Zero‐Inflated Promotion Cure‐Rate Models With Frailty and Flexible Activation Schemes

ABSTRACT

We propose a zero‐adjusted promotion cure‐rate regression model that jointly accommodates instantaneous failures (times at or near‐zero), a non‐susceptible cured fraction, competing risks, and latent heterogeneity modeled with a Gamma frailty. The number of latent causes follows a Poisson distribution and their activation times follow a Weibull distribution. We derive closed‐form survival and density functions under three activation schemes—minimum, maximum, and random—and allow covariates to act on the zero inflation and cure proportions as well as on the baseline time‐to‐event distribution. Parameters are estimated by maximum likelihood, and finite‐sample performance is assessed via Monte Carlo experiments. We illustrate the method with data on time to insulin initiation among 390 pregnant women with gestational diabetes, where the random activation scheme provides the best fit by Akaike Information Criterion and Bayesian Information Criterion and yields interpretable covariate effects. The proposed framework unifies and extends existing cure‐rate and zero‐adjusted survival models, providing a flexible tool for complex settings with cure fractions, competing risks, and frailty.

1. Introduction

Survival analysis is concerned with the time until an event of interest occurs. The foundational work by Kaplan and Meier (1958) introduced nonparametric estimators for survival functions, and (Cox 1972) proposed the semiparametric proportional hazards model. Both frameworks typically assume that every individual in the study is susceptible to the event and, given sufficiently long follow‐up, will eventually experience it. In practice, however, a subset of subjects may never undergo the event, either due to inherent immunity, effective treatment, or other factors, resulting in non‐susceptible, or cured, individuals.

To accommodate such long‐term survivors, (Berkson and Gage 1952) proposed the mixture cure‐rate model, which represents the population as a mixture of susceptible and cured subpopulations. The overall survival function then combines a point mass at infinity, the cured fraction, with a proper survival curve for the susceptible group. This approach not only estimates the cure proportion, but it also allows covariates to influence both the hazard of the event and the probability of cure. When no cure fraction is present, the model reduces to standard Kaplan–Meier or Cox regression methods (Kaplan and Meier 1958; Cox 1972). Applications of mixture cure‐rate models span many fields. In oncology, the outcome may be time to death or recurrence, recognizing that some patients achieve long‐term remission (Rahimzadeh et al. 2014). In criminology, researchers model time to recidivism for released offenders, allowing for those who never reoffend (Maller and Zhou 1996). In industrial reliability, the focus may be on time to failure of machinery components, where certain parts may effectively last indefinitely under normal conditions (Meeker et al. 2022). These examples demonstrate the versatility of cure‐rate models in contexts with a cured or immune fraction.

Despite their usefulness, standard cure‐rate models often assume a homogeneous susceptible population. In reality, unobserved factors, genetic, environmental, or behavioral, induce additional variability. Frailty models introduce an unobservable random effect into the hazard function to capture this latent heterogeneity (Vaupel et al. 1979). In the multiplicative frailty extension of the Cox model, each individual hazard is scaled by a frailty term, often assumed Gamma‐distributed, improving fit and interpretability when unexplained heterogeneity is present (Andersen et al. 1993; Hougaard 1995). Several authors have combined frailty with cure‐rate frameworks (Yu and Peng 2008; Calsavara et al. 2013; Leão et al. 2018). Moreover, in many applied settings, interventions or intrinsic processes impose a delay before failures become observable. For example, initiating medication may require a latent period before therapeutic benefits appear, or preventive maintenance may postpone mechanical breakdowns. Observations at or near time zero may, therefore, reflect this initial phase rather than true failure events. Zero‐adjusted, or zero‐inflated, cure‐rate models explicitly account for such instantaneous failures by allowing survival times equal to zero (de Oliveira et al. 2017; Calsavara, Rodrigues, Rocha, Tomazella, et al. 2019). While zeros frequently arise in real‐world survival data, the literature that simultaneously addresses zero inflation and cure fractions remains comparatively scarce.

The primary goal of this paper is to develop a comprehensive regression framework that simultaneously accommodates three key features often encountered in survival data, namely, a non‐negligible fraction of immediate failures, zero or near‐zero event times, a cure fraction of long‐term survivors, and unobserved heterogeneity among subjects. To this end, we extend the classical zero‐inflated mixture cure‐rate model by embedding a multiplicative frailty term into a promotion‐type hazard structure. Concretely, we assume that the number of latent competing causes follows a Poisson distribution, that their activation times follow a Weibull law, and that individual risk is further modulated by a Gamma‐distributed frailty. This unified approach yields closed‐form expressions for the survival and hazard functions under different activation schemes, minimum, maximum, and random, while accounting for latent variability. We estimate model parameters via maximum likelihood and investigate their finite‐sample behavior through Monte Carlo experiments. Finally, we illustrate the practical utility of the proposed framework by analyzing time to insulin initiation in a cohort of 390 pregnant women with gestational diabetes followed at a tertiary hospital in Brazil. These data exhibit both a non‐negligible proportion of very early insulin initiation times and a cured fraction corresponding to women who never require insulin during pregnancy, thereby providing a natural setting for a zero‐adjusted cure‐rate analysis. In this application, the proposed models offer improved flexibility and meaningful covariate interpretation relative to existing cure‐rate and zero‐adjusted survival formulations.

The remainder of this article is structured as follows. Section 2 reviews zero‐adjusted cure‐rate models under latent competing causes and activation schemes. Section 3 describes the formulation of the proposed frailty‐adjusted model. Section 4 outlines inference methods based on the likelihood function. Section 5 presents the simulation studies, including data generation, finite‐sample maximum likelihood estimates (MLE) performance, and misspecification and activation‐scheme identifiability. Section 6 illustrates the model using data on insulin use among pregnant women with gestational diabetes. Finally, Section 7 presents concluding remarks and future research directions.

2. Background

In this section, we present a brief description of the cure‐rate model proposed by Tsodikov et al. (1996) and by Tsodikov et al. (2003), later extended by Rodrigues et al. (2009), as well as a description of the zero‐adjusted promotion model and activation schemes.

2.1. Frailty Cure‐Rate Models

The statistical methods related to Tsodikov et al. (1996) and Chen et al. (1999) typically assume independence of individual survival times. Although this assumption is reasonable in many studies, it can be inappropriate when lifetimes are correlated, for example, when several recurrence times are observed for the same individual, characterizing multivariate survival data. In such cases, the independence assumption may fail. A common approach to incorporate this dependence is the frailty model, which has been widely used in the literature.

The frailty term is an unobservable, nonnegative random variable that is usually incorporated multiplicatively into the baseline hazard function. The multiplicative frailty model can be viewed as an extension of Cox (1972). Common choices for the frailty distribution include the Gamma and the log normal; the Gamma distribution is often preferred due to its algebraic tractability.

Following Elbers and Ridder (1982), identifiability requires that the frailty distribution has a finite mean. A convenient specification is a distribution with mean 1. In this paper, we assume a Gamma distribution with shape $1/\alpha$ and scale $\alpha$, so that the mean is 1 and $\alpha$ measures the amount of heterogeneity among individuals.

Let $V$ denote the frailty variable. With multiplicative frailty, the hazard function for the $i$th individual at time $t$ is

$$h_i(t)=h(t\mid v_i)=v_i{h}_0(t),i=1,\text{…},n,$$ (1)

where $v_1,v_2,\text{…},v_n$ are the individual frailty terms and $h_0(t)$ is the baseline hazard function, common to all individuals. Note that in (1) the individual hazard increases if $v_i>1$ and decreases if $v_i<1$. Conditionally on the frailty term $V$, the model has a proportional hazards structure and may be interpreted as a frailty extension of the Cox proportional hazards model. The standard Cox model is recovered only in the degenerate case $V\equiv 1$, that is, in the absence of unobserved heterogeneity. From (1), the conditional survival function is

$$S(t\mid v_i)=S_0{(t)}^{v_i},$$ (2)

where $S_0(·)$ is the baseline survival function. To obtain the unconditional survival function, eliminating the unobserved frailty, we integrate out $V$. The marginal survival function is

$$S(t)={\mathbb{E}}_V\left[S(t\mid V)\right]={\int }_0^{\infty }{e}^{-H_0(t)v}{f}_V(v)dv=L_V\left[H_0(t)\right],$$ (3)

where $f_V(·)$ is the density of $V$, $H_0(·)$ is the cumulative baseline hazard, and $L_V(·)$ denotes the Laplace transform. For a Gamma distribution with shape $1/\alpha$ and scale $\alpha$, the Laplace transform is

$$L_z(s)={(1+\alpha s)}^{-1/\alpha }.$$ (4)

Evaluating (4) at $s=H_0(t)$ yields the unconditional survival under Gamma frailty,

$$S(t)={\left(1+\alpha H_0(t)\right)}^{-1/\alpha }.$$ (5)

Consequently, from (5), the corresponding density function is

$$f(t)={\left(1+\alpha H_0(t)\right)}^{-(\alpha +1)/\alpha }{h}_0(t),$$ (6)

where $h_0(t)$ is the baseline hazard function.

