Installation

Install the latest released version fromCRAN

install.packages("GPTCM")

Install the latest development version fromGitHub:

#install.packages("remotes")remotes::install_github("ocbe-uio/GPTCM")

A simulation study

We provide insights about the parameter estimation of the proposedGPTCM by using Monte Carlo simulations. We consider sample sizes of\(n=200\). Each sample/patient has twoclinical covariates (i.e. one row of the clinical data matrix\(\mathbf X_0\in \mathbb R^{n\times 2}\)),and has cells belonging to\(L=3\)tumor cell subtypes with each subtype consisting of\(10\) cell-type-specific covariates(i.e. one row of data matrix\(\mathbf X_l\in\mathbb R^{n\times 10}\),\(l\in\{1,...,L\}\)). Each sample also has tumor cell subtypeproportions data (i.e. one row of the proportions data matrix\(\mathbf p\in [0,1]^{n\times 10}\))generated by\(10\) cell-type-specificcovariates based on the Dirichlet regression model. Every covariate isgenerated independently from the standard normal distribution except thefirst clinical variable generated from the Bernoulli distribution. Thetumor cell subtype proportions of each sample is generated independentfrom the Dirichlet distribution.

The survival times are generated based on the population survivalfunction using rate parameter\(\theta=\exp(\xi_0+\mathbfX_0\boldsymbol\xi)\), using the Weibull distributed survivalfunctions with mean parameters\(\boldsymbol\mu_{l}= \exp(\beta_{0l} + \mathbfX_{l} \boldsymbol \beta_l)\), and using the Dirichlet distributedcell type proportions with concentration parameters\(\boldsymbol\alpha_{l}= \exp(\zeta_{0l} + \mathbfX_{l} \boldsymbol \zeta_l)\),\(l\in\{1,...,L\}\). We use latent indicator variables for Bayesianvariable selection, i.e. \(\boldsymbol\gamma_l = 1\{\boldsymbol \beta_l\ne 0\}\) and\(\boldsymbol \eta_l = 1\{\boldsymbol \zeta_l\ne0\}\). Censoring is generated through an exponential distributionwith censoring rate\(20\%\). TheKaplan–Meier survival curve is as follows. See more details inZhao etal. (2025).

rm(list =ls())# simulate datan<-200# subjectsp<-10# variable selection predictorsL<-3# cell typeslibrary(GPTCM)set.seed(1234)dat<-simData(n, p, L)# KM curvelibrary(survival)library(survminer)fit.km<- survival::survfit(Surv(time, event)~1,data = dat$survObj)ggsurv<- survminer::ggsurvplot(fit.km,conf.int =TRUE,xlab ="Follow-up time (year)",ylab ="Survival probability (%)",legend ="none",risk.table =TRUE,cumevents =TRUE,palette ="jco",risk.table.title ="Number of patients at risk",tables.height =0.1,tables.theme =theme_cleantable(),tables.y.text =FALSE,ggtheme =theme_light())ggsurv$plot<- ggsurv$plot+theme(axis.text =element_text(size =15),axis.title =element_text(size =15,face ="bold")  )ggsurv

After fitting GPTCM-Ber2, we compare its the survival predictionperformance with other approaches (i.e. Kaplan-Meier method withoutcovariates, classic Cox model with two clinical covariates (Cox.clin),Cox model with mean aggregate covariates (Cox.X.mean), Cox model withmedian aggregate covariates (Cox.X.median), Cox model with clinical andmean aggregate covariates (Cox.clin.X.mean), and frequentistsemiparametric promotion time cure model with clinical variables(PTCM.clin)). The follow figure shows the prediction performance oftime-dependent Brier scores. It is clear that our GPTCM has much bettersurvival prognosis than other approaches.

## run Bayesian GPTCMset.seed(123)fit<-GPTCM(dat,nIter =1100,burnin =100)# draw time-dependent Brier scoresplotBrier(dat,datMCMC = fit,time.star =3,xlab ="Evalutation time points",ylab ="Prediction error")

The following figure shows the posterior distributions ofcell-type-specific effects associated with cell-type-specificprogression with the black diamonds indicating true effects, andmarginal posterior inclusion probabilities (mPIP) of Bayesian variableselection (BVS).

# show cel-type-specific effectsplotCoeff(dat,datMCMC = fit,estimator ="beta",bandwidth =0.02)# show BVSplotCoeff(dat,datMCMC = fit,estimator ="gamma")

The following figure shows the posterior distributions ofcell-type-specific effects associated with cell-type-specificproportions, and mPIP of Bayesian variable selection.

# show cel-type-specific effectsplotCoeff(dat,datMCMC = fit,estimator ="zeta",bandwidth =0.01)# show BVSplotCoeff(dat,datMCMC = fit,estimator ="eta")

References

Chen MH, Ibrahim JG, Sinha D (1999). A new Bayesian model forsurvival data with a surviving fraction. Journal of the AmericanStatistical Association, 94(447):909–919.

Yakovlev AY, Tsodikov AD, Asselain B (1996). Stochastic Models ofTumor Latency and Their Biostatistical Applications. World Scientific,Singapore.

Zhao Z, Kızılaslan F (2024). A note on promotion time cure modelswith a new biological consideration.arXiv:2408.17188.

Zhao Z, Kızılaslan F, Wang S, Zucknick M (2025). Generalizedpromotion time cure model: A new modeling framework to identifycell-type-specific genes and improve survival prognosis.arXiv:2509.01001.