Bayesian Progressive Three State Model with Censoring Due to Intervention

In screening programs, individuals are usually followed up and tested (screened) for the development of a disease, such as cancer. The target disease often develops progressively in stages; for example healthy (state 1), pre-state disease (state 2), and the disease state (state 3). When the pre-state disease is found during screening it is intervened upon, for example by surgical removal of a lesion, so that the progression of the pre-state disease to disease is interrupted. This is called censoring due to intervention. Researchers often want to estimate the time from baseline to the pre-state disease, the time from the pre-state disease to the disease, and the total time from baseline to the disease. In addition, researchers often want to regress these times on baseline covariates. To these ends, 'BayesTSM' estimates a progressive three-state model with censoring due to intervention using Bayesian estimation methods, as described in Klausch et al. (2023) .

BayesTSM

BayesTSM fits Bayesian progressive three-state semi-Markov models for screening panel data with censoring after intervention. It is designed for settings where individuals are repeatedly screened for a progressive disease process, for example:

state 1: disease-free,
state 2: pre-state disease / early disease,
state 3: disease / late disease.

The package models the latent transition time from baseline to state 2, denoted X, and the transition time from state 2 to state 3, denoted S. The total time from baseline to state 3 is Y = X + S.

BayesTSM uses accelerated failure time models for both transition times,

$$ \begin{aligned} \log X_i &= z_{Xi}^{\top}\beta_X + \sigma_X \epsilon_i, \ \log S_i &= z_{Si}^{\top}\beta_S + \sigma_S \xi_i . \end{aligned} $$

and estimates the model using Bayesian data augmentation and MCMC. Weibull, lognormal, and log-logistic transition time distributions are currently available.

Installation

Install the development version from GitHub with:

# install.packages("devtools")
devtools::install_github(
  "thomasklausch2/BayesTSM",
  build_vignettes = TRUE
)

Basic usage

library(BayesTSM)

set.seed(1)

dat <- gendat(
  n = 1000,
  p = 2,
  sigma.X = 0.3,
  mu.X    = 2,
  beta.X  = c(0.5, 0.5),
  sigma.S = 0.5,
  mu.S    = 1,
  beta.S  = c(0.5, 0.5),
  dist.X  = "weibull",
  dist.S  = "weibull",
  v.min   = 1,
  v.max   = 5,
  Tmax    = 200,
  mean.rc = 10
)

head(dat)
table(dat$d)

The variables passed to bayestsm() are:

d: event type, where 1 denotes right censoring, 2 a pre-state event, and 3 a terminal-state event;
L: left bound of the final screening interval;
R: right bound of the final screening interval, or Inf for right-censored observations;
Z.X: covariate matrix for the X transition model;
Z.S: covariate matrix for the S transition model.

d <- dat$d
L <- dat$L
R <- dat$R
Z <- dat[, c("Z.1", "Z.2")]

mod <- bayestsm(
  d = d,
  L = L,
  R = R,
  Z.X = Z,
  Z.S = Z,
  mc = 1e4,
  warmup = 5e2,
  thinning = 10,
  chains = 4,
  update_till_convergence = TRUE,
  mc_update = 1e4,
  MH = FALSE,
  dist.X = "weibull",
  dist.S = "weibull",
  seed_chains = 1:4
)

By default, BayesTSM uses slice sampling for the model parameters. Metropolis sampling can be used instead by setting MH = TRUE.

Posterior summaries

summary(mod, warmup = 500)
plot(mod, warmup = 500)

The summary method reports posterior medians, 95% credible intervals, R-hat values, and effective sample sizes.

Posterior predictive transition probabilities

Posterior predictive cumulative transition probabilities can be obtained with ppCIF().

ppCIF(
  mod,
  type = "quantiles",
  warmup = 500,
  quant = c(5, 10)
)

For plotting cumulative transition probability curves:

pp_grid <- ppCIF(
  mod,
  type = "quantiles",
  warmup = 500
)

plot(pp_grid, xlim = c(0, 50))

Conditional predictions can be obtained by fixing covariate values through fix_Z.X and fix_Z.S.

ppCIF(
  mod,
  type = "quantiles",
  warmup = 500,
  quant = c(5, 10),
  fix_Z.X = c(1, NA),
  fix_Z.S = c(1, NA)
)

Here, the first covariate is fixed at 1, while the second covariate is marginalized over its observed distribution.

Model comparison

Different transition time distributions can be compared using information criteria.

get_IC(mod, warmup = 500, cores = NULL)

get_IC() returns the deviance information criterion (DIC) and two versions of the widely applicable information criterion (WAIC-1 and WAIC-2).

References

Akwiwu, E. U., Coupé, V. M. H., Berkhof, J., & Klausch, T. (2026). A comparison of methods for modeling multistate cancer progression using screening data with censoring after intervention. Medical Decision Making. https://doi.org/10.1177/0272989X261422681

Klausch, T., Akwiwu, E. U., van de Wiel, M. A., Coupé, V. M. H., & Berkhof, J. (2023). A Bayesian accelerated failure time model for interval censored three-state screening outcomes. The Annals of Applied Statistics, 17(2), 1285–1306. https://doi.org/10.1214/22-AOAS1669

Vehtari, A., Gelman, A., Simpson, D., Carpenter, B., & Bürkner, P.-C. (2021). Rank-normalization, folding, and localization: An improved R-hat for assessing convergence of MCMC. Bayesian Analysis, 16(2), 667–718. https://doi.org/10.1214/20-BA1221

Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research, 11, 3571–3594.