# An Alternative Promotion Time Cure Model with Overdispersed Number of Competing Causes: An Application to Melanoma Data

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## Abstract

**:**

## 1. Introduction

- The distribution has only one parameter, being an alternative to traditional discrete distributions such as Poisson and Geometric.
- The probability generating function (pgf) of the distribution has a simple expression. In fact, if $M\sim \mathrm{Bell}\left(\theta \right)$, then its pgf is given by $G(s;\theta )=exp({e}^{\theta s}-{e}^{\theta})$, for $\left|s\right|<1$. See Proposition 1 in [12]. This fact is relevant from the point of view of cure rate models because the population survival function depends on this function.
- $P(M=0;\theta )=exp(-{e}^{\theta}+1)$ has a simple form. This fact is important because this probability is the cure rate of the model. The simplicity of this term allows, among other things, to reparameterize the model in terms of the cure rate.
- The distribution belongs to the power series family of distributions [13] with pmf $P(M=m;\theta )={a}_{m}{\theta}^{m}/A\left(\theta \right)$, where ${a}_{m}={B}_{m}/m!$, $m=0,1,2,\dots $, and $A\left(\theta \right)={\sum}_{m=0}^{\infty}{a}_{m}{\theta}^{m}=exp({e}^{\theta}-1)$. Recently, ref. [14] proposed an EM-type algorithm for a class of cure rate models based on the power series family. The maximization (M) step of the EM algorithm is decomposed in two steps involving the distribution of the number of competing causes M and the distribution of the latent times Z.
- To the best of our knowledge, the Bell distribution has not yet been proposed to model the number of competing causes in a cure rate models context.

## 2. Model

## 3. Inference

**E step**: For $i=1,\dots ,n$, compute ${\tilde{M}}_{i}^{\left(k\right)}={\mu}_{i}^{(k-1)}exp\left({\mu}_{i}^{(k-1)}\right)+{\delta}_{i}(1+{\mu}_{i}^{(k-1)})$, where ${\mu}_{i}^{(k-1)}={\theta}_{i}^{(k-1)}S({t}_{i};{\mathbf{\lambda}}^{(k-1)})$ with ${\theta}_{i}^{(k-1)}=log[1-log\left({p}_{i}^{(k-1)}\right)]$ and ${p}_{i}^{(k-1)}$ comes from (5).**M step 1**: Given ${\tilde{M}}^{\left(k\right)}={({\tilde{M}}_{1}^{\left(k\right)},\dots ,{\tilde{M}}_{n}^{\left(k\right)})}^{\top}$, find ${\mathbf{\beta}}^{\left(k\right)}$ that maximizes ${Q}_{1}\left(\mathbf{\beta}\right|{\mathbf{\psi}}^{\left(k\right)})$ with respect to $\mathbf{\beta}$, where$${Q}_{1}\left(\mathbf{\beta}\right|{\mathbf{\psi}}^{\left(k\right)})=\sum _{i=1}^{n}\left(\right)open="("\; close=")">{\tilde{M}}_{i}^{\left(k\right)}log\{log[1-log\left({p}_{i}\right)]\}-log\left({p}_{i}\right)$$**M step 2**: Given ${\tilde{M}}^{\left(k\right)}$, find ${\mathbf{\lambda}}^{\left(k\right)}$ that maximizes ${Q}_{2}\left(\mathbf{\lambda}\right|{\mathbf{\psi}}^{\left(k\right)})$ with respect to $\mathbf{\lambda}$, where$${Q}_{2}\left(\mathbf{\lambda}\right|{\mathbf{\psi}}^{\left(k\right)})=\sum _{i=1}^{n}\{{\tilde{M}}_{i}^{\left(k\right)}log\left[S({t}_{i};\mathbf{\lambda})\right]+{\delta}_{i}log\left[h({t}_{i};\mathbf{\lambda})\right]\}.$$

**Remark**

**1.**

## 4. Simulation Studies

#### 4.1. Estimation

#### 4.2. Model Comparison

## 5. Data Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Model Selection

## References

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**Figure 1.**Variance of the number of competing causes as a function of the cure rate for some models.

**Figure 2.**Scheme to draw the covariates in the simulation study. Bern (0.5) denotes the Bernoulli distribution with probability 0.5 and $U(a,b)$ denotes the uniform distribution on $(a,b$).

