Likelihood Inference for Flexible Cure Rate Models with Gamma Lifetimes

Abstract

A flexible cure rate survival model was developed by Rodrigues et al. (2009a) by assuming the competing cause variable to follow the Conway-Maxwell Poisson distribution. This model includes as special cases some of the well-known cure rate models. As the data obtained from cancer clinical trials are often right censored, the EM algorithm can be efficiently used to estimate the model parameters based on right censored data. In this paper, we consider the cure rate model developed by Rodrigues et al. (2009a) and by assuming the time-to-event to follow the gamma distribution, we develop exact likelihood inference based on the EM algorithm. An extensive Monte Carlo simulation study is performed to examine the method of inference developed. Model discrimination between different cure rate models is carried out by means of likelihood ratio test and Akaike and Bayesian information criteria. Finally, the proposed methodology is illustrated with a cutaneous melanoma data.

Keywords:

• Cure rate models
• Long-term survivor
• Conway-Maxwell Poisson (COM-Poisson) distribution
• Weighted Poisson distribution
• EM algorithm
• Profile likelihood
• Gamma distribution
• Likelihood ratio test
• Akaike information criterion (AIC)
• Bayesian information criterion (BIC)

Mathematics Subject Classifications:

• 62N02
• 62P10

Appendix A: First- and Second-Order Derivatives of the Q-function

Bernoulli cure rate model. The expressions of the first- and second-order derivatives of the function with respect to are as follows: for l, l′ = 0, 1, …, k, and x_i0 = 1∀i = 1, 2, …, n. The required first- and second-order derivatives of the function with respect to are as follows: where π^(k)_i is as defined in (11(11) ).

Poisson cure rate model. The required first- and second-order derivatives of the function with respect to and are as follows: for l, l′ = 0, 1, …, k, and x_i0 = 1∀i = 1, 2, …, n, where and π^(k)_i are as defined in (12(12) ) and (13(13) ), respectively.

Geometric cure rate model. The required first- and second-order derivatives of the function with respect to and are as follows: for l, l′ = 0, 1, …, k, and x_i0 = 1∀i = 1, 2, …, n, where (16) and π^(k)_i is as defined in (14(14) ).

COM-Poisson cure rate model. The required first- and second-order derivatives of the function with respect to and are as follows: for l, l′ = 0, 1, …, k, and x_i0 = 1∀i = 1, 2, …, n, where and π^(k)_i is as defined in (15(15) ).

The quantities z_1i, z_2i, z_21i, z_22i, z_01i, and z_02i are all computed by truncating the numerical series for each i. Let S_n denote the n-th order partial sum of the infinite series. Then, the infinite series can be approximated with the first n terms of the series if S_{n + 1} − S_n ⩽ ε, where ε is a pre-fixed small tolerance value. In our study, we chose ε = 0.01 for computational ease and efficiency.

Appendix B: Observed Information Matrix

Bernoulli cure rate model. The components of the score function are The observed information matrix has its components as The above are defined for l, l′ = 0, 1, …, k, x_i0 = 1∀i = 1, 2, …, n, where for i ∈ I₀.

Poisson cure rate model. The components of the score function are The observed information matrix has its components as The above are defined for l, l′ = 0, 1, …, k, x_i0 = 1∀i = 1, 2, …, n.

Geometric cure rate model. The components of the score function are The observed information matrix has its components as The above are defined for l, l′ = 0, 1, …, k, x_i0 = 1∀i = 1, 2, …, n, where for i ∈ I₀ and is as defined in (16(16) ).

COM-Poisson cure rate model. The components of the score function, for a fixed value of φ, are The observed information matrix has its components as The above are defined for l, l′ = 0, 1, …, k, x_i0 = 1∀i = 1, 2, …, n.