Lognormal lifetimes and likelihood-based inference for flexible cure rate models based on COM-Poisson family

A New Approach to Modeling the Cure Rate in the Presence of Interval Censored Data

We consider interval censored data with a cured subgroup that arises from longitudinal followup studies with a heterogeneous population where a certain proportion of subjects is not susceptible to the event of interest. We propose a two component mixture cure model, where the first component describing the probability of cure is modeled by a support vector machine-based approach and the second component describing the survival distribution of the uncured group is modeled by a proportional hazard structure. Our proposed model provides flexibility in capturing complex effects of covariates on the probability of cure unlike the traditional models that rely on modeling the cure probability using a generalized linear model with a known link function. For the estimation of model parameters, we develop an expectation maximization-based estimation algorithm. We conduct simulation studies and show that our proposed model performs better in capturing complex effects of covariates on the cure probability when compared to the traditional logit link-based two component mixture cure model. This results in more accurate (smaller bias) and precise (smaller mean square error) estimates of the cure probabilities, which in-turn improves the predictive accuracy of the latent cured status. We further show that our model's ability to capture complex covariate effects also improves the estimation results corresponding to the survival distribution of the uncured. Finally, we apply the proposed model and estimation procedure to an interval censored data on smoking cessation.

On the estimation of interval censored destructive negative binomial cure model

In this article, a competitive risk survival model is considered in which the initial number of risks, assumed to follow a negative binomial distribution, is subject to a destructive mechanism. Assuming the population of interest to have a cure component, the form of the data as interval-censored, and considering both the number of initial risks and risks remaining active after destruction to be missing data, we develop two distinct estimation algorithms for this model. Making use of the conditional distributions of the missing data, we develop an expectation maximization (EM) algorithm, in which the conditional expected complete log-likelihood function is decomposed into simpler functions which are then maximized independently. A variation of the EM algorithm, called the stochastic EM (SEM) algorithm, is also developed with the goal of avoiding the calculation of complicated expectations and improving performance at parameter recovery. A Monte Carlo simulation study is carried out to evaluate the performance of both estimation methods through calculated bias, root mean square error, and coverage probability of the asymptotic confidence interval. We demonstrate the proposed SEM algorithm as the preferred estimation method through simulation and further illustrate the advantage of the SEM algorithm, as well as the use of a destructive model, with data from a children's mortality study.

A support vector machine-based cure rate model for interval censored data

The mixture cure rate model is the most commonly used cure rate model in the literature. In the context of mixture cure rate model, the standard approach to model the effect of covariates on the cured or uncured probability is to use a logistic function. This readily implies that the boundary classifying the cured and uncured subjects is linear. In this article, we propose a new mixture cure rate model based on interval censored data that uses the support vector machine to model the effect of covariates on the uncured or the cured probability (i.e. on the incidence part of the model). Our proposed model inherits the features of the support vector machine and provides flexibility to capture classification boundaries that are nonlinear and more complex. The latency part is modeled by a proportional hazards structure with an unspecified baseline hazard function. We develop an estimation procedure based on the expectation maximization algorithm to estimate the cured/uncured probability and the latency model parameters. Our simulation study results show that the proposed model performs better in capturing complex classification boundaries when compared to both logistic regression-based and spline regression-based mixture cure rate models. We also show that our model's ability to capture complex classification boundaries improve the estimation results corresponding to the latency part of the model. For illustrative purpose, we present our analysis by applying the proposed methodology to the NASA's Hypobaric Decompression Sickness Database.

On the integration of decision trees with mixture cure model

The mixture cure model is widely used to analyze survival data in the presence of a cured subgroup. Standard logistic regression-based approaches to model the incidence may lead to poor predictive accuracy of cure, specifically when the covariate effect is non-linear. Supervised machine learning techniques can be used as a better classifier than the logistic regression due to their ability to capture non-linear patterns in the data. However, the problem of interpret-ability hangs in the balance due to the trade-off between interpret-ability and predictive accuracy. We propose a new mixture cure model where the incidence part is modeled using a decision tree-based classifier and the proportional hazards structure for the latency part is preserved. The proposed model is very easy to interpret, closely mimics the human decision-making process, and provides flexibility to gauge both linear and non-linear covariate effects. For the estimation of model parameters, we develop an expectation maximization algorithm. A detailed simulation study shows that the proposed model outperforms the logistic regression-based and spline regression-based mixture cure models, both in terms of model fitting and evaluating predictive accuracy. An illustrative example with data from a leukemia study is presented to further support our conclusion.

On the parameter estimation of Box-Cox transformation cure model

We propose an improved estimation method for the Box-Cox transformation (BCT) cure rate model parameters. Specifically, we propose a generic maximum likelihood estimation algorithm through a non-linear conjugate gradient (NCG) method with an efficient line search technique. We then apply the proposed NCG algorithm to BCT cure model. Through a detailed simulation study, we compare the model fitting results of the NCG algorithm with those obtained by the existing expectation maximization (EM) algorithm. First, we show that our proposed NCG algorithm allows simultaneous maximization of all model parameters unlike the EM algorithm when the likelihood surface is flat with respect to the BCT index parameter. Then, we show that the NCG algorithm results in smaller bias and noticeably smaller root mean square error of the estimates of the model parameters that are associated with the cure rate. This results in more accurate and precise inference on the cure rate. In addition, we show that when the sample size is large the NCG algorithm, which only needs the computation of the gradient and not the Hessian, takes less CPU time to produce the estimates. These advantages of the NCG algorithm allows us to conclude that the NCG method should be the preferred estimation method over the already existing EM algorithm in the context of BCT cure model. Finally, we apply the NCG algorithm to analyze a well-known melanoma data and show that it results in a better fit when compared to the EM algorithm.