**Figure 3.**Cox-Snell residuals (upper panels) and normalized quantile residuals plots (lower panels): (

**a**) Bellcr, (

**b**) Pocr, (

**c**) Locr, (

**d**) NBcr and (

**e**) Geocr models.

**Figure 4.**Estimate of the mean of the conditional distribution of ${M}_{i}\mid {t}_{i},{\delta}_{i};\widehat{\mathbf{\psi}}$ in melanoma data set using (

**a**) Bellcr, (

**b**) Locr and (

**c**) Geocr models.

**Figure 5.**Width of the 95% confidence interval for the conditional cure rate: (

**a**) without ulceration and ${t}_{0}=0$ year, (

**b**) without ulceration and ${t}_{0}=5$ years, (

**c**) with ulceration and ${t}_{0}=5$ years and (

**d**) with ulceration and ${t}_{0}=10$ years.

**Figure 6.**Estimates of the conditional cure rate (${p}_{{t}_{0}}$ in Equation (9)) for ${t}_{0}=5$ years: (

**a**) without ulceration and (

**b**) with ulceration. Estimated survival function for a tumor thickness of 10 cms: (

**c**) without ulceration and (

**d**) with ulceration.

**Table 1.**Variance of the number of competing causes as a function of the cure rate for some models parameterized in the cure rate (p).

Model | Var$\left(\mathit{M}\right)$ | Model | Var$\left(\mathit{M}\right)$ |
---|---|---|---|

Berncr | $p(1-p)$ | NBcr | $q(1-{p}^{1/q}){p}^{-2/q}$ |

Pocr | $-log\left(p\right)$ | Locr * | $\theta [\theta +log(1-\theta )]/{[(1-\theta )log(1-\theta )]}^{2}$ |

Geocr | $(1-p){p}^{-2}$ | Bellcr | $[1-log(p\left)\right]log[1-log(p\left)\right]\{1+log[1-log\left(p\right)\left]\right\}$ |

Cure Rate | ${\mathit{\beta}}_{0}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\beta}}_{3}$ | $\left(\mathbb{E}\right(\mathit{Z})$, Var$\left(\mathit{Z}\right))$ | $\mathit{\alpha}$ | $\mathit{\nu}$ | |
---|---|---|---|---|---|---|---|---|

Cure 1 | 2.197 | −0.811 | −1.024 | 1.770 | (7 , 4) | −8.018 | 3.920 | |

Cure 2 | 1.386 | −0.539 | −0.684 | 1.118 | (5 , 3) | −5.444 | 3.165 | |

Cure 3 | 0.847 | −0.442 | −0.547 | 0.843 | (2 , 1) | −1.712 | 2.101 |

**Table 3.**Simulated bias (Bias), average of the asymptotic standard errors (SE), root of the simulated mean squared error (RMSE) and coverage probability of the 95% asymptotic confidence intervals (CP) for the Bellcr model.

Cure Rate | $\left(\mathbb{E}\right(\mathbf{Z})$, Var$\left(\mathbf{Z}\right)$) | Parameter | $\mathit{n}=200$ | $\mathit{n}=400$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Bias | SE | RMSE | CP | Bias | SE | RMSE | CP | |||

Cure 1 | (7, 4) | ${\beta}_{0}$ | 0.109 | 0.594 | 0.659 | 0.945 | 0.046 | 0.403 | 0.422 | 0.952 |

${\beta}_{1}$ | −0.003 | 0.692 | 0.737 | 0.949 | −0.003 | 0.471 | 0.474 | 0.954 | ||

${\beta}_{2}$ | −0.067 | 0.210 | 0.257 | 0.963 | −0.027 | 0.138 | 0.143 | 0.950 | ||

${\beta}_{3}$ | 0.112 | 0.396 | 0.467 | 0.957 | 0.049 | 0.260 | 0.269 | 0.955 | ||

$\alpha $ | −0.335 | 0.823 | 0.898 | 0.953 | −0.221 | 0.572 | 0.647 | 0.951 | ||

$\nu $ | 0.164 | 0.446 | 0.487 | 0.951 | 0.107 | 0.307 | 0.339 | 0.951 | ||

(5, 3) | ${\beta}_{0}$ | 0.118 | 0.532 | 0.597 | 0.944 | 0.050 | 0.363 | 0.383 | 0.948 | |

${\beta}_{1}$ | 0.006 | 0.621 | 0.641 | 0.958 | 0.005 | 0.427 | 0.451 | 0.949 | ||

${\beta}_{2}$ | −0.057 | 0.183 | 0.208 | 0.950 | −0.021 | 0.123 | 0.132 | 0.950 | ||

${\beta}_{3}$ | 0.082 | 0.344 | 0.366 | 0.955 | 0.023 | 0.232 | 0.247 | 0.949 | ||