2.2. Gamma Frailty Zero‐Adjusted Promotion Cure‐Rate Model

Cavenague de Souza (2020) studied a zero‐adjusted cure‐rate model with frailty, also known as zero‐inflated cure‐rate with frailty (ZICRF) , as an alternative for modeling survival data with both a cure fraction and a zero fraction. This model extends the ZICR framework proposed by Louzada et al. (2018). In Cavenague de Souza (2020), Gamma and log‐normal distributions were considered for the frailty and the survival time, respectively. For the analyzed dataset, the authors reported significant heterogeneity, indicating that a zero‐adjusted model with frailty is an attractive option for settings with unobserved heterogeneity.

To model survival data with a cure fraction, a zero fraction, and unobserved heterogeneity, the population survival function of the zero‐adjusted cure‐rate model with frailty, ZICR F, is given by Cavenague de Souza (2020)

$$S_p(t\mid v)=p_1+\left(1-p_0-p_1\right)S_0(t\mid v),$$ (7)

where $p_0$ is the proportion of survival times equal to zero, $p_1$ is the proportion of cured individuals in the population, and $S_0(t\mid v)$ is the survival function for the susceptible subgroup conditional on the frailty variable $v$.

Integrating out the frailty via the Laplace transform yields the unconditional population survival function,

$$S_p(t)=p_1+\left(1-p_0-p_1\right)L_V\left[H_0(t)\right],$$ (8)

where $H_0(t)$ is the cumulative baseline hazard and $L_V[·]$ denotes the Laplace transform associated with the frailty distribution.

Assuming that $V$ follows a Gamma distribution with shape $1/\alpha$ and scale $\alpha$, the unconditional survival function of the ZICR F model is obtained by substituting (4) into (8),

$$S_p(t)=p_1+\left(1-p_0-p_1\right){\left(1+\alpha H_0(t)\right)}^{-1/\alpha }.$$ (9)

As a consequence of (9), the corresponding density is

$$f_p(t)=\left(1-p_0-p_1\right){\left(1+\alpha H_0(t)\right)}^{-(\alpha +1)/\alpha }{h}_0(t).$$

These expressions allow heterogeneity to be modeled through frailty, enabling estimation of frailty parameters and covariate effects on survival. Furthermore, the model in (9) satisfies the following limiting cases:

• If $p_0=0$ and $p_1=0$, one recovers the Gamma frailty survival function in (5).

• If $p_0=0$ and $\alpha \to 0$, the frailty degenerates at 1 and the model reduces to the standard Berkson and Gage mixture cure‐rate model (Berkson and Gage 1952).

2.3. Activation Schemes

In this section, we provide a brief description of activation schemes in the promotion cure‐rate model proposed by Yakovlev et al. (1993); Tsodikov et al. (1996), and Chen et al. (1999), where the modeling is grounded in a biological mechanism.

Following these studies, we assume that the subject under study has $M$ tumor cells competing for metastasis and that the $i$th carcinogenic cell has a random activation time $Z_i$, interpreted as an incubation time until a clinically detectable mass appears. The count $M$ may also be interpreted as the number of cells affected by a bacterial or viral infection. The random variables $Z_i$, $i=1,2,\text{…}$, are assumed independent and identically distributed and independent of $M$, and $M$ follows a Poisson distribution with parameter $\theta$.

To accommodate individuals who are not susceptible to the disease under study, that is, subjects with initial tumor burden equal to zero and, theoretically, unlimited survival time, we adopt $P(Z_0=\infty )=1$. Let $T$ denote the observed time to the event for exposed individuals. In line with (Cooner et al. 2007;;2006), we consider the following specifications:

1. $K=1$: activation of the first latent factor leads to the event, hence

   $$T=Z_{(1,M)}=\min \{Z_1,\text{…},Z_M\}.$$
2. $K=q$: activation occurs at the $q$th latent factor, hence

   $$T=Z_{(q,M)}.$$
3. $K=M$: activation of all latent factors is required, hence

   $$T=Z_{(M,M)}=\max \{Z_1,\text{…},Z_M\}.$$

3. Frailty Zero‐Adjusted Cure‐Rate Model Under Different Activation Schemes

In this section, we formulate the model under three activation schemes. For each individual, there are multiple latent causes competing for the event of interest, with count $M\sim \text{Poisson}(\theta )$. Let $Z_i$ denote the activation time of the $i$th latent cause. Following Section 2, we assume that $Z_1,Z_2,\text{…}$ are independent and identically distributed with distribution function $F_0(t)=1-S_0(t)$, independent of $M$. The observed time is defined according to the activation scheme: $T=\min \{Z_1,\text{…},Z_M\}$ (minimum), $T=\max \{Z_1,\text{…},Z_M\}$ (maximum), or $T=Z_{(K)}$ with $K$ uniformly distributed on $\{1,\text{…},M\}$ (random). These three activation mechanisms are summarized in Table 1.

TABLE 1.

Definition of the observed time $T$ under each activation scheme.

Activation scheme	$T$
Minimum	$T=\left\{\begin{array}{cc}\infty ,\hfill & \text{if}M=0,\hfill \\ Z_{(1,M)}=\min \{Z_1,\text{…},Z_M\},\hfill & \text{if}M\ge 1\hfill \end{array}\right.$
Maximum	$T=\left\{\begin{array}{cc}\infty ,\hfill & \text{if}M=0,\hfill \\ Z_{(M,M)}=\max \{Z_1,\text{…},Z_M\},\hfill & \text{if}M\ge 1\hfill \end{array}\right.$
Random	$T=Z_{(K,M)},K\mid M\sim \text{Uniform}\{1,\text{…},M\},k=1,\text{…},M$

According to Chen et al. (1999), the survival function for the full population, including susceptible and non‐susceptible individuals, is

$$\begin{array}{cc}\hfill S_p(t)& =\mathbb{P}(T>t,M\ge 0)\hfill \\ & =\mathbb{P}(M=0)+\mathbb{P}\left(T>t,M\ge 1\right)\hfill \\ & =e^{-\theta }+\sum _{k=1}^{\infty }\mathbb{P}(T>t\mid M=k)\mathbb{P}(M=k)\hfill \\ & =e^{-\theta }+\sum _{k=1}^{\infty }\mathbb{P}(T>t\mid M=k)\frac{{\theta }^k}{k!}{e}^{-\theta }.\hfill \end{array}$$ (10)

The term $\mathbb{P}(T>t\mid M)$ depends on the activation scheme:

$$\mathbb{P}(T>t\mid M)=\left\{\begin{array}{cc}{S}_0{(t)}^M,\hfill & \text{minimum},\hfill \\ 1-F_0{(t)}^M,\hfill & \text{maximum},\hfill \\ S_0(t),\hfill & \text{random, since}K\mid M\text{is uniform on}\{1,\text{…},M\}.\hfill \end{array}\right.$$

After obtaining $S_p(t)$, the density follows as $f_p(t)=-\frac{d}{dt}{S}_p(t)$. The resulting survival and density functions (without frailty) are summarized in Table 2.

TABLE 2.

Population survival $S_p(t)$ and density $f_p(t)$ under each activation scheme (no frailty).

Activation scheme	$S_p(t)$	$f_p(t)$
Minimum	$\exp \{-\theta F_0(t)\}$	$\theta f_0(t)\exp \{-\theta F_0(t)\}$
Maximum	$1-\exp \{-\theta [1-F_0(t)]\}+e^{-\theta }$	$\theta f_0(t)\exp \{-\theta [1-F_0(t)]\}$
Random	$e^{-\theta }+(1-e^{-\theta })S_0(t)$	$(1-e^{-\theta })f_0(t)$

Using $F_0(t)=1-S_0(t)$ and substituting $S_0(t)$ by the Gamma frailty survival in (5), we obtain the population densities and survivals with frailty shown in Table 3.

TABLE 3.

Population density $f_p(t)$ and survival $S_p(t)$ with Gamma frailty for each activation scheme.

Activation scheme	$f_p(t)$	$S_p(t)$
Minimum	$\theta {(1+\alpha H_0(t))}^{-(\alpha +1)/\alpha }{h}_0(t)\exp \left\{-\theta \left[1-{(1+\alpha H_0(t))}^{-1/\alpha }\right]\right\}$	$\exp \left\{-\theta \left[1-{(1+\alpha H_0(t))}^{-1/\alpha }\right]\right\}$
Maximum	$\theta {(1+\alpha H_0(t))}^{-(\alpha +1)/\alpha }{h}_0(t)\exp \left\{-\theta {(1+\alpha H_0(t))}^{-1/\alpha }\right\}$	$1-\exp \left\{-\theta {(1+\alpha H_0(t))}^{-1/\alpha }\right\}+e^{-\theta }$
Random	$(1-e^{-\theta }){(1+\alpha H_0(t))}^{-(\alpha +1)/\alpha }{h}_0(t)$	$e^{-\theta }+(1-e^{-\theta }){(1+\alpha H_0(t))}^{-1/\alpha }$

The survival functions $S_p(t)$ above are improper, with ${\lim }_{t\to \infty }{S}_p(t)=e^{-\theta }>0$, the cured fraction, irrespective of the activation scheme. Following Chen et al. (1999), we restrict attention to exposed individuals, $M\ge 1$, and define

$$S_p^{\ast }(t)=\mathbb{P}(T>t\mid M\ge 1).$$

For the ZICR model (no frailty), the survival and density for the non‐cured group are shown in Table 4.

