Quantifying rater variation for ordinal data using a rating scale model

We present a model‐based approach to the analysis of agreement between different raters in a situation where all raters have supplied ordinal ratings of the same cases in a sample. It is assumed that no “gold standard” is available. The model is an ordinal regression model with random effects—a so‐called rating scale model. The model includes case‐specific parameters that allow each case his or hers own level (disease severity). It also allows raters to have different propensities to score a given set of individuals more or less positively—the rater level. Based on the model, we suggest quantifying the rater variation using the median odds ratio. This allows expressing the variation on the same scale as the observed ordinal data. An important example that will serve to motivate and illustrate the proposed model is the study of breast cancer diagnosis based on screening mammograms. The purpose of the assessment is to detect early breast cancer in order to obtain improved cancer survival. In the study, mammograms from 148 women were evaluated by 110 expert radiologists. The experts were asked to rate each mammogram on a 5‐point scale ranging from “normal” to “probably malignant.”     

Estimating multilevel regional variation in excess mortality of cancer patients using integrated nested Laplace approximation

Models of excess mortality with random effects were used to estimate regional variation in relative or net survival of cancer patients. Statistical inference for these models based on the Markov chain Monte Carlo (MCMC) methods is computationally intensive and, therefore, not feasible for routine analyses of cancer register data. This study assessed the performance of the integrated nested Laplace approximation (INLA) in monitoring regional variation in cancer survival. Poisson regression model of excess mortality including both spatially correlated and unstructured random effects was fitted to the data of patients diagnosed with ovarian and breast cancer in Finland during 1955-2014 with follow up from 1960 through 2014 by using the period approach with five-year calendar time windows. We estimated standard deviations associated with variation (i) between hospital districts and (ii) between municipalities within hospital districts. Posterior estimates based on the INLA approach were compared to those based on the MCMC simulation. The estimates of the variation parameters were similar between the two approaches. Variation within hospital districts dominated in the total variation between municipalities. In 2000-2014, the proportion of the average variation within hospital districts was 68% (95% posterior interval: 35%-93%) and 82% (60%-98%) out of the total variation in ovarian and breast cancer, respectively. In the estimation of regional variation, the INLA approach was accurate, fast, and easy to implement by using the R-INLA package.     

Semiparametric accelerated failure time cure rate mixture models with competing risks

Modern medical treatments have substantially improved survival rates for many chronic diseases and have generated considerable interest in developing cure fraction models for survival data with a non‐ignorable cured proportion. Statistical analysis of such data may be further complicated by competing risks that involve multiple types of endpoints. Regression analysis of competing risks is typically undertaken via a proportional hazards model adapted on cause‐specific hazard or subdistribution hazard. In this article, we propose an alternative approach that treats competing events as distinct outcomes in a mixture. We consider semiparametric accelerated failure time models for the cause‐conditional survival function that are combined through a multinomial logistic model within the cure‐mixture modeling framework. The cure‐mixture approach to competing risks provides a means to determine the overall effect of a treatment and insights into how this treatment modifies the components of the mixture in the presence of a cure fraction. The regression and nonparametric parameters are estimated by a nonparametric kernel‐based maximum likelihood estimation method. Variance estimation is achieved through resampling methods for the kernel‐smoothed likelihood function. Simulation studies show that the procedures work well in practical settings. Application to a sarcoma study demonstrates the use of the proposed method for competing risk data with a cure fraction.     

Bayesian random threshold estimation in a Cox proportional hazards cure model

In this paper, we develop a Bayesian approach to estimate a Cox proportional hazards model that allows a threshold in the regression coefficient, when some fraction of subjects are not susceptible to the event of interest. A data augmentation scheme with latent binary cure indicators is adopted to simplify the Markov chain Monte Carlo implementation. Given the binary cure indicators, the Cox cure model reduces to a standard Cox model and a logistic regression model. Furthermore, the threshold detection problem reverts to a threshold problem in a regular Cox model. The baseline cumulative hazard for the Cox model is formulated non‐parametrically using counting processes with a gamma process prior. Simulation studies demonstrate that the method provides accurate point and interval estimates. Application to a data set of oropharynx cancer patients suggests a significant threshold in age at diagnosis such that the effect of gender on disease‐specific survival changes after the threshold. Copyright © 2013 John Wiley & Sons, Ltd.     

