Defective Gompertz Survival Model

• Defective Gompertz distribution is an improper survival model with a positive tail limit that represents a cure fraction.
• It integrates the cure rate directly into the lifetime law, eliminating the need for an explicit separate cure parameter found in mixture models.
• The model has been extended to regression, bivariate, and Bayesian joint frameworks, proving effective in various biomedical survival analyses.

The defective Gompertz distribution is an improper survival model obtained by relaxing the standard Gompertz requirement that survival vanish as time tends to infinity. In its cure-rate interpretation, the survival function converges instead to a positive constant, and that tail limit is identified with the proportion of individuals who are cured, immune, or long-term survivors. This construction has become a parsimonious alternative to explicit mixture cure models because the cure fraction is encoded directly in the lifetime law rather than introduced as a separate parameter. In recent work, the defective Gompertz family has been used in univariate cure modeling, competing-risks regression, bivariate dependence modeling, defective generalized Gompertz quantile regression, and Bayesian joint models for longitudinal and cure-survival data (Neto et al., 31 Jul 2025, K. et al., 2024, Rodrigues et al., 2021, Peres et al., 2020, Neto et al., 15 Jul 2025).

1. Definition and transition from the standard Gompertz law

A common starting point is the ordinary Gompertz distribution with density

$$f(t \, ; \alpha, \mu) = \mu\exp(\alpha t) \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$

and survival function

$$S(t \, ; \alpha, \mu) = \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$

for $\alpha>0$, $\mu>0$, and $t>0$. In this proper regime, the survival function converges to zero, so all probability mass is assigned to finite event times. The hazard is Gompertzian,

$h(t;\alpha,\mu)=\mu e^{\alpha t},$

and in the standard parameterization it is monotone increasing in time (Neto et al., 31 Jul 2025, Neto et al., 15 Jul 2025).

The defective version arises when the sign restriction on the shape parameter is reversed. If $\alpha<0$, then

$$p=\lim_{t\to\infty}S(t;\alpha,\mu)=\exp\left(\frac{\mu}{\alpha}\right)\in(0,1),$$

so the tail no longer collapses to zero. The survival law is then improper in the sense that part of the probability mass remains at infinity, and that missing finite-time mass is interpreted as permanent nonfailure or cure (Neto et al., 31 Jul 2025, Neto et al., 15 Jul 2025).

Another formulation used in cure-rate work writes the defective Gompertz survival as

$$S(t)=\exp\left\{-\frac{\alpha}{\beta}\left[1-\exp(-\beta t)\right]\right\}, \qquad t>0,$$

with $\alpha>0$ and $$S(t \, ; \alpha, \mu) = \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$0, giving cure fraction

$$S(t \, ; \alpha, \mu) = \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$1

density

$$S(t \, ; \alpha, \mu) = \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$2

and hazard

$$S(t \, ; \alpha, \mu) = \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$3

This suggests that sign conventions and parameter naming are not uniform across the literature, even though the central defective feature is always the same: the survival tail remains strictly positive (Peres et al., 2020).

For structural context, the standard Gompertz law on $$S(t \, ; \alpha, \mu) = \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$4 can also be written with parameters $$S(t \, ; \alpha, \mu) = \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$5 and $$S(t \, ; \alpha, \mu) = \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$6 as a proper distribution with survival

$$S(t \, ; \alpha, \mu) = \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$7

and hazard

$$S(t \, ; \alpha, \mu) = \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$8

That proper model is characterized in terms of extreme-value theory as a zero-truncated Gumbel minimum distribution, and those identities are often informative when defective variants are derived or checked algebraically (Bauckhage, 2014).

2. Cure-rate interpretation and mixture representation

The defining feature of a defective Gompertz model is not merely an altered hazard shape, but the presence of positive asymptotic survival. In cure-rate survival analysis this limit is interpreted directly as the cure fraction. In the univariate defective Gompertz model just described, the cure fraction is $$S(t \, ; \alpha, \mu) = \exp\left(-\frac{\mu}{\alpha}(\exp( \alpha t) - 1)\right),$$9 when $\alpha>0$0; in the alternative parameterization it is $\alpha>0$1 (Neto et al., 31 Jul 2025, Peres et al., 2020).

