Influence diagnostics for the Weibull-Negative-Binomial regression model with cure rate under latent failure causes

In this paper, we propose a flexible cure rate survival model by assuming that the number of competing causes of the event of interest follows the Negative Binomial distribution and the time to event follows a Weibull distribution. Indeed, we introduce the Weibull-Negative-Binomial (WNB) distribution, which can be used in order to model survival data when the hazard rate function is increasing, decreasing and some non-monotonous shaped. Another advantage of the proposed model is that it has some distributions commonly used in lifetime analysis as particular cases. Moreover, the proposed model includes as special cases some of the well-know cure rate models discussed in the literature. We consider a frequentist analysis for parameter estimation of a WNB model with cure rate. Then, we derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. Finally, the methodology is illustrated on a medical data.     

MINIMUM QUADRATIC DISTANCE ESTIMATION FOR A PARAMETRIC FAMILY OF DISCRETE DISTRIBUTIONS DEFINED RECURSIVELY

The distribution function of a random sum can easily be computed iteratively when the distribution of the number of independent identically distributed elements in the sum is itself defined recursively. Classical estimation procedures for such recursive parametric families often require specific distributional assumptions (e.g. Poisson, Negative Binomial). The minimum distance estimator proposed here is an estimator within a larger parametric family. The estimator is consistent, efficient when the parametric family is truncated, and can be made either robust or asymptotically efficient when the parametric family has infinite range. Its asymptotic distribution is derived. A brief illustration with Automobile Insurance data is included.     

On the Bayesian estimation and influence diagnostics for the Weibull-Negative-Binomial regression model with cure rate under latent failure causes

The purpose of this paper is to develop a Bayesian approach for the Weibull-Negative-Binomial regression model with cure rate under latent failure causes and presence of randomized activation mechanisms. We assume the number of competing causes of the event of interest follows a Negative Binomial (NB) distribution while the latent lifetimes are assumed to follow a Weibull distribution. Markov chain Monte Carlos (MCMC) methods are used to develop the Bayesian procedure. Model selection to compare the fitted models is discussed. Moreover, we develop case deletion influence diagnostics for the joint posterior distribution based on the ψ-divergence, which has several divergence measures as particular cases. The developed procedures are illustrated with a real data set.     

A flexible parametric survival model for fitting time to event data in clinical trials

Time‐to‐event data are common in clinical trials to evaluate survival benefit of a new drug, biological product, or device. The commonly used parametric models including exponential, Weibull, Gompertz, log‐logistic, log‐normal, are simply not flexible enough to capture complex survival curves observed in clinical and medical research studies. On the other hand, the nonparametric Kaplan Meier (KM) method is very flexible and successful on catching the various shapes in the survival curves but lacks ability in predicting the future events such as the time for certain number of events and the number of events at certain time and predicting the risk of events (eg, death) over time beyond the span of the available data from clinical trials. It is obvious that neither the nonparametric KM method nor the current parametric distributions can fulfill the needs in fitting survival curves with the useful characteristics for predicting. In this paper, a full parametric distribution constructed as a mixture of three components of Weibull distribution is explored and recommended to fit the survival data, which is as flexible as KM for the observed data but have the nice features beyond the trial time, such as predicting future events, survival probability, and hazard function.     

Parameter-driven state-space model for integer-valued time series with application

Time series of counts occur in many different contexts, the counts being usually of certain events or objects in specified time intervals. In this paper we introduce a model called parameter-driven state-space model to analyse integer-valued time series data. A key property of such model is that the distribution of the observed count data is independent, conditional on the latent process, although the observations are correlated marginally. Our simulation shows that the Monte Carlo Expectation Maximization (MCEM) algorithm and the particle method are useful for the parameter estimation of the proposed model. In the application to Malaysia dengue data, our model fits better when compared with several other models including that of Yang et al. (2015)     

Estimation of zero-inflated parameter-driven models via data cloning

ABSTRACT

In this paper, we propose the use of the Data Cloning (DC) approach to estimate parameter-driven zero-inflated Poisson and Negative Binomial models for time series of counts. The data cloning algorithm obtains the familiar maximum likelihood estimators and their standard errors via a fully Bayesian estimation. This provides some computational ease as well as inferential tools such as confidence intervals and diagnostic methods which, otherwise, are not readily available for parameter-driven models. To illustrate the performance of the proposed method, we use Monte Carlo Simulations and real data on asthma-related emergency department visits in the Canadian province of Ontario.     