Mixture cure model with random effects for the analysis of a multi-center tonsil cancer study

Cure models for clustered survival data have the potential for broad applicability. In this paper, we consider the mixture cure model with random effects and propose several estimation methods based on Gaussian quadrature, rejection sampling, and importance sampling to obtain the maximum likelihood estimates of the model for clustered survival data with a cure fraction. The methods are flexible to accommodate various correlation structures. A simulation study demonstrates that the maximum likelihood estimates of parameters in the model tend to have smaller biases and variances than the estimates obtained from the existing methods. We apply the model to a study of tonsil cancer patients clustered by treatment centers to investigate the effect of covariates on the cure rate and on the failure time distribution of the uncured patients. The maximum likelihood estimates of the parameters demonstrate strong correlation among the failure times of the uncured patients and weak correlation among cure statuses in the same center.     

Cure rate model with interval censored data

In cancer trials, a significant fraction of patients can be cured, that is, the disease is completely eliminated, so that it never recurs. In general, treatments are developed to both increase the patients' chances of being cured and prolong the survival time among non-cured patients. A cure rate model represents a combination of cure fraction and survival model, and can be applied to many clinical studies over several types of cancer. In this article, the cure rate model is considered in the interval censored data composed of two time points, which include the event time of interest. Interval censored data commonly occur in the studies of diseases that often progress without symptoms, requiring clinical evaluation for detection (Encyclopedia of Biostatistics. Wiley: New York, 1998; 2090-2095). In our study, an approximate likelihood approach suggested by Goetghebeur and Ryan (Biometrics 2000; 56:1139-1144) is used to derive the likelihood in interval censored data. In addition, a frailty model is introduced to characterize the association between the cure fraction and survival model. In particular, the positive association between the cure fraction and the survival time is incorporated by imposing a common normal frailty effect. The EM algorithm is used to estimate parameters and a multiple imputation based on the profile likelihood is adopted for variance estimation. The approach is applied to the smoking cessation study in which the event of interest is a smoking relapse and several covariates including an intensive care treatment are evaluated to be effective for both the occurrence of relapse and the non-smoking duration.     

Joint model for a diagnostic test without a gold standard in the presence of a dependent terminal event

Breast cancer patients after breast conservation therapy often develop ipsilateral breast tumor relapse (IBTR), whose classification (true local recurrence versus new ipsilateral primary tumor) is subject to error, and there is no available gold standard. Some patients may die because of breast cancer before IBTR develops. Because this terminal event may be related to the individual patient's unobserved disease status and time to IBTR, the terminal mechanism is non‐ignorable. This article presents a joint analysis framework to model the binomial regression with misclassified binary outcome and the correlated time to IBTR, subject to a dependent terminal event and in the absence of a gold standard. Shared random effects are used to link together two survival times. The proposed approach is evaluated by a simulation study and is applied to a breast cancer data set consisting of 4477 breast cancer patients. The proposed joint model can be conveniently fit using adaptive Gaussian quadrature tools implemented in SAS 9.3 (SAS Institute Inc., Cary, NC, USA) procedure NLMIXED . Copyright © 2014 John Wiley & Sons, Ltd.     

Estimating the proportion of cured patients in a censored sample

There has been a recurring interest in modelling survival data which hypothesize subpopulations of individuals highly susceptible to some types of adverse events while other individuals are assumed to be at much less risk, like recurrence of breast cancer. A binary random effect is assumed in this article to model the susceptibility of each individual. We propose a simple multiple imputation algorithm for the analysis of censored data which combines a binary regression formulation for the probability of occurrence of an event, say recurrence of the breast cancer tumour, and a Cox's proportional hazards regression model for the time to occurrence of the event if it does. The model distinguishes the effects of the covariates on the probability of cure and on the time to recurrence of the disease. A SAS macro has been written to implement the proposed multiple imputation algorithm so that sophisticated programming effort can be rendered into a user-friendly application. Simulation results show that the estimates are reasonably efficient. The method is applied to analyse the breast cancer recurrence data. The proposed method can be modified easily to accommodate more general random effects other than the binary random effects so that the random effects not only affect the probability of occurrence of the event, but also the heterogeneity of the time to recurrence of the event among the uncured patients.     

