Maximum likelihood methods for cure rate models with missing covariates

A generalized Gompertz promotion time cure model and its fitness to cancer data

The cure models based on standard distributions like exponential, Weibull, lognormal, Gompertz, gamma, are often used to analyze survival data from cancer clinical trials with long-term survivors. Sometimes, the data is simple, and the standard cure models fit them very well, however, most often the data are complex and the standard cure models don't fit them reasonably well. In this article, we offer a novel generalized Gompertz promotion time cure model and illustrate its fitness to gastric cancer data by three different methods. The generalized Gompertz distribution is as simple as the generalized Weibull distribution and is not computationally as intensive as the generalized F distribution. One detailed real data application is provided for illustration and comparison purposes.

Compatibility in imputation specification

Missing data such as data missing at random (MAR) are unavoidable in real data and have the potential to undermine the validity of research results. Multiple imputation is one of the most widely used MAR-based methods in education and behavioral science applications. Arbitrarily specifying imputation models can lead to incompatibility and cause biased estimation. Building on the recent developments of model-based imputation and Arnold's compatibility work, this paper systematically summarizes when the traditional fully conditional specification (FCS) is applicable and how to specify a model-based imputation model if needed. We summarize two Compatibility Requirements to help researchers check compatibility more easily and a decision tree to check whether the traditional FCS is applicable in a given scenario. Additionally, we present a clear overview of two types of model-based imputation: the sequential and separate specifications. We illustrate how to specify model-based imputation with examples. Additionally, we provide example code of a free software program, Blimp, for implementing model-based imputation.

Promotion time cure rate model with a neural network estimated nonparametric component

Promotion time cure rate models (PCM) are often used to model the survival data with a cure fraction. Medical images or biomarkers derived from medical images can be the key predictors in survival models. However, incorporating images in the PCM is challenging using traditional nonparametric methods such as splines. We propose to use neural network to model the nonparametric or unstructured predictors' effect in the PCM context. Expectation-maximization algorithm with neural network for the M-step is used for parameter estimation. Asymptotic properties of the proposed estimates are derived. Simulation studies show good performance in terms of both prediction and estimation. We finally apply our methods to analyze the brain images from open access series of imaging studies data.

Red propolis effect analysis of dairy calves health based on Weibull regression model with long-term survivors

Diarrhea is the most common cause of mortality and morbidity in dairy calves during the first weeks of life. It is responsible for the majority of costs related to animal death and treatments, as well as lower productivity due to reduced weight gain. Therefore, studies that focus on strategies to reduce diarrhea incidence and to improve animal welfare are very important for the dairy industry. For that reason, the beneficial effects of red propolis on the health status of preweaned dairy calves was studied. Animal disease data usually present incomplete observations of interest time, so-called censored observations and one of the statistical techniques for this modeling type is the survival analysis, hence it is a set of methods for analyzing data where the response variable is the time until the occurrence of an event of interest. We propose among some methods of survival data analysis, the long-term models. The motivation to study these models is the fact that part of the population is not susceptible to the event of interest during the period of the study, considered as immune or cured. In this paper, we studied the Weibull distribution in a structure of long-term model, including the covariates in the proportion of cured through the logistic link function. Besides, we used the residual analysis to check the assumptions of the model. The reason for the choice of the Weibull distribution was that this model is very flexibility to model a variety of data sets, among them animal science and long-term survival data. We illustrate its application with a case study from an animal experiment, which examined the time till the occurrence of diarrhea in Holstein calves, where a proportion of the animals were not susceptible to this health condition. This experiment aimed to verify the efficiency of red propolis in disease prevention and the influence of that on the proportion of animals that are not susceptible to diarrhea.

Bayesian joint analysis using a semiparametric latent variable model with non-ignorable missing covariates for CHNS data

Motivated by the China Health and Nutrition Survey (CHNS) data, a semiparametric latent variable model with a Dirichlet process (DP) mixtures prior on the latent variable is proposed to jointly analyse mixed binary and continuous responses. Non-ignorable missing covariates are considered through a selection model framework where a missing covariate model and a missing data mechanism model are included. The logarithm of the pseudo-marginal likelihood (LPML) is applied for selecting the priors, and the deviance information criterion measure focusing on the missing data mechanism model only is used for selecting different missing data mechanisms. A Bayesian index of local sensitivity to non-ignorability (ISNI) is extended to explore the local sensitivity of the parameters in our model. A simulation study is carried out to examine the empirical performance of the proposed methodology. Finally, the proposed model and the ISNI index are applied to analyse the CHNS data in the motivating example.