This direct encoding distinguishes defective Gompertz models from mixture cure models. Mixture formulations introduce an explicit cure component, for example

$\alpha>0$2

or

$\alpha>0$3

depending on whether $\alpha>0$4 is used for the cured or susceptible fraction. The defective approach instead lets the cure proportion emerge from the tail of the survival function itself, which avoids adding a separate cure-probability parameter to the baseline model (Neto et al., 31 Jul 2025, Neto et al., 15 Jul 2025, Peres et al., 2020).

In defective generalized Gompertz work, the improper survival law is often rewritten explicitly as a standard mixture to recover interpretable summaries for the susceptible subpopulation. Introducing an indicator $\alpha>0$5, with $\alpha>0$6 for immune and $\alpha>0$7 for susceptible, one sets

$\alpha>0$8

and decomposes the overall survival as

$\alpha>0$9

Here $\mu>0$0 is the proper survival function for susceptible individuals. This representation is especially important when the objective is not only to estimate the cure fraction but also to define quantiles and regression effects within the uncured group (Rodrigues et al., 2021).

In joint modeling, the same cure interpretation becomes subject-specific. With hazard

$\mu>0$1

the individual survival function is

$\mu>0$2

and when $\mu>0$3 its limit is

$\mu>0$4

The cure fraction is therefore not only a population quantity; it is shifted by the survival linear predictor and hence by shared random effects (Neto et al., 31 Jul 2025).

3. Generalized and reparameterized defective Gompertz families

A major development has been the move from the defective Gompertz model to defective generalized Gompertz families. One formulation starts from the generalized Gompertz distribution with density

$\mu>0$5

and survival

$\mu>0$6

In its proper form, $\mu>0$7, $\mu>0$8, and $\mu>0$9. The defective regime is obtained by taking $t>0$0, and the resulting cure fraction is

$t>0$1

Thus the generalized model retains the cure-rate interpretation while introducing an additional shape parameter (Rodrigues et al., 2021).

That same paper derives the quantile function of the generalized Gompertz distribution,

$t>0$2

and then shifts attention to the susceptible-group quantile

$t>0$3

obtained from the proper susceptible survival $t>0$4. The model is then reparameterized in terms of the susceptible-group $t>0$5-th quantile and linked to covariates through

$t>0$6

This construction yields a fully parametric cure-rate quantile regression model in which covariates affect both susceptible survival quantiles and the induced cure fraction (Rodrigues et al., 2021).

A second generalization uses a Lehmann-type power transformation. If $t>0$7 is a baseline CDF and $t>0$8, the generalized model is

$t>0$9

with survival

$h(t;\alpha,\mu)=\mu e^{\alpha t},$0

Applied to the defective Gompertz baseline, this produces the defective generalized Gompertz distribution with survival

$h(t;\alpha,\mu)=\mu e^{\alpha t},$1

and asymptotic cure fraction

$h(t;\alpha,\mu)=\mu e^{\alpha t},$2

Here the power parameter $h(t;\alpha,\mu)=\mu e^{\alpha t},$3 modifies the eventual cure probability while preserving defectiveness (Neto et al., 15 Jul 2025).

A further reparameterization makes the cure fraction itself explicit:

$h(t;\alpha,\mu)=\mu e^{\alpha t},$4

equivalently

$h(t;\alpha,\mu)=\mu e^{\alpha t},$5

This cure-centric form is designed for regression on long-term survival itself rather than on latent scale parameters (Neto et al., 15 Jul 2025).

4. Regression, dependence structures, and multivariate embeddings

Defective Gompertz models have been embedded in several regression architectures. In interval-censored competing risks, a cause-specific defective Gompertz survival is specified for each cause $h(t;\alpha,\mu)=\mu e^{\alpha t},$6 as

$h(t;\alpha,\mu)=\mu e^{\alpha t},$7

with $h(t;\alpha,\mu)=\mu e^{\alpha t},$8 and $h(t;\alpha,\mu)=\mu e^{\alpha t},$9. The cause-specific cure fraction is

$\alpha<0$0

and under independence the overall survival and overall cure fraction are

$\alpha<0$1

Covariates enter through

$\alpha<0$2

so both the hazard level and its time trend contribute to long-term nonfailure (K. et al., 2024).