A New Non-Linear Conjugate Gradient Algorithm for Destructive Cure Rate Model and a Simulation Study: Illustration with Negative Binomial Competing Risks

In this paper, we propose a new estimation methodology based on a projected non-linear conjugate gradient (PNCG) algorithm with an efficient line search technique. We develop a general PNCG algorithm for a survival model incorporating a proportion cure under a competing risks setup, where the initial number of competing risks are exposed to elimination after an initial treatment (known as destruction). In the literature, expectation maximization (EM) algorithm has been widely used for such a model to estimate the model parameters. Through an extensive Monte Carlo simulation study, we compare the performance of our proposed PNCG with that of the EM algorithm and show the advantages of our proposed method. Through simulation, we also show the advantages of our proposed methodology over other optimization algorithms (including other conjugate gradient type methods) readily available as R software packages. To show these, we assume the initial number of competing risks to follow a negative binomial distribution although our general algorithm allows one to work with any competing risks distribution. Finally, we apply our proposed algorithm to analyze a well-known melanoma data.

A simplified stochastic EM algorithm for cure rate model with negative binomial competing risks: An application to breast cancer data

In this article, a long-term survival model under competing risks is considered. The unobserved number of competing risks is assumed to follow a negative binomial distribution that can capture both over- and under-dispersion. Considering the latent competing risks as missing data, a variation of the well-known expectation maximization (EM) algorithm, called the stochastic EM algorithm (SEM), is developed. It is shown that the SEM algorithm avoids calculation of complicated expectations, which is a major advantage of the SEM algorithm over the EM algorithm. The proposed procedure also allows the objective function to be split into two simpler functions, one corresponding to the parameters associated with the cure rate and the other corresponding to the parameters associated with the progression times. The advantage of this approach is that each simple function, with lower parameter dimension, can be maximized independently. An extensive Monte Carlo simulation study is carried out to compare the performances of the SEM and EM algorithms. Finally, a breast cancer survival data is analyzed and it is shown that the SEM algorithm performs better than the EM algorithm.

A Flexible Multivariate Distribution for Correlated Count Data

Multivariate count data are often modeled via a multivariate Poisson distribution, but it contains an underlying, constraining assumption of data equi-dispersion (where its variance equals its mean). Real data are oftentimes over-dispersed and, as such, consider various advancements of a negative binomial structure. While data over-dispersion is more prevalent than under-dispersion in real data, however, examples containing under-dispersed data are surfacing with greater frequency. Thus, there is a demonstrated need for a flexible model that can accommodate both data types. We develop a multivariate Conway–Maxwell–Poisson (MCMP) distribution to serve as a flexible alternative for correlated count data that contain data dispersion. This structure contains the multivariate Poisson, multivariate geometric, and the multivariate Bernoulli distributions as special cases, and serves as a bridge distribution across these three classical models to address other levels of over- or under-dispersion. In this work, we not only derive the distributional form and statistical properties of this model, but we further address parameter estimation, establish informative hypothesis tests to detect statistically significant data dispersion and aid in model parsimony, and illustrate the distribution’s flexibility through several simulated and real-world data examples. These examples demonstrate that the MCMP distribution performs on par with the multivariate negative binomial distribution for over-dispersed data, and proves particularly beneficial in effectively representing under-dispersed data. Thus, the MCMP distribution offers an effective, unifying framework for modeling over- or under-dispersed multivariate correlated count data that do not necessarily adhere to Poisson assumptions.

Likelihood inference for COM-Poisson cure rate model with interval-censored data and Weibull lifetimes

In this paper, we consider a competing cause scenario and assume the number of competing causes to follow a Conway–Maxwell Poisson distribution which can capture both over and under dispersion that is usually encountered in discrete data. Assuming the population of interest having a component cure and the form of the data to be interval censored, as opposed to the usually considered right-censored data, the main contribution is in developing the steps of the expectation maximization algorithm for the determination of the maximum likelihood estimates of the model parameters of the flexible Conway–Maxwell Poisson cure rate model with Weibull lifetimes. An extensive Monte Carlo simulation study is carried out to demonstrate the performance of the proposed estimation method. Model discrimination within the Conway–Maxwell Poisson distribution is addressed using the likelihood ratio test and information-based criteria to select a suitable competing cause distribution that provides the best fit to the data. A simulation study is also carried out to demonstrate the loss in efficiency when selecting an improper competing cause distribution which justifies the use of a flexible family of distributions for the number of competing causes. Finally, the proposed methodology and the flexibility of the Conway–Maxwell Poisson distribution are illustrated with two known data sets from the literature: smoking cessation data and breast cosmesis data.

Proportional hazards under Conway–Maxwell-Poisson cure rate model and associated inference

Cure rate models or long-term survival models play an important role in survival analysis and some other applied fields. In this article, by assuming a Conway–Maxwell–Poisson distribution under a competing cause scenario, we study a flexible cure rate model in which the lifetimes of non-cured individuals are described by a Cox’s proportional hazard model with a Weibull hazard as the baseline function. Inference is then developed for a right censored data by the maximum likelihood method with the use of expectation-maximization algorithm and a profile likelihood approach for the estimation of the dispersion parameter of the Conway–Maxwell–Poisson distribution. An extensive simulation study is performed, under different scenarios including various censoring proportions, sample sizes, and lifetime parameters, in order to evaluate the performance of the proposed inferential method. Discrimination among some common cure rate models is then done by using likelihood-based and information-based criteria. Finally, for illustrative purpose, the proposed model and associated inferential procedure are applied to analyze a cutaneous melanoma data.