Missing data strategies for time-varying confounders in comparative effectiveness studies of non-missing time-varying exposures and right-censored outcomes

The treatment of missing data in comparative effectiveness studies with right-censored outcomes and time-varying covariates is challenging because of the multilevel structure of the data. In particular, the performance of an accessible method like multiple imputation (MI) under an imputation model that ignores the multilevel structure is unknown and has not been compared to complete-case (CC) and single imputation methods that are most commonly applied in this context. Through an extensive simulation study, we compared statistical properties among CC analysis, last value carried forward, mean imputation, the use of missing indicators, and MI-based approaches with and without auxiliary variables under an extended Cox model when the interest lies in characterizing relationships between non-missing time-varying exposures and right-censored outcomes. MI demonstrated favorable properties under a moderate missing-at-random condition (absolute bias <0.1) and outperformed CC and single imputation methods, even when the MI method did not account for correlated observations in the imputation model. The performance of MI decreased with increasing complexity such as when the missing data mechanism involved the exposure of interest, but was still preferred over other methods considered and performed well in the presence of strong auxiliary variables. We recommend considering MI that ignores the multilevel structure in the imputation model when data are missing in a time-varying confounder, incorporating variables associated with missingness in the MI models as well as conducting sensitivity analyses across plausible assumptions.

Identifying the source of food-borne disease outbreaks: An application of Bayesian variable selection

Early identification of contaminated food products is crucial in reducing health burdens of food-borne disease outbreaks. Analytic case-control studies are primarily used in this identification stage by comparing exposures in cases and controls using logistic regression. Standard epidemiological analysis practice is not formally defined and the combination of currently applied methods is subject to issues such as response misclassification, missing values, multiple testing problems and small sample estimation problems resulting in biased and possibly misleading results. In this paper, we develop a formal Bayesian variable selection method to account for misclassified responses and missing covariates, which are common complications in food-borne outbreak investigations. We illustrate the implementation and performance of our method on a Salmonella Thompson outbreak in the Netherlands in 2012. Our method is shown to perform better than the standard logistic regression approach with respect to earlier identification of contaminated food products. It also allows relatively easy implementation of otherwise complex methodological issues.

Evaluation of Alzheimer's disease progression based on clinical dementia rating scale with missing responses and covariates

In clinical trials, patient's disease severity is usually assessed on a Likert-type scale. Patients, however, may miss one or more follow-up visits (non-monotone missing). The statistical analysis of non-Gaussian longitudinal data with non-monotone missingness is difficult to handle, particularly when both response and time-dependent covariates are subject to such missingness. Even when the number of patients with intermittent missing data is small, ignoring those patients from analysis seems to be unsatisfactory. The focus of the current investigation is to study the progression of Alzheimer's disease by incorporating a non-ignorable missing data mechanism for both response and covariates in a longitudinal setup. Combining the cumulative logit longitudinal model for Alzheimer's disease progression with the bivariate binary model for the missing pattern, we develop a joint likelihood. The parameters are then estimated using the Monte Carlo Newton Raphson Expectation Maximization (MCNREM) method. This approach is quite easy to handle and the convergence of the estimates is attained in a reasonable amount of time. The study reveals that apolipo-protein plays a significant role in assessing a patient's disease severity. A detailed simulation has also been carried out for justifying the performance of our approach.

Variable selection for mixture and promotion time cure rate models

Failure-time data with cured patients are common in clinical studies. Data from these studies are typically analyzed with cure rate models. Variable selection methods have not been well developed for cure rate models. In this research, we propose two least absolute shrinkage and selection operators based methods, for variable selection in mixture and promotion time cure models with parametric or nonparametric baseline hazards. We conduct an extensive simulation study to assess the operating characteristics of the proposed methods. We illustrate the use of the methods using data from a study of childhood wheezing.