In bivariate survival analysis, defective Gompertz marginals have been coupled by a Clayton copula. With marginal survivals

$\alpha<0$3

the joint survival is

$\alpha<0$4

where $\alpha<0$5 controls positive dependence and

$\alpha<0$6

is Kendall’s tau for the Clayton copula. This produces a bivariate defective Gompertz distribution suitable for paired survival outcomes with cure fractions in both margins (Peres et al., 2020).

In joint longitudinal-survival modeling, the defective Gompertz baseline has been used as the survival component of a Bayesian model for multivariate longitudinal biomarkers and cure-survival outcomes. The repeated measurements are modeled as Poisson,

$\alpha<0$7

while the event-time hazard is

$\alpha<0$8

The same random effects that govern longitudinal trajectories enter the survival risk through

$\alpha<0$9

so latent biomarker heterogeneity affects not only the hazard but also the individual cure probability (Neto et al., 31 Jul 2025).

These developments show that the defective Gompertz distribution is not restricted to a single univariate cure model. It now functions as a baseline building block for quantile regression, competing risks, dependence modeling, and joint latent-variable formulations (Rodrigues et al., 2021, K. et al., 2024, Peres et al., 2020, Neto et al., 31 Jul 2025).

5. Inference and empirical performance

Inference for defective Gompertz models follows the censoring structure and model embedding. For right-censored quantile regression based on the defective generalized Gompertz model, the likelihood is

$$p=\lim_{t\to\infty}S(t;\alpha,\mu)=\exp\left(\frac{\mu}{\alpha}\right)\in(0,1),$$0

with truncated normal prior on $$p=\lim_{t\to\infty}S(t;\alpha,\mu)=\exp\left(\frac{\mu}{\alpha}\right)\in(0,1),$$1, gamma prior on $$p=\lim_{t\to\infty}S(t;\alpha,\mu)=\exp\left(\frac{\mu}{\alpha}\right)\in(0,1),$$2, and normal priors on regression coefficients. Because the posterior is not available in closed form, posterior simulation is performed with an Adaptive Metropolis MCMC scheme (Rodrigues et al., 2021).

For interval-censored competing risks, parameters are estimated by numerically maximizing the log-likelihood with nlminb, and the estimated covariance matrix is obtained from the inverse observed information. The paper states that the MLE is asymptotically multivariate normal, yielding Wald-type confidence intervals (K. et al., 2024).

For the bivariate copula model, maximum likelihood is carried out with the maxLik package in R using the Nelder–Mead algorithm, and a Bayesian analysis with independent uniform priors and MCMC sampling in R2jags is also described (Peres et al., 2020).

For joint longitudinal and cure-survival modeling, estimation is Bayesian but not MCMC-based. The model is cast as a latent Gaussian model with latent field

$$p=\lim_{t\to\infty}S(t;\alpha,\mu)=\exp\left(\frac{\mu}{\alpha}\right)\in(0,1),$$3

hyperparameters $$p=\lim_{t\to\infty}S(t;\alpha,\mu)=\exp\left(\frac{\mu}{\alpha}\right)\in(0,1),$$4, and posterior

$$p=\lim_{t\to\infty}S(t;\alpha,\mu)=\exp\left(\frac{\mu}{\alpha}\right)\in(0,1),$$5

The posterior marginals are then approximated by INLA, which is presented as computationally efficient and scalable compared with MCMC in multivariate joint models (Neto et al., 31 Jul 2025).

For the reparameterized defective generalized Gompertz regression model, Bayesian inference is performed with Hamiltonian Monte Carlo, implemented through Stan/rstan, using vague and independent priors on regression coefficients, $$p=\lim_{t\to\infty}S(t;\alpha,\mu)=\exp\left(\frac{\mu}{\alpha}\right)\in(0,1),$$6, and $$p=\lim_{t\to\infty}S(t;\alpha,\mu)=\exp\left(\frac{\mu}{\alpha}\right)\in(0,1),$$7 (Neto et al., 15 Jul 2025).