TABLE 4.

Survival $S_p^{\ast }(t)$ and density $f_p^{\ast }(t)$ for the non‐cured group (no frailty).

Activation scheme	$S_p^{\ast }(t)$	$f_p^{\ast }(t)$
Minimum	${\displaystyle \frac{\exp \{-\theta F_0(t)\}-e^{-\theta }}{1-e^{-\theta }}}$	$\left({\displaystyle \frac{\exp \{-\theta F_0(t)\}}{1-e^{-\theta }}}\right)\theta f_0(t)$
Maximum	${\displaystyle \frac{1-\exp \{-\theta [1-F_0(t)]\}}{1-e^{-\theta }}}$	$\left({\displaystyle \frac{\exp \{-\theta [1-F_0(t)]\}}{1-e^{-\theta }}}\right)\theta f_0(t)$
Random	$S_0(t)$	$f_0(t)$

Substituting $S_0(t)$ by ${(1+\alpha H_0(t))}^{-1/\alpha }$ yields $S_p^{\ast }(t)$ and $f_p^{\ast }(t)$ for the Gamma frailty case, summarized in Table 5.

TABLE 5.

Conditional density $f_p^{\ast }(t)$ and survival $S_p^{\ast }(t)$ with Gamma frailty under each activation scheme.

Activation scheme	$f_p^{\ast }(t)$	$S_p^{\ast }(t)$
Minimum	${\displaystyle \frac{\exp \left\{-\theta \left[1-{(1+\alpha H_0(t))}^{-1/\alpha }\right]\right\}}{1-e^{-\theta }}}\theta {(1+\alpha H_0(t))}^{-(\alpha +1)/\alpha }{h}_0(t)$	${\displaystyle \frac{\exp \left\{-\theta \left[1-{(1+\alpha H_0(t))}^{-1/\alpha }\right]\right\}-e^{-\theta }}{1-e^{-\theta }}}$
Maximum	${\displaystyle \frac{\exp \left\{-\theta {(1+\alpha H_0(t))}^{-1/\alpha }\right\}}{1-e^{-\theta }}}\theta {(1+\alpha H_0(t))}^{-(\alpha +1)/\alpha }{h}_0(t)$	${\displaystyle \frac{1-\exp \left\{-\theta {(1+\alpha H_0(t))}^{-1/\alpha }\right\}}{1-e^{-\theta }}}$
Random	${(1+\alpha H_0(t))}^{-(\alpha +1)/\alpha }{h}_0(t)$	${(1+\alpha H_0(t))}^{-1/\alpha }$

The functions in Table 5 are proper, satisfying $S_p^{\ast }(0)=1$ and $S_p^{\ast }(\infty )=0$. In what follows, we adopt the zero‐adjusted promotion cure‐rate specification

$$S_p(t)=p_1+(1-p_0-p_1)S_p^{\ast }(t),t\ge 0,$$ (11)

where $S_p^{\ast }(t)$ and $f_p^{\ast }(t)$ depend on the activation scheme as above. Different choices for the latent distribution of $Z_j$ under each scheme generate different parametric families; in this paper, we assume a Weibull baseline for $Z_j$.

4. Inference

Incorporating activation schemes for all competing causes into the zero‐adjusted promotion cure‐rate model allows us to decompose the population into three parts: the fraction that fails at the initial time ($p_0$), the fraction that is not susceptible and therefore cured ($p_1$), and the susceptible fraction $1-p_0-p_1$ governed by a proper survival function $S_p^{\ast }(t)$. We adopt the specification in (11), and model $p_0$ and $p_1$ via a multinomial logit so that $p_0,p_1,1-p_0-p_1\in (0,1)$ (see Pereira et al. 2013; Hosmer et al. 2013; Louzada et al. 2018). Let ${\mathbf{x}}_{1i}$ and ${\mathbf{x}}_{2i}$ be covariate vectors linked to the zero mass and the cure fraction, with coefficients ${\mathit{\beta }}_1$ and ${\mathit{\beta }}_2$. Then

$$\begin{array}{cc}& p_{0i}={\displaystyle \frac{\exp \{{\mathbf{x}}_{1i}^{\top }{\mathit{\beta }}_1\}}{1+\exp \{{\mathbf{x}}_{1i}^{\top }{\mathit{\beta }}_1\}+\exp \{{\mathbf{x}}_{2i}^{\top }{\mathit{\beta }}_2\}}},\hfill \\ & p_{1i}={\displaystyle \frac{\exp \{{\mathbf{x}}_{2i}^{\top }{\mathit{\beta }}_2\}}{1+\exp \{{\mathbf{x}}_{1i}^{\top }{\mathit{\beta }}_1\}+\exp \{{\mathbf{x}}_{2i}^{\top }{\mathit{\beta }}_2\}}}.\hfill \end{array}$$

Equivalently, writing $p_{0i}=e^{-{\nu }_i}$ and $p_{1i}=e^{-{\theta }_i}$ with ${\nu }_i,{\theta }_i>0$ yields

$$\begin{array}{cc}& {\nu }_i=-\log \left({\displaystyle \frac{\exp \{{\mathbf{x}}_{1i}^{\top }{\mathit{\beta }}_1\}}{1+\exp \{{\mathbf{x}}_{1i}^{\top }{\mathit{\beta }}_1\}+\exp \{{\mathbf{x}}_{2i}^{\top }{\mathit{\beta }}_2\}}}\right),\hfill \\ & {\theta }_i=-\log \left({\displaystyle \frac{\exp \{{\mathbf{x}}_{2i}^{\top }{\mathit{\beta }}_2\}}{1+\exp \{{\mathbf{x}}_{1i}^{\top }{\mathit{\beta }}_1\}+\exp \{{\mathbf{x}}_{2i}^{\top }{\mathit{\beta }}_2\}}}\right).\hfill \end{array}$$

To complete the specification of $S_p^{\ast }(t)$ under each activation scheme, we take the baseline for the latent times $Z_j$ to be Weibull with shape ${\alpha }_{1i}>0$ and scale ${\alpha }_{2i}>0$,

$$\begin{array}{cc}& F_0(z;{\alpha }_{1i},{\alpha }_{2i})=1-\exp \left\{-{\left({\displaystyle \frac{z}{{\alpha }_{2i}}}\right)}^{{\alpha }_{1i}}\right\},f_0(z;{\alpha }_{1i},\hfill \\ & {\alpha }_{2i})={\displaystyle \frac{{\alpha }_{1i}}{{\alpha }_{2i}}}{\left({\displaystyle \frac{z}{{\alpha }_{2i}}}\right)}^{{\alpha }_{1i}-1}\exp \left\{-{\left({\displaystyle \frac{z}{{\alpha }_{2i}}}\right)}^{{\alpha }_{1i}}\right\},\hfill \end{array}$$

and link these parameters to covariates ${\mathbf{x}}_{3i}$ and ${\mathbf{x}}_{4i}$ through

$${\alpha }_{1i}=\exp \{{\mathbf{x}}_{3i}^{\top }{\mathit{\beta }}_3\},{\alpha }_{2i}=\exp \{{\mathbf{x}}_{4i}^{\top }{\mathit{\beta }}_4\}.$$

When frailty is present, the Gamma variance parameter $\alpha >0$ enters $S_p^{\ast }(t)$ and $f_p^{\ast }(t)$ (see Table 5).

Finally, for observation $i$ the mixture form is

$$S_{p,i}(t)=e^{-{\theta }_i}+\left(1-e^{-{\nu }_i}-e^{-{\theta }_i}\right)S_{p,i}^{\ast }(t),$$

where $S_{p,i}^{\ast }(t)$ and $f_{p,i}^{\ast }(t)$ depend on the chosen activation scheme, as summarized in Table 5.

4.1. Likelihood Function

We estimate the parameters by maximum likelihood. Each subject belongs to one of three subgroups: (i) zero time, $t_i=0$, (ii) cured, and (iii) susceptible. Let the observed data be $\mathbb{D}={\{(t_i,{\delta }_i,{\mathbf{x}}_{1i},{\mathbf{x}}_{2i},{\mathbf{x}}_{3i},{\mathbf{x}}_{4i})\}}_{i=1}^n$, where ${\delta }_i=1$ denotes an observed event and ${\delta }_i=0$ denotes right censoring. The individual likelihood contribution is

$$\left\{\begin{array}{cc}{p}_{0i},\hfill & \text{if}{t}_i=0,\hfill \\ \left(1-p_{0i}-p_{1i}\right)f_{p,i}^{\ast }(t_i),\hfill & \text{if}{t}_i>0\text{and}{\delta }_i=1,\hfill \\ p_{1i}+\left(1-p_{0i}-p_{1i}\right)S_{p,i}^{\ast }(t_i),\hfill & \text{if}{t}_i>0\text{and}{\delta }_i=0,\hfill \end{array}\right.$$ (12)

where $S_{p,i}^{\ast }(t)$ and $f_{p,i}^{\ast }(t)$ are the proper survival and density for the susceptible group under the selected activation scheme.