On the comparison of risk of death according to different stages of breast cancer via the long-term exponentiated Weibull hazard model

Long-term survivor models have been extensively used for modelling time-to-event data with a significant proportion of patients who do not experience poor outcome. In this paper, we propose a new long-term survivor hazard model, which accommodates comprehensive families of cure rate models as particular cases, including modified Weibull, exponentiated Weibull, Weibull, exponential and Rayleigh distribution, among others. The maximum likelihood estimation procedure is presented. A simulation study evaluates bias and mean square error of the considered estimation procedure as well as the coverage probabilities of the parameters asymptotic and bootstrap confidence intervals. A real Brazilian dataset on breast cancer illustrates the methodology. From the practical point of view, under our modelling, we provide a parameter that works as a metric to quantify and compare the risk between different stages of the disease. We emphasize that, we developed an online platform for oncologists to calculate the probability of survival of patients diagnosed with breast cancer according to the stage of the disease in real time.

Dealing with missing covariates in epidemiologic studies: a comparison between multiple imputation and a full Bayesian approach

Incomplete data are generally a challenge to the analysis of most large studies. The current gold standard to account for missing data is multiple imputation, and more specifically multiple imputation with chained equations (MICE). Numerous studies have been conducted to illustrate the performance of MICE for missing covariate data. The results show that the method works well in various situations. However, less is known about its performance in more complex models, specifically when the outcome is multivariate as in longitudinal studies. In current practice, the multivariate nature of the longitudinal outcome is often neglected in the imputation procedure, or only the baseline outcome is used to impute missing covariates. In this work, we evaluate the performance of MICE using different strategies to include a longitudinal outcome into the imputation models and compare it with a fully Bayesian approach that jointly imputes missing values and estimates the parameters of the longitudinal model. Results from simulation and a real data example show that MICE requires the analyst to correctly specify which components of the longitudinal process need to be included in the imputation models in order to obtain unbiased results. The full Bayesian approach, on the other hand, does not require the analyst to explicitly specify how the longitudinal outcome enters the imputation models. It performed well under different scenarios. Copyright © 2016 John Wiley & Sons, Ltd.

Diagnostic Measures for the Cox Regression Model with Missing Covariates

This paper investigates diagnostic measures for assessing the influence of observations and model misspecification in the presence of missing covariate data for the Cox regression model. Our diagnostics include case-deletion measures, conditional martingale residuals, and score residuals. The Q-distance is proposed to examine the effects of deleting individual observations on the estimates of finite-dimensional and infinite-dimensional parameters. Conditional martingale residuals are used to construct goodness of fit statistics for testing possible misspecification of the model assumptions. A resampling method is developed to approximate the p-values of the goodness of fit statistics. Simulation studies are conducted to evaluate our methods, and a real data set is analyzed to illustrate their use.

Bayesian probability of success for clinical trials using historical data

Developing sophisticated statistical methods for go/no-go decisions is crucial for clinical trials, as planning phase III or phase IV trials is costly and time consuming. In this paper, we develop a novel Bayesian methodology for determining the probability of success of a treatment regimen on the basis of the current data of a given trial. We introduce a new criterion for calculating the probability of success that allows for inclusion of covariates as well as allowing for historical data based on the treatment regimen, and patient characteristics. A new class of prior distributions and covariate distributions is developed to achieve this goal. The methodology is quite general and can be used with univariate or multivariate continuous or discrete data, and it generalizes Chuang-Stein's work. This methodology will be invaluable for informing the scientist on the likelihood of success of the compound, while including the information of covariates for patient characteristics in the trial population for planning future pre-market or post-market trials.

Improved survival modeling in cancer research using a reduced piecewise exponential approach

Statistical models for survival data are typically nonparametric, for example, the Kaplan-Meier curve. Parametric survival modeling, such as exponential modeling, however, can reveal additional insights and be more efficient than nonparametric alternatives. A major constraint of the existing exponential models is the lack of flexibility due to distribution assumptions. A flexible and parsimonious piecewise exponential model is presented to best use the exponential models for arbitrary survival data. This model identifies shifts in the failure rate over time based on an exact likelihood ratio test, a backward elimination procedure, and an optional presumed order restriction on the hazard rate. Such modeling provides a descriptive tool in understanding the patient survival in addition to the Kaplan-Meier curve. This approach is compared with alternative survival models in simulation examples and illustrated in clinical studies.

