Piecewise Linear Approximations for Cure Rate Models and Associated Inferential Issues

A flexible model based on piecewise linear approximation for the analysis of left truncated right censored data with covariates, and applications to Worcester Heart Attack Study data and Channing House data

Left truncated right censored (LTRC) data arise quite commonly from survival studies. In this article, a model based on piecewise linear approximation is proposed for the analysis of LTRC data with covariates. Specifically, the model involves a piecewise linear approximation for the cumulative baseline hazard function of the proportional hazards model. The principal advantage of the proposed model is that it does not depend on restrictive parametric assumptions while being flexible and data-driven. Likelihood inference for the model is developed. Through detailed simulation studies, the robustness property of the model is studied by fitting it to LTRC data generated from different processes covering a wide range of lifetime distributions. A sensitivity analysis is also carried out by fitting the model to LTRC data generated from a process with a piecewise constant baseline hazard. It is observed that the performance of the model is quite satisfactory in all those cases. Analyses of two real LTRC datasets by using the model are provided as illustrative examples. Applications of the model in some practical prediction issues are discussed. In summary, the proposed model provides a comprehensive and flexible approach to model a general structure for LTRC lifetime data.

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.

On a reparameterization of a flexible family of cure models

The existence of items not susceptible to the event of interest is of both theoretical and practical importance. Although researchers may provide, for example, biological, medical, or sociological evidence for the presence of such items (cured), statistical models performing well under the existence or not of a cured proportion, frequently offer a necessary flexibility. This work introduces a new reparameterization of a flexible family of cure models, which not only includes among its special cases, the most studied cure models (such as the mixture, bounded cumulative hazard, and negative binomial cure model) but also classical survival models (ie, without cured items). One of the main properties of the proposed family, apart from its computationally tractable closed form, is that the case of zero cured proportion is not found at the boundary of the parameter space, as it typically happens to other families. A simulation study examines the (finite) performance of the suggested methodology, focusing to the estimation through EM algorithm and model discrimination, by the aid of the likelihood ratio test and Akaike information criterion; for illustrative purposes, analysis of two real life datasets (on recidivism and cutaneous melanoma) is also carried out.

A two-way flexible generalized gamma transformation cure rate model

We propose a two-way flexible cure rate model. The first flexibility is provided by considering a family of Box-Cox transformation cure models that include the commonly used cure models as special cases. The second flexibility is provided by proposing the wider class of generalized gamma distributions to model the associated lifetime. The advantage of this two-way flexibility is that it allows us to carry out tests of hypotheses to select an adequate cure model (within the family of Box-Cox transformation cure models) and a suitable lifetime distribution (within the wider class of generalized gamma distributions) that jointly provides the best fit to a given data. First, we study the maximum likelihood estimation of the generalized gamma Box-Cox transformation (GGBCT) model parameters. Then, we use the flexibility of our proposed model to carry out power studies to demonstrate the power of likelihood ratio test in rejecting mis-specified models. Furthermore, we study the bias and efficiency of the estimators of the cure rates under model mis-specification. Our findings strongly suggest the importance of selecting a correct lifetime distribution and a correct cure rate model, which can be achieved through the proposed two-way flexible model. Finally, we illustrate the applicability of our proposed model using a data from a breast cancer study and show that our model provides a better fit than the existing semiparametric Box-Cox transformation cure model with piecewise exponential approximation to the lifetime distribution.

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.

On a new piecewise regression model with cure rate: Diagnostics and application to medical data

In this article, we discuss an extension of the classical negative binomial cure rate model with piecewise exponential distribution of the time to event for concurrent causes, which enables the modeling of monotonic and non-monotonic hazard functions (ie, the shape of the hazard function is not assumed as in traditional parametric models). This approach produces a flexible cure rate model, depending on the choice of time partition. We discuss local influence on this negative binomial power piecewise exponential model. We report on Monte Carlo simulation studies and application of the model to real melanoma and leukemia datasets.

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.

Power laws in the Roman Empire: a survival analysis

The Roman Empire shaped western civilization, and many Roman principles are embodied in modern institutions. Although its political institutions proved both resilient and adaptable, allowing it to incorporate diverse populations, the Empire suffered from many conflicts. Indeed, most emperors died violently, from assassination, suicide or in battle. These conflicts produced patterns in the length of time that can be identified by statistical analysis. In this paper, we study the underlying patterns associated with the reign of the Roman emperors by using statistical tools of survival data analysis. We consider all the 175 Roman emperors and propose a new power-law model with change points to predict the time-to-violent-death of the Roman emperors. This model encompasses data in the presence of censoring and long-term survivors, providing more accurate predictions than previous models. Our results show that power-law distributions can also occur in survival data, as verified in other data types from natural and artificial systems, reinforcing the ubiquity of power-law distributions. The generality of our approach paves the way to further related investigations not only in other ancient civilizations but also in applications in engineering and medicine.

A novel Bayesian continuous piecewise linear log-hazard model, with estimation and inference via reversible jump Markov chain Monte Carlo

We present a reversible jump Bayesian piecewise log-linear hazard model that extends the Bayesian piecewise exponential hazard to a continuous function of piecewise linear log hazards. A simulation study encompassing several different hazard shapes, accrual rates, censoring proportion, and sample sizes showed that the Bayesian piecewise linear log-hazard model estimated the true mean survival time and survival distributions better than the piecewsie exponential hazard. Survival data from Wake Forest Baptist Medical Center is analyzed by both methods and the posterior results are compared.

On a class of non-linear transformation cure rate models

In this paper, we propose a generalization of the mixture (binary) cure rate model, motivated by the existence of a zero-modified (inflation or deflation) distribution, on the initial number of causes, under a competing cause scenario. This non-linear transformation cure rate model is in the same form of models studied in the past; however, following our approach, we are able to give a realistic interpretation to a specific class of proper transformation functions, for the cure rate modeling. The estimation of the parameters is then carried out using the maximum likelihood method along with a profile approach. A simulation study examines the accuracy of the proposed estimation method and the model discrimination based on the likelihood ratio test. For illustrative purposes, analysis of two real life data-sets, one on recidivism and another on cutaneous melanoma, is also carried out.

A flexible family of transformation cure rate models

In this paper, we introduce a flexible family of cure rate models, mainly motivated by the biological derivation of the classical promotion time cure rate model and assuming that a metastasis-competent tumor cell produces a detectable-tumor mass only when a specific number of distinct biological factors affect the cell. Special cases of the new model are, among others, the promotion time (proportional hazards), the geometric (proportional odds), and the negative binomial cure rate model. In addition, our model generalizes specific families of transformation cure rate models and some well-studied destructive cure rate models. Exact likelihood inference is carried out by the aid of the expectationŰmaximization algorithm; a profile likelihood approach is exploited for estimating the parameters of the model while model discrimination problem is analyzed by the aid of the likelihood ratio test. A simulation study demonstrates the accuracy of the proposed inferential method. Finally, as an illustration, we fit the proposed model to a cutaneous melanoma data-set. Copyright © 2017 John Wiley & Sons, Ltd.

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.