Let $\mathit{\zeta }$ collect all free parameters. With a Weibull baseline and Gamma frailty, we write

$$\mathit{\zeta }={\left({\mathit{\beta }}_1^{\top },{\mathit{\beta }}_2^{\top },{\mathit{\beta }}_3^{\top },{\mathit{\beta }}_4^{\top },\alpha \right)}^{\top },$$

where the Weibull shape and scale are determined by ${\mathit{\beta }}_3$ and ${\mathit{\beta }}_4$, respectively, and $\alpha$ is the frailty variance parameter. The full likelihood is

$$\begin{array}{cc}& L(\mathit{\zeta };\mathbb{D})\propto \prod _{i:t_i=0}{p}_{0i}\prod _{i:t_i>0}\hfill \\ & \times \left\{{\left[(1-p_{0i}-p_{1i})f_{p,i}^{\ast }(t_i)\right]}^{{\delta }_i}{\left[p_{1i}+(1-p_{0i}-p_{1i})S_{p,i}^{\ast }(t_i)\right]}^{1-{\delta }_i}\right\}.\hfill \end{array}$$

The MLE $\widehat{\mathit{\zeta }}$ solves the score equations $U(\mathit{\zeta })=\partial \ell (\mathit{\zeta })/\partial \mathit{\zeta }=\mathbf{0}$. Since closed‐form solutions are not available, we use a quasi‐Newton method (Shanno 1970). In practice, all optimizations were carried out in R using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) implementation in the optim function. In the simulation study, we initialized the algorithm at the true parameter values with small random perturbations, and in the insulin data application we used estimates from simpler submodels (without frailty and/or without zero inflation) as starting values. Convergence was monitored by the default optim criteria, requiring small relative changes in both the log‐likelihood and the parameter vector, and, when necessary, by checking that different starting points led to essentially identical solutions. For the models considered here, computing times were on the order of a few seconds on a standard laptop.

Under standard regularity and for sufficiently large $n$, Wald inference is based on the inverse observed information,

$$\widehat{\mathrm{Var}}(\widehat{\mathit{\zeta }})={\left[-\frac{{\partial }^2\ell (\mathit{\zeta })}{\partial \mathit{\zeta }\partial {\mathit{\zeta }}^{\top }}\right]}_{\mathit{\zeta }=\widehat{\mathit{\zeta }}}^{-1}$$

(see Migon et al. 2014; Ospina and Ferrari 2012). Thus, approximate $(1-\alpha )\times 100\%$ confidence intervals are

$$\begin{array}{cc}& {\widehat{\alpha }}_1\pm z_{1-\alpha /2}\mathrm{se}({\widehat{\alpha }}_1),{\widehat{\alpha }}_2\pm z_{1-\alpha /2}\mathrm{se}({\widehat{\alpha }}_2),\widehat{\alpha }\pm z_{1-\alpha /2}\mathrm{se}(\widehat{\alpha }),\hfill \\ & {\widehat{\beta }}_{·}\pm z_{1-\alpha /2}\mathrm{se}({\widehat{\beta }}_{·}),\hfill \end{array}$$

where $z_{1-\alpha /2}$ is the standard normal quantile and ${\widehat{\beta }}_{·}$ denotes any component of ${\widehat{\mathit{\beta }}}_1,{\widehat{\mathit{\beta }}}_2,{\widehat{\mathit{\beta }}}_3,{\widehat{\mathit{\beta }}}_4$.

5. Simulation Studies

This section has three parts. First, we describe the common data‐generation mechanism used in all simulations. Second, we assess the finite‐sample performance of the MLE when the fitted model is correctly specified. Third, we evaluate misspecification and practical identifiability, including whether information criteria can distinguish the true activation scheme when the sample size changes.

5.1. Data Generation

In the first simulation experiment, we assess the finite‐sample performance of the MLE for the proposed models under different activation schemes and sample sizes $n\in \{100,250,500,750,1000\}$. To evaluate covariate effects, we generate a single binary covariate $X\sim \mathrm{Bernoulli}(0.5)$ indicating group membership. In these simulations, the four covariate vectors used in the linear predictors are scalar and take the same value, that is, ${\mathbf{x}}_{1i}={\mathbf{x}}_{2i}={\mathbf{x}}_{3i}={\mathbf{x}}_{4i}=x_i$, where $x_i\in \{0,1\}$. In the correctly specified MLE experiment, the failure time follows a Weibull distribution with parameters $({\alpha }_{1i},{\alpha }_{2i})$. We link the zero inflation, cure fraction, and Weibull parameters to $X$ by

$${\nu }_i=-\log \left(\frac{\exp \{{\beta }_{10}+x_i{\beta }_{11}\}}{1+\exp \{{\beta }_{10}+x_i{\beta }_{11}\}+\exp \{{\beta }_{20}+x_i{\beta }_{21}\}}\right),$$

$${\theta }_i=-\log \left(\frac{\exp \{{\beta }_{20}+x_i{\beta }_{21}\}}{1+\exp \{{\beta }_{10}+x_i{\beta }_{11}\}+\exp \{{\beta }_{20}+x_i{\beta }_{21}\}}\right),$$ (13)

$${\alpha }_{1i}=\exp \{{\beta }_{30}+x_i{\beta }_{31}\},{\alpha }_{2i}=\exp \{{\beta }_{40}+x_i{\beta }_{41}\}.$$

Algorithm 1 generates samples from the zero‐adjusted promotion cure‐rate model under a chosen activation scheme and under the distributional specifications required by each simulation study. For the MLE performance study, $G$ is Weibull and $H$ is Gamma. For the misspecification study in Subsection 5.3, the same algorithm is used after replacing $G$ by Gompertz or generalized exponential distributions and/or replacing $H$ by an inverse Gaussian frailty.

ALGORITHM 1. Generator of random samples from the zero‐adjusted promotion cure‐rate model under a general activation scheme.

1:	Choose the sample size $n$, the activation scheme $A\in \{\mathrm{minimum},\mathrm{maximum},\mathrm{random}\}$, the latent activation‐time distribution $G$, the frailty distribution $H$, and the censoring distribution.
2:	Set the parameters for $p_{0i}$, $p_{1i}$, $G$, and, when present, $H$. In the correctly specified MLE study, $G$ is Weibull, and $H$ is Gamma.
3:	for $i=1,\text{…},n$ do
4:	Generate $x_i\sim \mathrm{Bernoulli}(0.5)$ and compute $p_{0i}=e^{-{\nu }_i}$, $p_{1i}=e^{-{\theta }_i}$ and the parameters of $G$ and $H$ for individual $i$.
5:	Generate $u_i\sim \mathrm{Uniform}(0,1)$.
6:	if $u_i\le p_{0i}$ then
7:	Set $s_i=0$.
8:	else if $u_i>1-p_{1i}$ then
9:	Set $s_i=\infty$.
10:	else
11:	Set $s_i=F_{A,G,H,i}^{\ast -1}\{(u_i-p_{0i})/(1-p_{0i}-p_{1i})\}$, equivalently the root of $F_i(s_i)-u_i=0$, where $F_i(t)=p_{0i}+(1-p_{0i}-p_{1i})F_{A,G,H,i}^{\ast }(t)$.
12:	end if
13:	end for
14:	Choose a finite censoring upper bound using the finite generated $s_i$ values and generate $w_i$ from the censoring distribution.
15:	Set $t_i=\min (s_i,w_i)$ and ${\delta }_i=\mathbb{I}(t_i<w_i)$ for $i=1,\text{…},n$.

5.2. Finite‐Sample Performance of the MLE

We consider three parameter scenarios shown in Table 6.

TABLE 6.

Parameter values under different scenarios.

Parameters	Scenarios
	1	2	3
${\beta }_{10}$	$-2.00$	$-0.50$	$-0.50$
${\beta }_{11}$	0.50	1.50	0.50
${\beta }_{20}$	$-1.00$	$-0.75$	$-1.00$
${\beta }_{21}$	0.75	3.00	1.00
${\beta }_{30}$	0.50	$-2.00$	$-0.75$
${\beta }_{31}$	0.50	1.50	1.00
${\beta }_{40}$	1.50	$-1.25$	1.25
${\beta }_{41}$	2.00	1.00	1.00
$\alpha$	$(0.25,0.50)$	$(0.25,0.50)$	$(0.25,0.50)$

From Table 6, scenario 1 yields $p_0\approx 9\%$ for $x=0$ and $11\%$ for $x=1$, while $p_1\approx 24.47\%$ and $38.90\%$, respectively. In scenario 2, $p_0\approx 18.63\%$ and $21.19\%$, with $p_1\approx 30.72\%$ and $57.61\%$. In scenario 3, $p_0\approx 30.72\%$ and $33.33\%$, with $p_1\approx 18.63\%$ and $33.33\%$. These values represent low, medium, and high levels of zero inflation and cure fraction. Scenario 1 has few early failures, Scenario 2 is intermediate, and Scenario 3 has many early failures.