Addressing missing covariates for the regression analysis of competing risks: Prognostic modelling for triaging patients diagnosed with prostate cancer

Competing risks arise in medical research when subjects are exposed to various types or causes of death. Data from large cohort studies usually exhibit subsets of regressors that are missing for some study subjects. Furthermore, such studies often give rise to censored data. In this article, a carefully formulated likelihood-based technique for the regression analysis of right-censored competing risks data when two of the covariates are discrete and partially missing is developed. The approach envisaged here comprises two models: one describes the covariate effects on both long-term incidence and conditional latencies for each cause of death, whilst the other deals with the observation process by which the covariates are missing. The former is formulated with a well-established mixture model and the latter is characterised by copula-based bivariate probability functions for both the missing covariates and the missing data mechanism. The resulting formulation lends itself to the empirical assessment of non-ignorability by performing sensitivity analyses using models with and without a non-ignorable component. The methods are illustrated on a 20-year follow-up involving a prostate cancer cohort from the National Cancer Institutes Surveillance, Epidemiology, and End Results program.

A new threshold regression model for survival data with a cure fraction

Due to the fact that certain fraction of the population suffering a particular type of disease get cured because of advanced medical treatment and health care system, we develop a general class of models to incorporate a cure fraction by introducing the latent number N of metastatic-competent tumor cells or infected cells caused by bacteria or viral infection and the latent antibody level R of immune system. Various properties of the proposed models are carefully examined and a Markov chain Monte Carlo sampling algorithm is developed for carrying out Bayesian computation for model fitting and comparison. A real data set from a prostate cancer clinical trial is analyzed in detail to demonstrate the proposed methodology.

A hands-on approach for fitting long-term survival models under the GAMLSS framework

In many data sets from clinical studies there are patients insusceptible to the occurrence of the event of interest. Survival models which ignore this fact are generally inadequate. The main goal of this paper is to describe an application of the generalized additive models for location, scale, and shape (GAMLSS) framework to the fitting of long-term survival models. In this work the number of competing causes of the event of interest follows the negative binomial distribution. In this way, some well known models found in the literature are characterized as particular cases of our proposal. The model is conveniently parameterized in terms of the cured fraction, which is then linked to covariates. We explore the use of the gamlss package in R as a powerful tool for inference in long-term survival models. The procedure is illustrated with a numerical example.

Maximum Likelihood Inference for the Cox Regression Model with Applications to Missing Covariates

In this paper, we carry out an in-depth theoretical investigation for existence of maximum likelihood estimates for the Cox model (Cox, 1972, 1975) both in the full data setting as well as in the presence of missing covariate data. The main motivation for this work arises from missing data problems, where models can easily become difficult to estimate with certain missing data configurations or large missing data fractions. We establish necessary and sufficient conditions for existence of the maximum partial likelihood estimate (MPLE) for completely observed data (i.e., no missing data) settings as well as sufficient conditions for existence of the maximum likelihood estimate (MLE) for survival data with missing covariates via a profile likelihood method. Several theorems are given to establish these conditions. A real dataset from a cancer clinical trial is presented to further illustrate the proposed methodology.

Missing data methods in longitudinal studies: a review

Incomplete data are quite common in biomedical and other types of research, especially in longitudinal studies. During the last three decades, a vast amount of work has been done in the area. This has led, on the one hand, to a rich taxonomy of missing-data concepts, issues, and methods and, on the other hand, to a variety of data-analytic tools. Elements of taxonomy include: missing data patterns, mechanisms, and modeling frameworks; inferential paradigms; and sensitivity analysis frameworks. These are described in detail. A variety of concrete modeling devices is presented. To make matters concrete, two case studies are considered. The first one concerns quality of life among breast cancer patients, while the second one examines data from the Muscatine children's obesity study.

Estimating the grid of time-points for the piecewise exponential model

One of the greatest challenges related to the use of piecewise exponential models (PEMs) is to find an adequate grid of time-points needed in its construction. In general, the number of intervals in such a grid and the position of their endpoints are ad-hoc choices. We extend previous works by introducing a full Bayesian approach for the piecewise exponential model in which the grid of time-points (and, consequently, the endpoints and the number of intervals) is random. We estimate the failure rates using the proposed procedure and compare the results with the non-parametric piecewise exponential estimates. Estimates for the survival function using the most probable partition are compared with the Kaplan-Meier estimators (KMEs). A sensitivity analysis for the proposed model is provided considering different prior specifications for the failure rates and for the grid. We also evaluate the effect of different percentage of censoring observations in the estimates. An application to a real data set is also provided. We notice that the posteriors are strongly influenced by prior specifications, mainly for the failure rates parameters. Thus, the priors must be fairly built, say, really disclosing the expert prior opinion.