Semiparametric estimation of the cure fraction in population-based cancer survival analysis

With rapid development in medical research, the treatment of diseases including cancer has progressed dramatically and those survivors may die from causes other than the one under study, especially among elderly patients. Motivated by the Surveillance, Epidemiology, and End Results (SEER) female breast cancer study, background mortality is incorporated into the mixture cure proportional hazards (MCPH) model to improve the cure fraction estimation in population-based cancer studies. Here, that patients are “cured” is defined as when the mortality rate of the individuals in diseased group returns to the same level as that expected in the general population, where the population level mortality is presented by the mortality table of the United States. The semiparametric estimation method based on the EM algorithm for the MCPH model with background mortality (MCPH+BM) is further developed and validated via comprehensive simulation studies. Real data analysis shows that the proposed semiparametric MCPH+BM model may provide more accurate estimation in population-level cancer study.     

Mixture cure model with random effects for clustered interval-censored survival data

The mixture cure model is an effective tool for analysis of survival data with a cure fraction. This approach integrates the logistic regression model for the proportion of cured subjects and the survival model (either the Cox proportional hazards or accelerated failure time model) for uncured subjects. Methods based on the mixture cure model have been extensively investigated in the literature for data with exact failure/censoring times. In this paper, we propose a mixture cure modeling procedure for analyzing clustered and interval-censored survival time data by incorporating random effects in both the logistic regression and PH regression components. Under the generalized linear mixed model framework, we develop the REML estimation for the parameters, as well as an iterative algorithm for estimation of the survival function for interval-censored data. The estimation procedure is implemented via an EM algorithm. A simulation study is conducted to evaluate the performance of the proposed method in various practical situations. To demonstrate its usefulness, we apply the proposed method to analyze the interval-censored relapse time data from a smoking cessation study whose subjects were recruited from 51 zip code regions in the southeastern corner of Minnesota.     

Sample size calculation for the proportional hazards cure model

In clinical trials with time‐to‐event endpoints, it is not uncommon to see a significant proportion of patients being cured (or long‐term survivors), such as trials for the non‐Hodgkins lymphoma disease. The popularly used sample size formula derived under the proportional hazards (PH) model may not be proper to design a survival trial with a cure fraction, because the PH model assumption may be violated. To account for a cure fraction, the PH cure model is widely used in practice, where a PH model is used for survival times of uncured patients and a logistic distribution is used for the probability of patients being cured. In this paper, we develop a sample size formula on the basis of the PH cure model by investigating the asymptotic distributions of the standard weighted log‐rank statistics under the null and local alternative hypotheses. The derived sample size formula under the PH cure model is more flexible because it can be used to test the differences in the short‐term survival and/or cure fraction. Furthermore, we also investigate as numerical examples the impacts of accrual methods and durations of accrual and follow‐up periods on sample size calculation. The results show that ignoring the cure rate in sample size calculation can lead to either underpowered or overpowered studies. We evaluate the performance of the proposed formula by simulation studies and provide an example to illustrate its application with the use of data from a melanoma trial. Copyright © 2012 John Wiley & Sons, Ltd.     

Hierarchical likelihood inference on clustered competing risks data

The frailty model, an extension of the proportional hazards model, is often used to model clustered survival data. However, some extension of the ordinary frailty model is required when there exist competing risks within a cluster. Under competing risks, the underlying processes affecting the events of interest and competing events could be different but correlated. In this paper, the hierarchical likelihood method is proposed to infer the cause‐specific hazard frailty model for clustered competing risks data. The hierarchical likelihood incorporates fixed effects as well as random effects into an extended likelihood function, so that the method does not require intensive numerical methods to find the marginal distribution. Simulation studies are performed to assess the behavior of the estimators for the regression coefficients and the correlation structure among the bivariate frailty distribution for competing events. The proposed method is illustrated with a breast cancer dataset. Copyright © 2015 John Wiley & Sons, Ltd.     