Regression for doubly inflated multivariate Poisson distributions

Dependent multivariate count data occur in several research studies. These data can be modelled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula-based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.     

Interval-censored data with misclassification: a Bayesian approach

Survival data involving silent events are often subject to interval censoring (the event is known to occur within a time interval) and classification errors if a test with no perfect sensitivity and specificity is applied. Considering the nature of this data plays an important role in estimating the time distribution until the occurrence of the event. In this context, we incorporate validation subsets into the parametric proportional hazard model, and show that this additional data, combined with Bayesian inference, compensate the lack of knowledge about test sensitivity and specificity improving the parameter estimates. The proposed model is evaluated through simulation studies, and Bayesian analysis is conducted within a Gibbs sampling procedure. The posterior estimates obtained under validation subset models present lower bias and standard deviation compared to the scenario with no validation subset or the model that assumes perfect sensitivity and specificity. Finally, we illustrate the usefulness of the new methodology with an analysis of real data about HIV acquisition in female sex workers that have been discussed in the literature.     

Buckley–James-type estimation of quantile regression with recurrent gap time data

In longitudinal studies, an individual may potentially undergo a series of repeated recurrence events. The gap times, which are referred to as the times between successive recurrent events, are typically the outcome variables of interest. Various regression models have been developed in order to evaluate covariate effects on gap times based on recurrence event data. The proportional hazards model, additive hazards model, and the accelerated failure time model are all notable examples. Quantile regression is a useful alternative to the aforementioned models for survival analysis since it can provide great flexibility to assess covariate effects on the entire distribution of the gap time. In order to analyze recurrence gap time data, we must overcome the problem of the last gap time subjected to induced dependent censoring, when numbers of recurrent events exceed one time. In this paper, we adopt the Buckley–James-type estimation method in order to construct a weighted estimation equation for regression coefficients under the quantile model, and develop an iterative procedure to obtain the estimates. We use extensive simulation studies to evaluate the finite-sample performance of the proposed estimator. Finally, analysis of bladder cancer data is presented as an illustration of our proposed methodology.     

A comparison of parametric and semi-parametric survival models with artificial neural networks

Survival models are used to examine data in the event of an occurrence. These are discussed in various types including parametric, non-parametric and semi-parametric models. Parametric models require a clear distribution of survival time, and semi-parametric models assume proportional hazards. Among these models, the non-parametric model of artificial neural network has the fewest assumptions and can be often replaced by other models. Given the importance of distribution Weibull survival models in this study of simulation shape parameter of the Weibull distribution have been assumed as 1, 2 and 3, and also the average rate at levels of 0%–75% have been censored. The values predicted by the neural network forecasting model with parametric survival and Cox regression models were compared. This comparison considering levels of complexity due to the hazard model using the ROC curve and the corresponding tests have been carried out.     

Modeling marginal features in studies of recurrent events in the presence of a terminal event

We study models for recurrent events with special emphasis on the situation where a terminal event acts as a competing risk for the recurrent events process and where there may be gaps between periods during which subjects are at risk for the recurrent event. We focus on marginal analysis of the expected number of events and show that an Aalen–Johansen type estimator proposed by Cook and Lawless is applicable in this situation. A motivating example deals with psychiatric hospital admissions where we supplement with analyses of the marginal distribution of time to the competing event and the marginal distribution of the time spent in hospital. Pseudo-observations are used for the latter purpose.

     

Evolution of recurrent asthma event rate over time in frailty models

Summary. To model the time evolution of the event rate in recurrent event data a crucial role is played by the timescale that is used. Depending on the timescale selected the interpretation of the time evolution will be entirely different, both in parametric and semiparametric frailty models. The gap timescale is more appropriate when studying the recurrent event rate as a function of time since the last event, whereas the calendar timescale keeps track of actual time. We show both timescales in action on data from an asthma prevention trial in young children. The frailty model is further extended to include both timescales simultaneously as this might be most relevant in practice.     