Semiparametric Models for Missing Covariate and Response Data in Regression Models

We consider a class of semiparametric models for the covariate distribution and missing data mechanism for missing covariate and/or response data for general classes of regression models including generalized linear models and generalized linear mixed models. Ignorable and nonignorable missing covariate and/or response data are considered. The proposed semiparametric model can be viewed as a sensitivity analysis for model misspecification of the missing covariate distribution and/or missing data mechanism. The semiparametric model consists of a generalized additive model (GAM) for the covariate distribution and/or missing data mechanism. Penalized regression splines are used to express the GAMs as a generalized linear mixed effects model, in which the variance of the corresponding random effects provides an intuitive index for choosing between the semiparametric and parametric model. Maximum likelihood estimates are then obtained via the EM algorithm. Simulations are given to demonstrate the methodology, and a real data set from a melanoma cancer clinical trial is analyzed using the proposed methods.

Bayesian approaches to joint cure-rate and longitudinal models with applications to cancer vaccine trials

Complex issues arise when investigating the association between longitudinal immunologic measures and time to an event, such as time to relapse, in cancer vaccine trials. Unlike many clinical trials, we may encounter patients who are cured and no longer susceptible to the time-to-event endpoint. If there are cured patients in the population, there is a plateau in the survival function, S(t), after sufficient follow-up. If we want to determine the association between the longitudinal measure and the time-to-event in the presence of cure, existing methods for jointly modeling longitudinal and survival data would be inappropriate, since they do not account for the plateau in the survival function. The nature of the longitudinal data in cancer vaccine trials is also unique, as many patients may not exhibit an immune response to vaccination at varying time points throughout the trial. We present a new joint model for longitudinal and survival data that accounts both for the possibility that a subject is cured and for the unique nature of the longitudinal data. An example is presented from a cancer vaccine clinical trial.

Estimating Cure Rates From Survival Data: An Alternative to Two-Component Mixture Models

This article considers the utility of the bounded cumulative hazard model in cure rate estimation, which is an appealing alternative to the widely used two-component mixture model. This approach has the following distinct advantages: (1) It allows for a natural way to extend the proportional hazards regression model, leading to a wide class of extended hazard regression models. (2) In some settings the model can be interpreted in terms of biologically meaningful parameters. (3) The model structure is particularly suitable for semiparametric and Bayesian methods of statistical inference. Notwithstanding the fact that the model has been around for less than a decade, a large body of theoretical results and applications has been reported to date. This review article is intended to give a big picture of these modeling techniques and associated statistical problems. These issues are discussed in the context of survival data in cancer.

A mixture model for bovine abortion and foetal survival

The effect of spontaneous abortion on the dairy industry is substantial, costing the industry on the order of US dollars 200 million per year in California alone. We analyse data from a cohort study of nine dairy herds in Central California. A key feature of the analysis is the observation that only a relatively small proportion of cows will abort (around 10;15 per cent), so that it is inappropriate to analyse the time-to-abortion (TTA) data as if it were standard censored survival data, with cows that fail to abort by the end of the study treated as censored observations. We thus broaden the scope to consider the analysis of foetal lifetime distribution (FLD) data for the cows, with the dual goals of characterizing the effects of various risk factors on (i). the likelihood of abortion and, conditional on abortion status, on (ii). the risk of early versus late abortion. A single model is developed to accomplish both goals with two sets of specific herd effects modelled as random effects. Because multimodal foetal hazard functions are expected for the TTA data, both a parametric mixture model and a non-parametric model are developed. Furthermore, the two sets of analyses are linked because of anticipated dependence between the random herd effects. All modelling and inferences are accomplished using modern Bayesian methods.

The analysis of survival data with a non-susceptible fraction and dual censoring mechanisms

It is known that the ages of onset of many diseases are determined by both a genetic predisposition to disease as well as environmental risk factors that are capable of either triggering or hastening the onset of disease. Difficulties in modelling onset ages arise when a large fraction fail to inherit the disease-causing gene, and multiple reasons for censoring result in unobserved onset ages. We present a parametric Bayesian model that includes subjects with missing age information, non-susceptible subjects and allows for regression on risk factor information. The model is fit using Markov chain Monte Carlo simulation from the posterior distribution, and allows the simultaneous estimation of the proportion of the population at risk of disease, the mean onset age of disease, survival after disease onset, and the association of risk factors with susceptibility, onset age and survival after onset. An example employing Huntington's disease data is presented.