and the impact of mixed effects and frailty can be considered.

data(LeukSurv)
dat<-LeukSurv
dat$status<-dat$cens
dat$arm[dat$age>=65]<-"Greater than or equal to 65 years old"
dat$arm[dat$age<65]<-"Less than 65 years old"
dat[c(1,1:5*200), c("time", "status", "arm", "age", "district")]

Sample code

First of all, we calculate the naive Weibull model with two variables arm and sex.

fit_x <- brm_surv(time="time", cnsr="1-status", 
                  var=c("factor(arm)", "factor(sex)"), 
                  data=dat, family="Weibull", random="fixed")

The WAIC and loo can be illustrated as below:

fit_x$waic
fit_x$loo

	Estimate	SE
elpd_waic	-6056.5	73.6
p_waic	4.8	0.4
waic	12113.0	147.1

	Estimate	SE
elpd_loo	-6056.5	73.6
p_loo	4.8	0.4
looic	12113.0	147.1

Although main purpose of rmstBayespara is to present group differences in RMST, the Bayesian parametric model also can illustrate the posterior distribution of covariate parameters.

estimate <- apply(fit_x$post_sample, 2, summary)
round(estimate, 4)
boxplot(fit_x$post_sample, cex.axis=0.8)

	b_intercept	b_factor(arm)	b_factor(sex)	shape
Min.	5.4637	1.1233	-0.5284	0.5114
1st Qu.	5.7489	1.5012	-0.1710	0.5446
Median	5.8212	1.5867	-0.0865	0.5542
Mean	5.8219	1.5860	-0.0873	0.5547
3rd Qu.	5.8930	1.6692	-0.0058	0.5648
Max.	6.2475	2.0499	0.3431	0.6007

The difference in RMST between arms of posterior samples from Weibull model can be obtained as below:

ps_x<-fit_x$post_sample
rmst_diff_x<-as.numeric(
  rmstpara(tau=100, var=ps_x[,'b_intercept']+ps_x[,'b_factor(arm)'], 
           shape=ps_x[,'shape'], family="Weibull",random="fixed") - 
    rmstpara(tau=100, var=ps_x[,'b_intercept'],
             shape=ps_x[,'shape'], family="Weibull",random="fixed")
  )
c(summary(rmst_diff_x),quantile(rmst_diff_x,c(0.025,0.975)))
hist(rmst_diff_x, breaks=100, freq=F)

Min.	1st Qu.	Median	Mean	3rd Qu.	Max.	2.5%	97.5%
10.18390	13.89870	14.77579	14.77010	15.59717	19.53334	12.25126	17.27399

The 95% confidence interval can be obtained from the 2.5% and 97.5% results.

Next, the analysis of mixed effects model can be implemented as follows.　By setting random="normal", it is possible to adapt to a mixed effects model.

fit_x_r <- brm_surv(time="time", cnsr="1-status", var=c("factor(arm)", "factor(sex)"), 
                    rvar="district", data=dat, family="Weibull", random="normal")
fit_x_r$waic
fit_x_r$loo

	Estimate	SE
elpd_waic	-6050.0	73.8
p_waic	21.0	1.6
waic	12100.0	147.6

	Estimate	SE
elpd_loo	-6050.1	73.8
p_loo	21.0	1.6
looic	12100.1	147.6

The results of the difference in RMST with mixed effect, taking heterogeneity into account, can be calculated as follows.

ps_x_r<-fit_x_r$post_sample
rmst_diff_x_r<-as.numeric(
  rmstpara(tau=100, var=ps_x_r[,'b_intercept']+ps_x_r[,'b_factor(arm)'],
           rvar=ps_x_r[,'district'], shape=ps_x_r[,'shape'], 
           family="Weibull", random="normal") - 
    rmstpara(tau=100, var=ps_x_r[,'b_intercept'], 
           rvar=ps_x_r[,'district'], shape=ps_x_r[,'shape'], 
           family="Weibull", random="normal")
  )
c(summary(rmst_diff_x_r),quantile(rmst_diff_x_r,c(0.025,0.975)))

