Bayesian defective Marshall-Olkin Gompertz model: an integrated approach to identifying cure fraction

Regression models have a substantial impact on interpretation of treatments, genetic characteristics and other potential risk factors in survival analysis. In many applications, the description of censoring and survival curve reveals the presence of cure fraction on data, which leads to alternative modeling. The most common approach to introduce covariates under a parameter estimation is the cure rate model and its variations, although the use of defective distributions have introduced a more parsimonious and integrated approach. Defective distributions are given by a density function whose integration is not one after changing the domain of one of the parameters, making them appropriate for survival curves with an evident plateau. In this work, we introduce a new Bayesian defective regression model for long-term survival outcomes using the Marshall-Olkin Gompertz distribution. The estimation process is under the Bayesian paradigm. We evaluate the asymptotic properties of our proposal under the vague prior scheme in Monte Carlo studies. We present a motivating real-world application using data from patients diagnosed with testicular cancer in São Paulo, Brazil, in which long-term survivors were identified. Scenarios of cure with uncertainty estimates via credible intervals are provided to evaluate characteristics such as risk age, presence of treatment, and cancer stage.

💡 Research Summary

This paper introduces a novel Bayesian defective regression framework for long‑term survival data by extending the Marshall‑Olkin Gompertz distribution (MOGD) to a defective form. Traditional cure‑rate models separate the population into cured and uncured groups and require an explicit cure‑fraction parameter, which adds complexity and demands a priori decision about the presence of a cure. Defective distributions, by contrast, become improper when a parameter leaves its conventional domain; the limiting value of the survival function then directly yields the cure fraction.

The authors first describe the MOGD, a three‑parameter distribution that augments the classic Gompertz model with a Marshall‑Olkin parameter Υ. The density, survival, and hazard functions are given in equations (1)–(3). By allowing the shape parameter α to take any real value, the model becomes defective when α < 0. In that case the survival function converges to

p = limₜ→∞ S(t) = Υ exp(μα) /