For each combination of scenario, activation scheme, and sample size, we run $1000$ Monte Carlo replications. We compute MLEs and standard errors and summarize performance by bias, root mean squared error (RMSE), and empirical coverage probability (CP) of nominal $95\%$ Wald intervals. We use

$$\mathrm{RMSE}(\zeta )=\sqrt{\mathrm{Bias}{(\zeta )}^2+\mathrm{Var}(\zeta )},\mathrm{Bias}(\zeta )=\widehat{\zeta }-\zeta .$$

Due to Monte Carlo variability, CP will not be exactly 0.95; with $1000$ replications, the expected $95\%$ binomial interval is approximately $[0.936,0.964]$.

Standard errors for $p_0$ and $p_1$ are obtained by the delta method with first‐order Taylor approximation. Optimization uses optim in R with the BFGS algorithm; see R Core Team RStudio Team (2020).

Figures 1, 2, 3 display bias, RMSE, and CP across sample sizes for the maximum, random, and minimum activation schemes, respectively. Main findings are as follows:

1. Across schemes and scenarios, RMSEs of the MLEs shrink toward zero with $n$, indicating consistency. In scenario 1 under the maximum scheme (Figure 1), the parameter ${\beta }_{21}$ needs about $n\ge 500$ for stable estimation. Under the minimum scheme (Figure 3), ${\beta }_{11}$ and ${\beta }_{21}$ in Scenarios 1 and 2 also require larger $n$.

2. Biases tend to zero as $n$ increases, and CPs are generally close to $95\%$. Under the random scheme (Figure 2), ${\beta }_{31}$ approaches nominal CP more slowly than ${\beta }_{40}$. Scenario 3 typically attains nominal CP from $n\approx 250$.

3. Scenarios with larger cured and zero fractions (Scenarios 2 and 3) improve estimation of the regression parameters linked to $p_0=e^{-\nu }$ and $p_1=e^{-\theta }$, yielding smaller bias and better CP than scenario 1, due to more information in the zero and cured components.

4. Conversely, in Scenario 1, with fewer zeros and cured individuals and thus longer observed times, estimation of the Weibull parameters $({\alpha }_1,{\alpha }_2)$ shows smaller bias and improved CP relative to Scenarios 2 and 3.

FIGURE 1.

Finite‐sample performance of the MLE under the maximum activation scheme. Each panel shows the bias, RMSE, and empirical coverage of nominal 95% Wald confidence intervals for the model parameters as a function of the sample size $n$. Colors identify the three simulation scenarios in Table 6: Scenario 1 (dark green), Scenario 2 (dark blue), and Scenario 3 (dark red); solid and dashed lines correspond to $\alpha =0.25$ and $\alpha =0.50$, respectively. All results are based on 1000 Monte Carlo replications for each combination of $n$, scenario, and frailty variance.

FIGURE 2.

Finite‐sample performance of the MLE under the random activation scheme. Each panel shows the bias, RMSE, and empirical coverage of nominal 95% Wald confidence intervals for the model parameters as a function of the sample size $n$. Colors identify the three simulation scenarios in Table 6: Scenario 1 (dark green), Scenario 2 (dark blue), and Scenario 3 (dark red); solid and dashed lines correspond to $\alpha =0.25$ and $\alpha =0.50$, respectively. All results are based on 1000 Monte Carlo replications for each combination of $n$, scenario, and frailty variance.

FIGURE 3.

Finite‐sample performance of the MLE under the minimum activation scheme. Each panel shows the bias, RMSE, and empirical coverage of nominal 95% Wald confidence intervals for the model parameters as a function of the sample size $n$. Colors identify the three simulation scenarios in Table 6: Scenario 1 (dark green), Scenario 2 (dark blue), and Scenario 3 (dark red); solid and dashed lines correspond to $\alpha =0.25$ and $\alpha =0.50$, respectively. All results are based on 1000 Monte Carlo replications for each combination of $n$, scenario, and frailty variance.

The conclusions from this MLE experiment concern estimation accuracy under correct specification. First, sample sizes in the range $n\ge 250$–500 are typically desirable for stable estimation of covariate effects on the zero‐inflation and cure components when the corresponding proportions are small. Second, moderate zero and cure fractions provide more information about the multinomial logit parameters, whereas datasets with fewer zeros and cured individuals are more informative about the Weibull baseline parameters. The separate question of whether the activation scheme itself can be distinguished from the data is addressed in Subsection 5.3.

5.3. Evaluating Misspecification and Activation‐Scheme Identifiability

This subsection addresses a different question from the MLE experiment: whether the model‐selection procedure can recover the correct distributional specification or activation mechanism. The proposed model includes several components: latent heterogeneity represented by a Gamma frailty, a Poisson number of latent causes, an activation scheme, and a Weibull distribution for the activation times, where all components were specified without covariates. We, therefore, considered misspecification scenarios, following the strategy of Louzada et al. (2025), in which the fitted model is compared with alternatives that use different activation‐time distributions and/or a different frailty distribution. The latent cause count was kept Poisson in all scenarios to preserve the competing‐risks interpretation.

Overall, six scenarios were considered:

• Scenario I: Generate samples from the model with Gamma frailty ($\alpha =0.25$) and a Weibull distribution (${\alpha }_1=2$, ${\alpha }_2=1$) for the activation times and fit the data using the same frailty, but a Gompertz and a generalized exponential distribution for the activation times.

• Scenario II: Generate samples from the model with Gamma frailty ($\alpha =0.25$) and a Gompertz distribution (${\alpha }_1=2$, ${\alpha }_2=1$) for the activation times and fit the data using the same frailty, but a Weibull and a generalized exponential distribution for the activation times.

• Scenario III: Generate samples from the model with Gamma frailty ($\alpha =0.25$) and a generalized exponential distribution (${\alpha }_1=2$, ${\alpha }_2=1$) for the activation times and fit the data using the same frailty, but a Weibull and a Gompertz distribution for the activation times.

• Scenario IV: Generate samples from the model with Gamma frailty ($\alpha =0.75$) and a Weibull distribution (${\alpha }_1=2$, ${\alpha }_2=1$) for the activation times and fit the data using the same activation times, but using an inverse Gaussian distribution for the frailty.

• Scenario V: Generate samples from the model with Gamma frailty ($\alpha =0.75$) and a Weibull distribution (${\alpha }_1=2$, ${\alpha }_2=1$) for the activation times and fit the data using an inverse Gaussian distribution for the frailty and a generalized exponential distribution for the activation times.

• Scenario VI: Generate samples from the model with Gamma frailty ($\alpha =0.75$) and a Weibull distribution (${\alpha }_1=2$, ${\alpha }_2=1$) for the activation times and fit the data using an inverse Gaussian distribution for the frailty and a Gompertz distribution for the activation times.

For each scenario 1000 samples were generated, and the Akaike Information Criterion (AIC) (Berkson and Gage 1952) was calculated for the correctly specified model and for the competing misspecified models. Table 7 reports the proportion of replications in which AIC selected the correctly specified model.

TABLE 7.

Proportion of samples in which AIC selected the correctly specified model in each misspecification scenario.

$n$	I	II	III	IV	V	VI
50	0.390	0.728	0.398	0.561	0.596	0.851
150	0.597	0.798	0.554	0.650	0.583	0.995
300	0.743	0.865	0.611	0.777	0.624	1.000
500	0.841	0.936	0.642	0.860	0.649	1.000
1000	0.902	0.974	0.690	0.954	0.743	1.000

For activation‐time distributions, Scenario I shows that Weibull activation times are harder to recover in small samples, but the selection probability increases from 0.390 at $n=50$ to 0.902 at $n=1000$. Scenario II shows that a Gompertz activation‐time model is selected with probability 0.728 even at $n=50$, increasing to 0.974 at $n=1000$. Scenario III indicates that the generalized exponential distribution is the hardest to distinguish among the three alternatives, with a correct selection probability of 0.69 at $n=1000$. For frailty misspecification, Scenario IV shows that distinguishing Gamma from inverse Gaussian frailty can be difficult in small samples but improves markedly as $n$ increases.

In Scenarios V and VI, both the frailty and activation‐time specifications are changed. Scenario V shows that when the activation time follows a generalized exponential distribution, the same difficulty seen in Scenario III persists. In contrast, Scenario VI shows that the original model is easily distinguished when the activation time follows a Gompertz distribution, with correct selection once the sample size exceeds 300 observations.