Min.	1st Qu.	Median	Mean	3rd Qu.	Max.	2.5%	97.5%
8.837232	12.074202	12.917372	12.974916	13.844946	17.774116	10.443351	15.619410

The results of the difference in RMST for distinct 1, taking heterogeneity into account, can be calculated as follows.

rmst_diff_x_r_1<-as.numeric(
  rmstpara(tau=100, var=ps_x_r[,'b_intercept']+ps_x_r[,'b_factor(arm)'], 
           shape=ps_x_r[,'shape'], rvar=ps_x_r[,'sd_district[1]'], 
           family="Weibull",random="normal") - 
    rmstpara(tau=100, var=ps_x_r[,'b_intercept'], 
             shape=ps_x_r[,'shape'], rvar=ps_x_r[,'sd_district[1]'], 
             family="Weibull",random="normal")
  )
c(summary(rmst_diff_x_r_1),quantile(rmst_diff_x_r_1,c(0.025,0.975)))

Min.	1st Qu.	Median	Mean	3rd Qu.	Max.	2.5%	97.5%
0.2712211	0.4375881	0.4887243	0.4945920	0.5460238	0.9143291	0.3492974	0.6731644

Finally, frailty effects model can be implemented as follows.

fit_x_f <- brm_surv(time="time", cnsr="1-status", var=c("factor(arm)", "factor(sex)"), 
                    rvar="district", data=dat, family="Weibull", random="frailty")
fit_x_f$waic
fit_x_f$loo

	Estimate	SE
elpd_waic	-6050.0	73.9
p_waic	22.0	1.7
waic	12100.0	147.7

	Estimate	SE
elpd_loo	-6050.1	73.9
p_loo	22.1	1.7
looic	12100.2	147.7

The results of the difference in RMST with frailty effect, taking heterogeneity into account, can be calculated as follows.

ps_x_f<-fit_x_f$post_sample
rmst_diff_x_f<-as.numeric(
  rmstpara(tau=100, var=ps_x_f[,'b_intercept']+ps_x_f[,'b_factor(arm)'],
           rvar=ps_x_f[,'sd_district'], shape=ps_x_f[,'shape'], 
           family="Weibull", random="frailty") - 
    rmstpara(tau=100, var=ps_x_f[,'b_intercept'], 
           rvar=ps_x_f[,'sd_district'], shape=ps_x_f[,'shape'], 
           family="Weibull", random="frailty")
  )
c(summary(rmst_diff_x_f),quantile(rmst_diff_x_f,c(0.025,0.975)))

Min.	1st Qu.	Median	Mean	3rd Qu.	Max.	2.5%	97.5%
0.08846989	0.68157877	1.02491134	1.13388377	1.44250681	4.85410016	0.27837340	2.70644121

The results of the difference in RMST for distinct 1, taking heterogeneity into account, can be calculated as follows.

rmst_diff_x_f_1<-as.numeric(
  rmstpara(tau=100, var=ps_x_f[,'b_intercept']+ps_x_f[,'b_factor(arm)'], 
           shape=ps_x_f[,'shape'],rvar=ps_x_f[,"sd_district[1]"],  
           family="Weibull",random="frailty") - 
    rmstpara(tau=100, var=ps_x_f[,'b_intercept'], 
             shape=ps_x_f[,'shape'], rvar=ps_x_f[,"sd_district[1]"], 
             family="Weibull",random="frailty")
  )
c(summary(rmst_diff_x_f_1),quantile(rmst_diff_x_f_1,c(0.025,0.975)))

Min.	1st Qu.	Median	Mean	3rd Qu.	Max.	2.5%	97.5%
-25.039281591	8.385407514	10.359507614	9.357048164	11.463067134	14.833317067	0.005389485	12.930702706

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paper.md

Summary

Statement of need

Illustration of rmstBayespara

Example data

Sample code

Acknowledgements

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Summary

Statement of need

Illustration of rmstBayespara

Example data

Sample code

Acknowledgements

References