Unified likelihood model tackles double censoring in multistate survival analysis

Statisticians have proposed a new parametric framework that simultaneously handles correlated transition times and double censoring in progressive multistate survival data, and they have demonstrated it on retinal vascular development in premature infants with retinopathy of prematurity. The work, published in BMC Medical Research Methodology, addresses a gap the authors identify: existing approaches typically treat within-subject dependence and left-plus-right censoring separately, while joint analysis of progressive multistate transitions, shared frailty and double censoring in a single likelihood framework has received limited attention.

The model describes transition intensities between states using Weibull distributions. A shared gamma frailty term captures the within-subject dependence among an individual’s successive transition times, meaning that unobserved heterogeneity common to one subject’s whole disease or development pathway is explicitly accounted for rather than ignored. Parameters are estimated by maximum likelihood, with log-stabilized numerical integration over the frailty distribution introduced to keep the computations stable.

A defining feature of the method is its treatment of double censoring. In multistate studies, some transition times may be known only to fall within an interval, so they are subject to left as well as right censoring. By building both censoring mechanisms and the frailty dependence into one likelihood, the model avoids the biases that arise when either feature is dropped from the analysis.

The authors evaluated the approach through simulations under a range of conditions: sample sizes of 100, 200 and 600; frailty variances of 0.5, 1 and 1.5; and proportions of left censoring of 20%, 40% and 60%. Performance was assessed using bias, root mean square error, and the coverage probability of 95% confidence intervals.

Under the simulation settings examined, the proposed model showed reduced bias and improved estimation accuracy and confidence-interval coverage compared with models that ignore frailty and/or left censoring. The gains became more pronounced as the frailty variance and the proportion of left censoring increased, indicating that the benefit of the unified approach grows precisely where simpler models struggle most.

To illustrate the method in practice, the researchers applied it to data on retinal vascular development in premature infants with retinopathy of prematurity, a condition in which the stages of disease progression can be modeled as a sequence of states. The application, the authors report, demonstrates the framework’s utility for correlated multistate processes in biomedical studies.

The authors conclude that the framework provides a practical approach for progressive multistate data with correlated transition times and double censoring. Because the model is parametric, with Weibull transition intensities and a gamma frailty distribution, it offers a computationally tractable route to joint estimation that earlier separate treatments of these data features did not provide.

The article is published open access, and the version currently available is shared early to provide faster access to peer-reviewed, accepted research; it is citable and carries a permanent DOI, with a final Version of Record to replace it after further edits.

Subject of Research: A parametric progressive multistate frailty model that jointly handles shared frailty and double censoring, applied to retinopathy of prematurity data.

Article Title: A unified likelihood approach for progressive multistate frailty models under double censoring: application to retinopathy of prematurity

Article References: Dadgar, A., Zeraati, H., Bazvand, F., & Yaseri, M. (2026). A unified likelihood approach for progressive multistate frailty models under double censoring: application to retinopathy of prematurity. BMC Medical Research Methodology. https://doi.org/10.1186/s12874-026-03042-5

Image Credits: AI Generated

DOI: 10.1186/s12874-026-03042-5

Keywords: multistate models, frailty, double censoring, maximum likelihood, Weibull distribution, simulation study, retinopathy of prematurity, survival analysis, biostatistics, premature infants, unified, likelihood