$\alpha $ | −0.206 | 0.518 | 0.563 | 0.943 | −0.124 | 0.361 | 0.397 | 0.946 | ||

$\nu $ | 0.111 | 0.321 | 0.348 | 0.943 | 0.073 | 0.223 | 0.237 | 0.945 | ||

Cure 2 | (7, 4) | ${\beta}_{0}$ | 0.043 | 0.471 | 0.501 | 0.945 | 0.021 | 0.323 | 0.323 | 0.953 |

${\beta}_{1}$ | 0.005 | 0.590 | 0.600 | 0.955 | −0.003 | 0.408 | 0.434 | 0.953 | ||

${\beta}_{2}$ | −0.035 | 0.139 | 0.147 | 0.950 | −0.015 | 0.095 | 0.100 | 0.950 | ||

${\beta}_{3}$ | 0.058 | 0.268 | 0.288 | 0.941 | 0.028 | 0.183 | 0.191 | 0.945 | ||

$\alpha $ | −0.276 | 0.820 | 0.889 | 0.940 | −0.132 | 0.569 | 0.590 | 0.949 | ||

$\nu $ | 0.137 | 0.431 | 0.458 | 0.947 | 0.069 | 0.298 | 0.309 | 0.952 | ||

(5, 3) | ${\beta}_{0}$ | 0.051 | 0.421 | 0.433 | 0.948 | 0.020 | 0.291 | 0.288 | 0.952 | |

${\beta}_{1}$ | −0.019 | 0.534 | 0.552 | 0.956 | −0.001 | 0.369 | 0.390 | 0.951 | ||

${\beta}_{2}$ | −0.026 | 0.124 | 0.127 | 0.964 | −0.013 | 0.086 | 0.087 | 0.950 | ||

${\beta}_{3}$ | 0.045 | 0.240 | 0.248 | 0.951 | 0.022 | 0.165 | 0.171 | 0.950 | ||

$\alpha $ | −0.146 | 0.514 | 0.534 | 0.944 | −0.078 | 0.358 | 0.360 | 0.953 | ||

$\nu $ | 0.090 | 0.311 | 0.324 | 0.948 | 0.051 | 0.216 | 0.220 | 0.952 | ||

Cure 3 | (7, 4) | ${\beta}_{0}$ | 0.005 | 0.420 | 0.414 | 0.957 | 0.004 | 0.290 | 0.300 | 0.949 |

${\beta}_{1}$ | 0.000 | 0.542 | 0.541 | 0.957 | −0.003 | 0.375 | 0.387 | 0.950 | ||

${\beta}_{2}$ | −0.024 | 0.116 | 0.119 | 0.956 | −0.011 | 0.079 | 0.083 | 0.953 | ||

${\beta}_{3}$ | 0.050 | 0.227 | 0.234 | 0.951 | 0.013 | 0.155 | 0.160 | 0.950 | ||

$\alpha $ | −0.192 | 0.765 | 0.814 | 0.957 | −0.132 | 0.534 | 0.567 | 0.953 | ||

$\nu $ | 0.096 | 0.401 | 0.421 | 0.956 | 0.071 | 0.279 | 0.297 | 0.948 | ||

(5, 3) | ${\beta}_{0}$ | 0.013 | 0.376 | 0.387 | 0.954 | 0.013 | 0.261 | 0.275 | 0.948 | |

${\beta}_{1}$ | −0.010 | 0.490 | 0.500 | 0.950 | −0.005 | 0.341 | 0.336 | 0.950 | ||

${\beta}_{2}$ | −0.018 | 0.104 | 0.110 | 0.955 | −0.012 | 0.072 | 0.074 | 0.948 | ||

${\beta}_{3}$ | 0.029 | 0.203 | 0.211 | 0.946 | 0.017 | 0.141 | 0.145 | 0.950 | ||

$\alpha $ | −0.129 | 0.479 | 0.515 | 0.939 | −0.061 | 0.334 | 0.339 | 0.953 | ||

$\nu $ | 0.081 | 0.289 | 0.307 | 0.951 | 0.035 | 0.201 | 0.200 | 0.950 |

**Table 4.**Rejection rate (in %) of the Bellcr model when compared with different models using the Vuong’s test statistic—Part 1.