A Semi‐parametric Transformation Frailty Model for Semi‐competing Risks Survival Data

In the analysis of semi‐competing risks data interest lies in estimation and inference with respect to a so‐called non‐terminal event, the observation of which is subject to a terminal event. Multi‐state models are commonly used to analyse such data, with covariate effects on the transition/intensity functions typically specified via the Cox model and dependence between the non‐terminal and terminal events specified, in part, by a unit‐specific shared frailty term. To ensure identifiability, the frailties are typically assumed to arise from a parametric distribution, specifically a Gamma distribution with mean 1.0 and variance, say, σ². When the frailty distribution is misspecified, however, the resulting estimator is not guaranteed to be consistent, with the extent of asymptotic bias depending on the discrepancy between the assumed and true frailty distributions. In this paper, we propose a novel class of transformation models for semi‐competing risks analysis that permit the non‐parametric specification of the frailty distribution. To ensure identifiability, the class restricts to parametric specifications of the transformation and the error distribution; the latter are flexible, however, and cover a broad range of possible specifications. We also derive the semi‐parametric efficient score under the complete data setting and propose a non‐parametric score imputation method to handle right censoring; consistency and asymptotic normality of the resulting estimators is derived and small‐sample operating characteristics evaluated via simulation. Although the proposed semi‐parametric transformation model and non‐parametric score imputation method are motivated by the analysis of semi‐competing risks data, they are broadly applicable to any analysis of multivariate time‐to‐event outcomes in which a unit‐specific shared frailty is used to account for correlation. Finally, the proposed model and estimation procedures are applied to a study of hospital readmission among patients diagnosed with pancreatic cancer.     

Untangle the structural and random zeros in statistical modelings

Count data with structural zeros are common in public health applications. There are considerable researches focusing on zero-inflated models such as zero-inflated Poisson (ZIP) and zero-inflated Negative Binomial (ZINB) models for such zero-inflated count data when used as response variable. However, when such variables are used as predictors, the difference between structural and random zeros is often ignored and may result in biased estimates. One remedy is to include an indicator of the structural zero in the model as a predictor if observed. However, structural zeros are often not observed in practice, in which case no statistical method is available to address the bias issue. This paper is aimed to fill this methodological gap by developing parametric methods to model zero-inflated count data when used as predictors based on the maximum likelihood approach. The response variable can be any type of data including continuous, binary, count or even zero-inflated count responses. Simulation studies are performed to assess the numerical performance of this new approach when sample size is small to moderate. A real data example is also used to demonstrate the application of this method.     

The definition of start time in cancer treatment studies analysed by non-mixture cure models

Non-mixture cure models are derived from a simplified representation of the biological process that takes place after treatment for cancer. These models are intended to represent the time from the end of treatment to the time of first recurrence of the cancer in studies when a proportion of those treated are completely cured. However, for many studies, other start times are more relevant. In a clinical trial, it may be more natural to model the time from randomisation rather than the time from the end of treatment and in an epidemiological study, the time from diagnosis might be more meaningful. Some simulations and two real studies of childhood cancer are presented to show that starting from time of diagnosis or randomisation can affect the estimates of the cure fraction. The susceptibility of different parametric kernels to errors caused by using start times other than the end of treatment is also assessed. Analysing failures on treatment and relapse after completing the treatment as two processes offers a simple way of overcoming many of these problems.     

Weibull prediction of event times in clinical trials

In clinical trials with interim analyses planned at pre-specified event counts, one may wish to predict the times of these landmark events as a tool for logistical planning. Currently available methods use either a parametric approach based on an exponential model for survival (Bagiella and Heitjan, Statistics in Medicine 2001; 20:2055) or a non-parametric approach based on the Kaplan-Meier estimate (Ying et al., Clinical Trials 2004; 1:352). Ying et al. (2004) demonstrated the trade-off between bias and variance in these models; the exponential method is highly efficient when its assumptions hold but potentially biased when they do not, whereas the non-parametric method has minimal bias and is well calibrated under a range of survival models but typically gives wider prediction intervals and may fail to produce useful predictions early in the trial. As a potential compromise, we propose here to make predictions under a Weibull survival model. Computations are somewhat more difficult than with the simpler exponential model, but Monte Carlo studies show that predictions are robust under a broader range of assumptions. We demonstrate the method using data from a trial of immunotherapy for chronic granulomatous disease.     