To provide direct evidence about activation‐scheme identifiability, we also generated data from the same general model but under the maximum activation scheme and compared it with the corresponding model under the minimum activation scheme using AIC. Figure 4 shows the proportion of replications in which the true activation scheme was selected as the sample size increases, based on 5000 replications. The correct‐selection rate increases substantially with $n$, but it is lower for small and moderate samples. This provides evidence for the practical finding that activation schemes may be weakly identifiable in finite samples when they induce similar survival patterns. Thus, the conclusions from this subsection are about robustness and identifiability rather than MLE accuracy. Information criteria are useful for comparing activation schemes and distributional assumptions, but their ability to select the correct specification depends on sample size and on how distinct the induced survival patterns are. In applications where competing schemes produce similar fitted curves, information criteria should therefore be complemented with substantive knowledge about the latent activation process and with diagnostic checks.

FIGURE 4.

Simulation results comparing activation schemes using 5000 replications. The x‐axis represents the sample size, while the y‐axis indicates the rate at which AIC selected the true activation scheme.

6. Application With Insulin Dataset

The dataset analyzed here, referred to as the insulin dataset, was introduced by Calsavara, Rodrigues, Rocha, Louzada, et al. (2019). It comprises retrospective times to initiation of insulin therapy for 390 pregnant women diagnosed with gestational diabetes mellitus (GDM) before 24 weeks of pregnancy. Follow‐up occurred during prenatal care at Hospital das Clínicas, University of São Paulo, Brazil, from 2012 to 2015. The GDM diagnosis was based on abnormal fasting plasma glucose in the first trimester (92–125 mg/dL), and insulin was started when glycemic targets were not achieved (fasting glucose $>$ 95 mg/dL and 1 h postprandial glucose $>$ 140 mg/dL), according to Souza et al. (2019). All women received nutritional and physical activity counseling and were instructed in self‐monitoring of blood glucose. Some women initiated insulin within the first 7 days, before the counseling program could take effect, so their survival times were recorded as zero. Conversely, a subset did not require insulin during pregnancy, characterizing a cured fraction.

This application matches a setting with a latent number of causal factors, as outlined in the Introduction. Accordingly, we consider three activation schemes. If a single causal factor is sufficient to trigger insulin use, the minimum scheme applies; if all factors must occur, the maximum scheme applies; otherwise, a random activation scheme is appropriate. The application uses the no‐frailty special case ($\alpha \to 0$); the Gamma frailty component is evaluated in the simulation studies.

We investigate whether the following covariates affect the time from the first obstetric visit to insulin initiation: family history of diabetes (0: no, $n=143$; 1: yes, $n=247$), prior history of GDM (0: primigravida, $n=94$; 1: no, $n=249$; 2: yes, $n=47$), smoking history (0: no, $n=356$; 1: yes, $n=34$), prior fetal macrosomia (0: no, $n=356$; 1: yes, $n=34$), prior chronic hypertension (0: no, $n=282$; 1: yes, $n=108$), and pre‐pregnancy body mass index (BMI) category (0: normal, BMI $\le 25$, $n=84$; 1: overweight, $25<\mathrm{BMI}\le 30$, $n=120$; 2: obesity, $\mathrm{BMI}>30$, $n=186$). Our goal is to assess whether these variables influence the probability of insulin use at the first visit and the probability of cure. The parameters $p_0$ and $p_1$ are linked to covariates by

$$\begin{array}{cc}& p_0=\frac{\exp ({\mathbf{x}}_i^{\top }{\mathit{\beta }}_1)}{1+\exp ({\mathbf{x}}_i^{\top }{\mathit{\beta }}_1)+\exp ({\mathbf{x}}_i^{\top }{\mathit{\beta }}_2)},\hfill \\ & p_1=\frac{\exp ({\mathbf{x}}_i^{\top }{\mathit{\beta }}_2)}{1+\exp ({\mathbf{x}}_i^{\top }{\mathit{\beta }}_1)+\exp ({\mathbf{x}}_i^{\top }{\mathit{\beta }}_2)},\hfill \end{array}$$

where ${\mathbf{x}}_i^{\top }=(x_{1i},x_{2i},x_{3i},x_{4i},x_{5i},x_{6i})$ for $i=1,\text{…},390$. Here, ${\mathit{\beta }}_1^{\top }=({\beta }_{10},{\beta }_{11},{\beta }_{12},{\beta }_{13},{\beta }_{14},{\beta }_{15},{\beta }_{16})$ are coefficients for the zero inflation component, and ${\mathit{\beta }}_2^{\top }=({\beta }_{20},{\beta }_{21},{\beta }_{22},{\beta }_{23},{\beta }_{24},{\beta }_{25},{\beta }_{26})$ are coefficients for the cure fraction.

We first fitted 18 models, one per covariate and activation scheme (six covariates by three schemes), using a Weibull baseline for time in all models. Results are reported in Tables A1 and A2. The three activation schemes yield very similar AIC and Bayesian Information Criterion (BIC) values, indicating that, for these data, the overall goodness‐of‐fit is not highly sensitive to the choice of activation mechanism. Nonetheless, the random activation scheme consistently provides the smallest AIC/BIC in almost all comparisons, and we, therefore, adopt it as our preferred specification. This choice is also clinically appealing, since insulin indication in gestational diabetes is known to be multifactorial and may require different combinations of factors across patients (Souza et al. 2019).

From the single covariate analyses, family history of diabetes, prior hypertension, and pre‐pregnancy BMI are important for explaining the cure fraction. In contrast, no individual risk factor shows a statistically significant association with insulin use at the first visit, as reflected in the ${\beta }_{11}$ estimates. Tables A1 and A2 also report point estimates and $95\%$ confidence intervals for $p_0$ and $p_1$. Standard errors for $p_0$ and $p_1$ were obtained via the delta method with first‐order Taylor approximation. For all activation schemes, both proportions are statistically relevant in the fitted models.

We then fitted a full model including all six covariates under each activation scheme. Estimates for these models are presented in Table A3. Prior fetal macrosomia is significant for explaining insulin use within the first 7 days, whereas family history of diabetes and smoking history primarily affect the cure component. For categorical covariates with three levels (e.g., pre‐pregnancy BMI and prior GDM), some activation schemes lead to slightly different patterns of statistical significance across categories. In these cases, we focus our substantive interpretation on the best‐fitting scheme (random activation) and on the direction and magnitude of the estimated effects, treating discrepancies across schemes as an indication of residual model uncertainty rather than as conflicting evidence.

Figure 5 compares, for selected covariates, the Kaplan–Meier estimates with the fitted population survival functions under the maximum, minimum, and random activation schemes. Table 8 reports the final selected model, with prior fetal macrosomia linked to $p_0$ and history of diabetes and smoking linked to $p_1$. The selected covariates are jointly relevant to describe early insulin use and the cure fraction across schemes. Prior fetal macrosomia has a positive effect (${\beta }_{11}>0$), increasing the probability of insulin at the first visit. History of diabetes shows a negative effect on the cure fraction (${\beta }_{22}<0$), indicating a lower chance of remaining insulin free during pregnancy. The QQ plot of the normalized randomized quantile residuals in Figure A1 supports the adequacy of the final specification.

FIGURE 5.

Kaplan–Meier estimates and fitted survival curves from the zero‐adjusted promotion cure‐rate model under the minimum, maximum, and random activation schemes, stratified by prior history of GDM, smoking, prior fetal macrosomia, prior chronic hypertension, pre‐pregnancy BMI, and family history of diabetes.

TABLE 8.

Maximum likelihood estimates (MLE), standard error (SE), $95\%$ confidence interval (CI), and information criteria (AIC, BIC) for the final zero‐adjusted promotion cure‐rate model under the three activation mechanisms, insulin dataset.

	Minimum		Maximum		Random
Parameter	MLE	SE	MLE	SE	MLE	SE
${\beta }_{10}$	$-0.770$	0.197	$-0.769$	0.197	$-0.769$	0.197
${\beta }_{11}$ (Prior macrosomia)	1.118	0.428	1.111	0.428	1.115	0.428
${\beta }_{20}$	1.582	0.207	1.586	0.209	1.584	0.208
${\beta }_{21}$ (Smoking)	$-0.447$	0.368	$-0.493$	0.369	$-0.473$	0.369
${\beta }_{22}$ (History diabetes)	$-0.675$	0.235	$-0.674$	0.235	$-0.674$	0.235
AIC	1187.2404		1187.2572		1187.1498
BIC	1238.8003		1238.8171		1238.7097

Across all covariate patterns, the three fitted curves closely follow the empirical estimates and lie very near to each other. Thus, the conclusion from the application is that, for this dataset, the observable information is not sufficient to strongly favor one activation mechanism in terms of goodness‐of‐fit. The random‐activation model has the smallest AIC and BIC, but the differences are small; consequently, the activation schemes are best interpreted here as a sensitivity analysis for the latent competing‐risks structure. Importantly, the estimated covariate effects and the substantive conclusions about early insulin initiation and the cure fraction remain essentially unchanged across schemes. This application‐specific conclusion is distinct from the simulation conclusion in Subsection 5.3, which studies when activation schemes become distinguishable as sample size and distributional separation increase.