Across these studies, simulation evidence is broadly consistent. Bias and mean squared error decrease as sample size increases; interval coverage or 95% credible-interval coverage is close to nominal in many scenarios; estimation becomes harder under strong dependence, high cure fractions, or small samples; and low-information or vague-prior Bayesian procedures can still perform satisfactorily in moderate samples (Neto et al., 31 Jul 2025, K. et al., 2024, Peres et al., 2020, Neto et al., 15 Jul 2025).

6. Applications, related Gompertz-family models, and common misconceptions

Applications are predominantly biomedical. The defective generalized Gompertz quantile regression model was illustrated with male breast cancer data from Brazil, where the focus was the impact of clinical stage and age on different survival quantiles as well as on the cure rate (Rodrigues et al., 2021). In the SANAD epilepsy trial, the defective Gompertz joint model produced $$p=\lim_{t\to\infty}S(t;\alpha,\mu)=\exp\left(\frac{\mu}{\alpha}\right)\in(0,1),$$8 with $$p=\lim_{t\to\infty}S(t;\alpha,\mu)=\exp\left(\frac{\mu}{\alpha}\right)\in(0,1),$$9, supporting a defective baseline; it estimated a hazard ratio $$S(t)=\exp\left\{-\frac{\alpha}{\beta}\left[1-\exp(-\beta t)\right]\right\}, \qquad t>0,$$0 for lamotrigine relative to carbamazepine, and average cure fractions of $$S(t)=\exp\left\{-\frac{\alpha}{\beta}\left[1-\exp(-\beta t)\right]\right\}, \qquad t>0,$$1 for the carbamazepine group and $$S(t)=\exp\left\{-\frac{\alpha}{\beta}\left[1-\exp(-\beta t)\right]\right\}, \qquad t>0,$$2 for the lamotrigine group (Neto et al., 31 Jul 2025). In interval-censored HIV data, the defective Gompertz regression model fit better than the defective inverse Gaussian according to AIC and BIC, and yielded cure-fraction estimates such as $$S(t)=\exp\left\{-\frac{\alpha}{\beta}\left[1-\exp(-\beta t)\right]\right\}, \qquad t>0,$$3 for females and $$S(t)=\exp\left\{-\frac{\alpha}{\beta}\left[1-\exp(-\beta t)\right]\right\}, \qquad t>0,$$4 for males in the gender analysis (K. et al., 2024). In uterine cancer data from São Paulo, the reparameterized defective generalized Gompertz model identified surgery as protective and age over 50, metastatic stage, and chemotherapy as factors associated with reduced cure probability (Neto et al., 15 Jul 2025). The bivariate defective Gompertz model has also been applied to breast cancer, diabetic retinopathy, cervical cancer, and tobacco-stained fingers data (Peres et al., 2020).

Several related models are often confused with the defective Gompertz distribution but are conceptually distinct. The standard Gompertz distribution is proper, not defective, and its structural interpretations as a zero-truncated Gumbel minimum law and as a model with hazard $$S(t)=\exp\left\{-\frac{\alpha}{\beta}\left[1-\exp(-\beta t)\right]\right\}, \qquad t>0,$$5 do not by themselves imply a cure fraction (Bauckhage, 2014). The Beta-Gompertz distribution and the Marshall-Olkin extended generalized Gompertz distribution are also proper lifetime models: they are used to enrich density and hazard shapes, including decreasing, increasing, bathtub-shaped, upside-down bathtub, or constant hazards, but they do not encode a point mass at infinity or an improper survival tail (Jafari et al., 2014, Benkhelifa, 2016). In another direction, “Soft bounds on diffusion produce skewed distributions and Gompertz growth” studies Gompertz dynamics for the maximum of a support, not a defective Gompertz probability distribution for lifetimes (Mandrà et al., 2014). Likewise, the Gompertz-Pareto income distribution uses a Gompertz component for the lower-income bulk and a Pareto tail for the richest fraction; it is a truncated or incomplete Gompertz construction in an income-distribution setting rather than a cure-rate survival model (Figueira et al., 2010).

The central misconception, therefore, is to equate any Gompertzian hazard, Gompertz growth curve, or truncated Gompertz component with a defective Gompertz distribution. In the survival-analysis literature, “defective Gompertz” has a narrower meaning: an improper Gompertz-type survival law whose positive tail limit is interpreted as cure, long-term survival, or permanent immunity.