True Model | ($\mathbb{E}\left(\mathbf{Z}\right)$, Var$\left(\mathbf{Z}\right)$) | Model Compared with Bellcr | $\mathit{n}=200$ | $\mathit{n}=400$ | ||||
---|---|---|---|---|---|---|---|---|

Cure Rate | Cure Rate | |||||||

Cure 1 | Cure 2 | Cure 3 | Cure 1 | Cure 2 | Cure 3 | |||

Bellcr | (7, 4) | Pocr | 2.7 | 2.4 | 2.9 | 1.8 | 2.1 | 1.6 |

Locr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

NBcr | 4.2 | 3.3 | 3.7 | 3.8 | 3.6 | 3.1 | ||

Geocr | 0.4 | 0.3 | 0.2 | 0.0 | 0.0 | 0.5 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

(5, 3) | Pocr | 1.7 | 2.1 | 2.8 | 1.0 | 1.0 | 1.8 | |

Locr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

NBcr | 3.6 | 2.9 | 3.5 | 2.3 | 2.2 | 2.8 | ||

Geocr | 0.2 | 0.6 | 0.5 | 0.0 | 0.0 | 0.3 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

(2, 1) | Pocr | 1.3 | 1.5 | 2.8 | 0.7 | 0.9 | 0.7 | |

Locr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

NBcr | 2.7 | 2.7 | 3.7 | 2.3 | 2.0 | 1.6 | ||

Geocr | 0.5 | 0.7 | 0.8 | 0.0 | 0.0 | 0.1 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

Pocr | (7, 4) | Pocr | 13.6 | 11.4 | 11.8 | 16.6 | 13.6 | 14.2 |

Locr | 0.0 | 0.0 | 0.1 | 0.0 | 0.0 | 0.0 | ||

NBcr | 16.1 | 12.4 | 12.9 | 19.6 | 15.5 | 16.0 | ||

Geocr | 0.2 | 0.1 | 0.0 | 0.0 | 0.0 | 0.0 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

(5, 3) | Pocr | 12.9 | 14.5 | 11.7 | 21.7 | 17.2 | 15.2 | |

Locr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

NBcr | 14.9 | 15.3 | 12.2 | 24.4 | 18.6 | 17.2 | ||

Geocr | 0.0 | 0.2 | 0.1 | 0.0 | 0.0 | 0.0 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

(2, 1) | Pocr | 15.6 | 15.0 | 13.9 | 20.9 | 18.3 | 15.1 | |

Locr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

NBcr | 17.2 | 16.4 | 14.6 | 22.8 | 20.1 | 16.8 | ||

Geocr | 0.1 | 0.0 | 0.1 | 0.0 | 0.0 | 0.0 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

Locr | (7, 4) | Pocr | 6.2 | 1.6 | 0.4 | 9.5 | 0.0 | 0.3 |

Locr | 79.9 | 79.7 | 79.6 | 80.9 | 80.8 | 80.8 | ||

NBcr | 5.9 | 5.0 | 6.0 | 9.5 | 2.3 | 6.3 | ||

Geocr | 0.5 | 3.7 | 9.7 | 0.0 | 4.5 | 11.3 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

(5, 3) | Pocr | 6.8 | 0.8 | 0.7 | 6.7 | 0.6 | 0.2 | |

Locr | 76.1 | 75.9 | 75.8 | 80.5 | 80.4 | 80.3 | ||

NBcr | 7.4 | 3.4 | 4.8 | 6.7 | 3.9 | 7.9 | ||

Geocr | 0.3 | 4.3 | 9.4 | 0.1 | 4.4 | 14.7 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

(2, 1) | Pocr | 5.7 | 0.9 | 0.4 | 7.2 | 0.3 | 0.1 | |

Locr | 80.0 | 79.9 | 79.7 | 78.6 | 78.5 | 78.6 | ||

NBcr | 6.1 | 4.7 | 6.7 | 7.2 | 4.1 | 10.9 | ||

Geocr | 0.5 | 6.7 | 10.7 | 0.1 | 6.3 | 18.2 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

**Table 5.**Rejection rate (in %) of the Bellcr model when compared with different models using the Vuong’s test statistic—Part 2.

True Model | ($\mathbb{E}\left(\mathbf{Z}\right)$, Var$\left(\mathbf{Z}\right)$) | Model Compared with Bellcr | $\mathit{n}=200$ | $\mathit{n}=400$ | ||||
---|---|---|---|---|---|---|---|---|

Cure Rate | Cure Rate | |||||||

Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 | |||

NBcr ($q=3$) | (7, 4) | Pocr | 0.9 | 2.3 | 3.4 | 0.4 | 1.4 | 1.6 |

Locr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

NBcr | 4.8 | 3.5 | 5.3 | 6.5 | 4.6 | 3.9 | ||

Geocr | 0.8 | 1.4 | 0.6 | 0.3 | 0.5 | 0.5 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