7. Concluding Remarks

This paper proposed a zero‐adjusted promotion cure‐rate regression framework with Gamma frailty under three activation schemes, minimum, maximum, and random. The construction unifies early failures and long‐term survivors within a single likelihood, and it nests relevant models as special cases, for example, the Berkson and Gage mixture cure model when $p_0=0$ and $\alpha \to 0$ (Berkson and Gage 1952). Closed‐form expressions for survival and density functions were derived for each activation mechanism, which facilitates inference and interpretation, and the parameterization allows covariates to act on excess zeros, on the cure fraction, and on the failure time distribution.

The Monte Carlo study showed that maximum likelihood estimation performs well across activation schemes. Bias and RMSE decreased with sample size, and empirical coverage was close to the nominal level in most scenarios. The study also highlighted practical aspects of design; for instance, larger samples may be required for stable estimation of some covariate effects on $p_0$ and $p_1$ when the corresponding proportions are small.

The data application with time to insulin initiation in gestational diabetes illustrated the practical value of the approach. Information criteria favored the random activation scheme, which is consistent with a multifactorial clinical decision process. Prior fetal macrosomia increased the probability of insulin use at the first visit, while history of diabetes and smoking mainly explained the cure component. Randomized quantile residual diagnostics supported the adequacy of the final specification.

This work suggests several directions for further research. First, the Weibull baseline and the Poisson model for the number of latent causes are convenient and interpretable, yet alternative baselines, spline‐based hazards, or semiparametric specifications could improve robustness. Second, other frailty distributions, shared frailty structures for clustered data, or covariate‐dependent frailty could better capture unobserved heterogeneity. Third, model selection among activation schemes could be enhanced with predictive scoring rules or cross‐validation. Fourth, time‐varying covariates and dynamic zero inflation could accommodate evolving clinical states. Finally, extensions to handle misclassification of early events, interval censoring, or left truncation would broaden applicability.

An additional point concerns the assessment of model assumptions in practical applications. Although the quantile‐residual diagnostics in our case study indicate that the proposed model fits the insulin data well, this may not hold in other settings. For this reason, we emphasize that the use of our model should be accompanied by routine diagnostic checks, including the inspection of quantile residuals, comparison with simpler alternative specifications using information criteria, and examination of component‐specific estimates such as the zero probability and the cure fraction. These procedures provide a practical way to detect potential violations of the assumptions and to ensure that the chosen specification is supported by the empirical features of the data.

Although the model assumes a Gamma frailty distribution, this choice is widely used and captures a broad range of heterogeneity patterns through a single variance parameter. Alternative frailty specifications, such as log‐normal or inverse Gaussian frailties, may lead to similar population‐level behavior when the frailty variance is moderate. For this reason, the practical impact of the frailty distribution should be assessed empirically through diagnostic checks and model comparison rather than assumed a priori. The insulin application illustrates the no‐frailty special case, while the simulation studies evaluate estimation and misspecification of the frailty component.

In summary, the proposed family of zero‐adjusted promotion cure‐rate models with frailty provides a coherent and flexible framework for settings with instantaneous failures, cure fractions, and unobserved heterogeneity. Theoretical tractability, satisfactory finite‐sample behavior, and interpretability in the insulin application indicate that these models are a useful addition to the survival analysis toolkit.

Conflicts of Interest

The authors declare no conflicts of interest.

Open Research Badges

This article has earned an Open Data badge for making publicly available the digitally‐shareable data necessary to reproduce the reported results. The data is available in the Supporting Information section.

This article has earned an open data badge “Reproducible Research” for making publicly available the code necessary to reproduce the reported results. The results reported in this article could fully be reproduced.

Supporting information

Supporting File: bimj70162‐sup‐0001‐Datacode.zip.

Appendix A. Other Simulation Results

TABLE A1.

Maximum likelihood estimates (MLE), standard error (SE), $95\%$ confidence interval (CI), and information criterion (AIC, BIC) values obtained by a zero‐adjusted promotion cure‐rate model under different activation mechanisms fitted for the insulin dataset

	Minimum					Maximum					Random
Parameter	MLE	Lower	Upper	SE		MLE	Lower	Upper	SE		MLE	Lower	Upper	SE
${\beta }_{10}$ (intercept)	−0.650	−1.348	0.048	0.356		−0.651	−1.349	0.047	0.356		−0.653	−1.351	0.045	0.356
${\beta }_{11}$ (family diabetes)	0.031	−0.782	0.844	0.415		0.032	−0.781	0.846	0.415		0.034	−0.779	0.848	0.415
${\beta }_{20}$ (intercept)	1.547	1.097	1.998	0.230		1.546	1.096	1.996	0.230		1.545	1.095	1.995	0.230
${\beta }_{21}$ (family diabetes)	−0.682	−1.220	−0.144	0.275		−0.681	−1.219	−0.143	0.274		−0.680	−1.218	−0.142	0.274
AIC	1188.0048					1187.9497					1187.9162
BIC	1219.7339					1219.6788					1219.6454
$p_0(No)$	0.084	0.038	0.129	0.023		0.084	0.038	0.129	0.023		0.084	0.038	0.129	0.023
$p_0(Yes)$	0.138	0.095	0.181	0.022		0.138	0.095	0.181	0.022		0.138	0.095	0.181	0.022
$p_1(No)$	0.755	0.685	0.826	0.036		0.755	0.685	0.826	0.036		0.755	0.685	0.826	0.036
$p_1(Yes)$	0.607	0.546	0.668	0.031		0.607	0.546	0.668	0.031		0.607	0.546	0.668	0.031
${\beta }_{10}$ (intercept)	−0.736	−1.128	−0.343	0.200		−0.735	−1.127	−0.343	0.200		−0.735	−1.127	−0.343	0.200
${\beta }_{11}$ (prior macrosomia)	0.736	−0.268	1.739	0.512		0.735	−0.268	1.739	0.512		0.735	−0.268	1.739	0.512
${\beta }_{20}$ (intercept)	1.143	0.886	1.400	0.131		1.142	0.885	1.399	0.131		1.142	0.885	1.399	0.131
${\beta }_{21}$ (prior macrosomia)	−0.572	−1.428	0.284	0.437		−0.559	−1.414	0.297	0.436		−0.567	−1.423	0.290	0.437
AIC	1188.5019					1188.6249					1188.4775
BIC	1220.2311					1220.3540					1220.2067
$p_0$ (No)	0.104	0.072	0.136	0.016		0.104	0.072	0.136	0.016		0.104	0.072	0.136	0.016
$p_0$ (Yes)	0.265	0.117	0.414	0.076		0.264	0.116	0.412	0.075		0.265	0.116	0.413	0.076
$p_1$ (No)	0.679	0.631	0.728	0.025		0.679	0.631	0.728	0.025		0.679	0.631	0.728	0.025
$p_1$ (Yes)	0.469	0.302	0.637	0.086		0.473	0.305	0.640	0.086		0.471	0.303	0.638	0.086
${\beta }_{10}$ (intercept)	−0.787	−1.251	−0.324	0.237		−0.787	−1.251	−0.323	0.237		−0.788	−1.251	−0.324	0.237
${\beta }_{11}$ (prior hypertension)	0.415	−0.320	1.150	0.375		0.413	−0.321	1.148	0.375		0.415	−0.319	1.150	0.375
${\beta }_{20}$ (intercept)	1.246	0.952	1.541	0.150		1.251	0.956	1.545	0.150		1.247	0.953	1.542	0.150
${\beta }_{21}$ (prior hypertension)	−0.530	−1.063	0.003	0.272		−0.550	−1.083	−0.016	0.272		−0.538	−1.072	−0.005	0.272
AIC	1186.9748					1186.4715					1186.6558
BIC	1218.7040					1218.2006					1218.3850
$p_0$ (No)	0.092	0.058	0.126	0.017		0.092	0.058	0.126	0.017		0.092	0.058	0.126	0.017
$p_0$ (Yes)	0.184	0.111	0.257	0.037		0.186	0.112	0.259	0.037		0.185	0.112	0.258	0.037
$p_1$ (No)	0.705	0.652	0.758	0.027		0.706	0.653	0.759	0.027		0.705	0.652	0.759	0.027
$p_1$ (Yes)	0.548	0.454	0.642	0.048		0.544	0.450	0.638	0.048		0.546	0.452	0.640	0.048
${\beta }_{10}$ (intercept)	−0.620	−0.999	−0.240	0.194		−0.619	−0.999	−0.240	0.194		−0.620	−1.000	−0.240	0.194
${\beta }_{11}$ (smoker)	−0.074	−1.212	1.065	0.581		−0.074	−1.213	1.065	0.581		−0.071	−1.210	1.067	0.581
${\beta }_{20}$ (intercept)	1.143	0.884	1.401	0.132		1.144	0.885	1.402	0.132		1.143	0.884	1.401	0.132
${\beta }_{21}$ (smoker)	−0.491	−1.298	0.315	0.411		−0.504	−1.311	0.304	0.412		−0.499	−1.308	0.309	0.412
AIC	1194.0061					1193.8797					1193.8464
BIC	1225.7352					1225.6088					1225.5756
$p_0$ (No)	0.115	0.082	0.148	0.017		0.115	0.082	0.148	0.017		0.115	0.082	0.148	0.017
$p_0$ (Yes)	0.146	0.028	0.265	0.060		0.147	0.028	0.266	0.061		0.147	0.028	0.266	0.061
$p_1$ (No)	0.671	0.622	0.720	0.025		0.671	0.622	0.720	0.025		0.671	0.622	0.720	0.025
$p_1$ (Yes)	0.561	0.395	0.727	0.085		0.558	0.392	0.725	0.085		0.559	0.392	0.726	0.085

TABLE A2.