Tests for outliers in the inverse Gaussian distribution,with application to first hitting time models

The inverse Gaussian (IG) distribution is often applied in statistical modelling, especially with lifetime data. We present tests for outlying values of the parameters (μ, λ) of this distribution when data are available from a sample of independent units and possibly with more than one event per unit. Outlier tests are constructed from likelihood ratio tests for equality of parameters. The test for an outlying value of λ is based on an F-distributed statistic that is transformed to an approximate normal statistic when there are unequal numbers of events per unit. Simulation studies are used to confirm that Bonferroni tests have accurate size and to examine the powers of the tests. The application to first hitting time models, where the IG distribution is derived from an underlying Wiener process, is described. The tests are illustrated on data concerning the strength of different lots of insulating material.     

Predictive event modelling in multicenter clinical trials with waiting time to response

A new analytic statistical technique for predictive event modeling in ongoing multicenter clinical trials with waiting time to response is developed. It allows for the predictive mean and predictive bounds for the number of events to be constructed over time, accounting for the newly recruited patients and patients already at risk in the trial, and for different recruitment scenarios. For modeling patient recruitment, an advanced Poisson-gamma model is used, which accounts for the variation in recruitment over time, the variation in recruitment rates between different centers and the opening or closing of some centers in the future. A few models for event appearance allowing for 'recurrence', 'death' and 'lost-to-follow-up' events and using finite Markov chains in continuous time are considered. To predict the number of future events over time for an ongoing trial at some interim time, the parameters of the recruitment and event models are estimated using current data and then the predictive recruitment rates in each center are adjusted using individual data and Bayesian re-estimation. For a typical scenario (continue to recruit during some time interval, then stop recruitment and wait until a particular number of events happens), the closed-form expressions for the predictive mean and predictive bounds of the number of events at any future time point are derived under the assumptions of Markovian behavior of the event progression. The technique is efficiently applied to modeling different scenarios for some ongoing oncology trials. Case studies are considered.     

Planning and analyzing clinical trials with competing risks: Recommendations for choosing appropriate statistical methodology

In the analysis of time‐to‐event data, competing risks occur when multiple event types are possible, and the occurrence of a competing event precludes the occurrence of the event of interest. In this situation, statistical methods that ignore competing risks can result in biased inference regarding the event of interest. We review the mechanisms that lead to bias and describe several statistical methods that have been proposed to avoid bias by formally accounting for competing risks in the analyses of the event of interest. Through simulation, we illustrate that Gray's test should be used in lieu of the logrank test for nonparametric hypothesis testing. We also compare the two most popular models for semiparametric modelling: the cause‐specific hazards (CSH) model and Fine‐Gray (F‐G) model. We explain how to interpret estimates obtained from each model and identify conditions under which the estimates of the hazard ratio and subhazard ratio differ numerically. Finally, we evaluate several model diagnostic methods with respect to their sensitivity to detect lack of fit when the CSH model holds, but the F‐G model is misspecified and vice versa. Our results illustrate that adequacy of model fit can strongly impact the validity of statistical inference. We recommend analysts incorporate a model diagnostic procedure and contingency to explore other appropriate models when designing trials in which competing risks are anticipated.     

Regression splines in the quasi-likelihood analysis of recurrent event data

In many longitudinal studies of recurrent events there is an interest in assessing how recurrences vary over time and across treatments or strata in the population. Usual analyses of such data assume a parametric form for the distribution of the recurrences over time. Here, we consider a semiparametric model for the analysis of such longitudinal studies where data are collected as panel counts. The model is a non-homogeneous Poisson process with a multiplicative intensity incorporating covariates through a proportionality assumption. Heterogeneity is accounted for in the model through subject-specific random effects. The key feature of the model is the use of regression splines to model the distribution of recurrences over time. This provides a flexible and robust method of relaxing parametric assumptions. In addition, quasi-likelihood methods are proposed for estimation, requiring only first and second moment assumptions to obtain consistent estimates. Simulations demonstrate that the method produces estimators of the rate with low bias and whose standardized distributions are well approximated by the normal. The usefulness of this approach, especially as an exploratory tool, is illustrated by analyzing a study designed to assess the effectiveness of a pheromone treatment in disturbing the mating habits of the Cherry Bark Tortrix moth.