(5, 3) | Pocr | 0.8 | 2.5 | 3.0 | 0.4 | 0.4 | 1.5 | |

Locr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

NBcr | 7.2 | 4.8 | 4.2 | 11.2 | 3.4 | 4.1 | ||

Geocr | 1.7 | 1.0 | 0.7 | 0.5 | 0.3 | 0.7 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

(2, 1) | Pocr | 0.7 | 1.4 | 2.8 | 0.1 | 1.0 | 1.7 | |

Locr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

NBcr | 5.0 | 4.2 | 3.9 | 9.8 | 4.6 | 4.7 | ||

Geocr | 1.0 | 0.8 | 0.5 | 0.6 | 0.2 | 0.5 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

Geocr | (7, 4) | Pocr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

Locr | 0.4 | 0.8 | 0.5 | 0.1 | 0.1 | 0.4 | ||

NBcr | 64.7 | 19.4 | 13.3 | 92.4 | 49.8 | 34.3 | ||

Geocr | 58.7 | 25.7 | 18.8 | 83.5 | 49.1 | 38.8 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

(5, 3) | Pocr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |

Locr | 0.0 | 0.1 | 0.3 | 0.0 | 0.0 | 0.1 | ||

NBcr | 69.8 | 25.8 | 15.8 | 94.8 | 55.6 | 36.8 | ||

Geocr | 62.5 | 30.8 | 22.1 | 87.4 | 52.0 | 39.5 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

(2, 1) | Pocr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |

Locr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | ||

NBcr | 72.2 | 29.4 | 17.3 | 96.2 | 60.4 | 40.7 | ||

Geocr | 63.9 | 33.4 | 24.2 | 89.6 | 56.7 | 42.7 | ||

Berncr | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

**Table 6.**Parameter estimates, standard errors (SE) and maximum value of the log-likelihood function $\left(\right)$ for different models.

Parameter | Model | |||||||
---|---|---|---|---|---|---|---|---|

Bellcr | Pocr | Locr | Geocr | |||||

Estimate | SE | Estimate | SE | Estimate | SE | Estimate | SE | |

${\beta}_{0}$ | 1.8759 | 0.2449 | 1.9039 | 0.3431 | 1.6771 | 0.3566 | 1.8127 | 0.3501 |

${\beta}_{1}$ | −1.4536 | 0.2843 | −1.4814 | 0.3901 | −1.4816 | 0.3278 | −1.4807 | 0.3569 |

${\beta}_{2}$ | −0.1908 | 0.0416 | −0.1960 | 0.0705 | −0.1412 | 0.0356 | −0.1785 | 0.0537 |

$\alpha $ | −3.2461 | 0.3124 | −2.9435 | 0.3411 | −4.1863 | 0.5154 | −3.4907 | 0.3916 |

$\nu $ | 1.8287 | 0.1808 | 1.7368 | 0.2178 | 2.2110 | 0.2725 | 1.9132 | 0.2357 |

$exp\left({\beta}_{1}\right)$ | 0.2337 | 0.0664 | 0.2273 | 0.0887 | 0.2273 | 0.0745 | 0.2275 | 0.0812 |

$exp\left({\beta}_{2}\right)$ | 0.8263 | 0.0344 | 0.8220 | 0.0580 | 0.8683 | 0.0309 | 0.8365 | 0.0449 |

$\ell \left(\widehat{\mathbf{\psi}}\right)$ | −206.3 | −207.5 | −203.7 | −205.4 |

**Table 7.**Vuong’s statistic and p-value when testing the Bell cure rate model against some alternatives.

Model | Pocr | Locr | Geocr |
---|---|---|---|

Vuong’s statistic | −2.473 | 1.238 | 1.146 |

p-Value | 0.013 | 0.216 | 0.252 |

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Gallardo, D.I.; de Castro, M.; Gómez, H.W.
An Alternative Promotion Time Cure Model with Overdispersed Number of Competing Causes: An Application to Melanoma Data. *Mathematics* **2021**, *9*, 1815.
https://doi.org/10.3390/math9151815

**AMA Style**

Gallardo DI, de Castro M, Gómez HW.
An Alternative Promotion Time Cure Model with Overdispersed Number of Competing Causes: An Application to Melanoma Data. *Mathematics*. 2021; 9(15):1815.
https://doi.org/10.3390/math9151815

**Chicago/Turabian Style**

Gallardo, Diego I., Mário de Castro, and Héctor W. Gómez.
2021. "An Alternative Promotion Time Cure Model with Overdispersed Number of Competing Causes: An Application to Melanoma Data" *Mathematics* 9, no. 15: 1815.
https://doi.org/10.3390/math9151815