Maximum likelihood estimates (MLE), standard error (SE), $95\%$ confidence interval (CI), and information criterion (AIC, BIC) values obtained by zero‐adjusted promotion cure‐rate model under different activation mechanisms fitted for the insulin dataset.

	Minimum				Maximum				Random
Parameter	MLE	Lower	Upper	SE	MLE	Lower	Upper	SE	MLE	Lower	Upper	SE
${\beta }_{10}$ (intercept)	−0.533	−1.314	0.248	0.399	−0.532	−1.313	0.249	0.399	−0.532	−1.313	0.249	0.399
${\beta }_{11}$ (primigesta)	0.453	−0.654	1.560	0.565	0.452	−0.655	1.559	0.565	0.452	−0.655	1.559	0.565
${\beta }_{{11}_1}$ (No)	−0.317	−1.233	0.599	0.467	−0.318	−1.234	0.598	0.467	−0.318	−1.234	0.598	0.467
${\beta }_{20}$ (intercept)	0.163	−0.484	0.810	0.330	0.169	−0.477	0.815	0.330	0.160	−0.488	0.807	0.330
${\beta }_{21}$ (primigesta)	1.506	0.629	2.383	0.448	1.502	0.625	2.378	0.447	1.510	0.632	2.387	0.448
${\beta }_{{21}_1}$ (No)	0.939	0.225	1.654	0.364	0.932	0.219	1.645	0.364	0.942	0.227	1.656	0.365
AIC	1188.0339				1188.1137				1187.9113
BIC	1235.6276				1235.7075				1235.5050
$p_0$ (Primigesta)	0.128	0.060	0.195	0.034	0.128	0.060	0.195	0.034	0.128	0.060	0.195	0.034
$p_1$ (Primigesta)	0.734	0.645	0.823	0.046	0.734	0.645	0.824	0.046	0.734	0.645	0.823	0.046
$p_0$ (No)	0.096	0.060	0.133	0.019	0.096	0.060	0.133	0.019	0.096	0.060	0.133	0.019
$p_1$ (No)	0.678	0.620	0.736	0.030	0.678	0.620	0.736	0.030	0.678	0.620	0.736	0.030
$p_0$ (Yes)	0.212	0.096	0.329	0.060	0.212	0.095	0.329	0.059	0.213	0.096	0.330	0.060
$p_1$ (Yes)	0.426	0.284	0.567	0.072	0.427	0.286	0.568	0.072	0.425	0.283	0.566	0.072
${\beta }_{10}$ (intercept)	−0.548	−1.006	−0.091	0.233	−0.548	−1.005	−0.090	0.233	−0.548	−1.005	−0.090	0.233
${\beta }_{21}$ (Normal)	−0.463	−1.696	0.769	0.629	−0.464	−1.697	0.768	0.629	−0.464	−1.696	0.769	0.629
${\beta }_{{11}_1}$ (Overweight)	−0.108	−0.920	0.704	0.414	−0.108	−0.919	0.704	0.414	−0.108	−0.919	0.704	0.414
${\beta }_{20}$ (intercept)	0.758	0.421	1.095	0.172	0.755	0.418	1.091	0.172	0.756	0.420	1.093	0.172
${\beta }_{21}$ (Normal)	1.081	0.361	1.801	0.367	1.080	0.360	1.799	0.367	1.080	0.360	1.800	0.367
${\beta }_{{21}_1}$ (Overweight)	0.425	−0.135	0.986	0.286	0.434	−0.126	0.995	0.286	0.429	−0.131	0.990	0.286
AIC	1185.8927				1185.7000				1185.7147
BIC	1233.4865				1233.2937				1233.3084
$p_0$ (Normal)	0.047	0.002	0.093	0.023	0.048	0.002	0.093	0.023	0.048	0.002	0.093	0.023
$p_1$ (Normal)	0.822	0.740	0.904	0.042	0.821	0.739	0.903	0.042	0.821	0.740	0.903	0.042
$p_0$ (Overweight)	0.108	0.053	0.164	0.028	0.108	0.053	0.164	0.028	0.108	0.053	0.164	0.028
$p_1$ (Overweight)	0.683	0.599	0.766	0.043	0.684	0.601	0.767	0.042	0.683	0.600	0.766	0.043
$p_0$ (Obesity)	0.156	0.104	0.208	0.027	0.156	0.104	0.208	0.027	0.156	0.104	0.208	0.027
$p_1$ (Obesity)	0.575	0.504	0.646	0.036	0.574	0.503	0.645	0.036	0.574	0.503	0.646	0.036

TABLE A3.

Maximum likelihood estimates (MLE), standard error (SE), $95\%$ confidence interval (CI), and information criterion (AIC, BIC) values obtained considering all risk factors by zero‐adjusted promotion cure‐rate model under different activation mechanisms fitted for the insulin dataset

	Minimum				Maximum				Random
Parameter	MLE	Lower	Upper	SE	MLE	Lower	Upper	SE	MLE	Lower	Upper	SE
${\beta }_{10}$ (intercept)	−1.405	−2.542	−0.268	0.580	−0.961	−2.087	0.166	0.575	−0.987	−2.103	0.129	0.569
${\beta }_{11}$ (History diabetes)	0.414	−0.421	1.250	0.426	0.036	−0.794	0.865	0.423	0.058	−0.768	0.884	0.421
${\beta }_{12}$ (smoking)	0.197	−0.980	1.374	0.601	−0.268	−1.410	0.873	0.582	−0.244	−1.381	0.892	0.580
${\beta }_{{13}_1}$ (primigesta)	1.081	−0.110	2.272	0.608	0.343	−0.899	1.585	0.634	0.245	−0.977	1.467	0.623
${\beta }_{{13}_2}$ (No)	−0.221	−1.188	0.747	0.494	−0.405	−1.389	0.579	0.502	−0.474	−1.457	0.510	0.502
${\beta }_{{14}_1}$ (normal)	−0.810	−2.103	0.483	0.660	−0.077	−1.381	1.228	0.666	−0.009	−1.316	1.299	0.667
${\beta }_{{14}_2}$ (Overweight)	−0.126	−0.975	0.722	0.433	0.161	−0.698	1.020	0.438	0.170	−0.682	1.022	0.435
${\beta }_{15}$ (Prior fetal Macrosomia)	1.032	−0.033	2.097	0.543	1.047	−0.026	2.119	0.547	1.073	0.006	2.141	0.545
${\beta }_{16}$ (Chronic hypertension)	0.494	−0.300	1.288	0.405	0.208	−0.584	0.999	0.404	0.258	−0.532	1.048	0.403
${\beta }_{20}$ (Intercept)	−0.350	−1.457	0.756	0.565	0.456	−0.438	1.351	0.456	0.382	−0.505	1.268	0.452
${\beta }_{21}$ (History diabetes)	−0.197	−0.824	0.429	0.319	−0.799	−1.442	−0.156	0.328	−0.791	−1.429	−0.153	0.325
${\beta }_{22}$ (smoking)	−0.341	−1.209	0.528	0.443	−1.298	−2.589	−0.007	0.659	−1.300	−2.578	−0.022	0.652
${\beta }_{{23}_1}$ (primigesta)	1.990	0.864	3.117	0.575	0.838	−0.353	2.030	0.608	0.718	−0.439	1.875	0.590
${\beta }_{{23}_2}$ (No)	1.195	0.251	2.139	0.482	0.747	−0.092	1.586	0.428	0.682	−0.166	1.529	0.432
${\beta }_{{24}_1}$ (normal)	0.368	−0.495	1.230	0.440	1.396	0.417	2.376	0.500	1.538	0.532	2.544	0.513
${\beta }_{{24}_2}$ (Overweight)	0.167	−0.458	0.792	0.319	0.635	−0.076	1.347	0.363	0.685	−0.026	1.396	0.363
${\beta }_{25}$ (Prior fetal Macrosomia)	0.172	−0.772	1.116	0.482	0.175	−0.825	1.175	0.510	0.275	−0.711	1.261	0.503
${\beta }_{26}$ (Chronic hypertension)	−0.139	−0.752	0.474	0.313	−0.658	−1.405	0.089	0.381	−0.605	−1.335	0.124	0.372
AIC	1181.7417				1191.4407				1189.2752
BIC	1324.5230				1334.2220				1332.0565

FIGURE A1.

QQ plot of the normalized randomized quantile residuals with identity line for the final model.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.