A Bayesian model for misclassified binary outcomes and correlated survival data with applications to breast cancer

Breast cancer patients may experience ipsilateral breast tumor relapse (IBTR) after breast conservation therapy. IBTR is classified as either true local recurrence or new ipsilateral primary tumor. The correct classification of IBTR status has significant implications in therapeutic decision‐making and patient management. However, the diagnostic tests to classify IBTR are imperfect and prone to misclassification. In addition, some observed survival data (e.g., time to relapse, time from relapse to death) are strongly correlated with IBTR status. We present a Bayesian approach to model the potentially misclassified IBTR status and the correlated survival information. We conduct the inference using a Bayesian framework via Markov chain Monte Carlo simulation implemented in WinBUGS . Extensive simulation shows that the proposed method corrects biases and provides more efficient estimates for the covariate effects on the probability of IBTR and the diagnostic test accuracy. Moreover, our method provides useful subject‐specific patient prognostic information. Our method is motivated by, and applied to, a dataset of 397 breast cancer patients. Copyright © 2012 John Wiley & Sons, Ltd.     

Analyzing semi‐competing risks data with missing cause of informative terminal event

Cancer studies frequently yield multiple event times that correspond to landmarks in disease progression, including non‐terminal events (i.e., cancer recurrence) and an informative terminal event (i.e., cancer‐related death). Hence, we often observe semi‐competing risks data. Work on such data has focused on scenarios in which the cause of the terminal event is known. However, in some circumstances, the information on cause for patients who experience the terminal event is missing; consequently, we are not able to differentiate an informative terminal event from a non‐informative terminal event. In this article, we propose a method to handle missing data regarding the cause of an informative terminal event when analyzing the semi‐competing risks data. We first consider the nonparametric estimation of the survival function for the terminal event time given missing cause‐of‐failure data via the expectation–maximization algorithm. We then develop an estimation method for semi‐competing risks data with missing cause of the terminal event, under a pre‐specified semiparametric copula model. We conduct simulation studies to investigate the performance of the proposed method. We illustrate our methodology using data from a study of early‐stage breast cancer. Copyright © 2016 John Wiley & Sons, Ltd.     

A bivariate survival model with compound Poisson frailty

A correlated frailty model is suggested for analysis of bivariate time‐to‐event data. The model is an extension of the correlated power variance function (PVF) frailty model (correlated three‐parameter frailty model) (J. Epidemiol. Biostat. 1999; 4:53–60). It is based on a bivariate extension of the compound Poisson frailty model in univariate survival analysis (Ann. Appl. Probab. 1992; 4:951–972). It allows for a non‐susceptible fraction (of zero frailty) in the population, overcoming the common assumption in survival analysis that all individuals are susceptible to the event under study. The model contains the correlated gamma frailty model and the correlated inverse Gaussian frailty model as special cases. A maximum likelihood estimation procedure for the parameters is presented and its properties are studied in a small simulation study. This model is applied to breast cancer incidence data of Swedish twins. The proportion of women susceptible to breast cancer is estimated to be 15 per cent. Copyright © 2009 John Wiley & Sons, Ltd.     

Multilevel mixed effects parametric survival models using adaptive Gauss–Hermite quadrature with application to recurrent events and individual participant data meta‐analysis

Multilevel mixed effects survival models are used in the analysis of clustered survival data, such as repeated events, multicenter clinical trials, and individual participant data (IPD) meta‐analyses, to investigate heterogeneity in baseline risk and covariate effects. In this paper, we extend parametric frailty models including the exponential, Weibull and Gompertz proportional hazards (PH) models and the log logistic, log normal, and generalized gamma accelerated failure time models to allow any number of normally distributed random effects. Furthermore, we extend the flexible parametric survival model of Royston and Parmar, modeled on the log‐cumulative hazard scale using restricted cubic splines, to include random effects while also allowing for non‐PH (time‐dependent effects). Maximum likelihood is used to estimate the models utilizing adaptive or nonadaptive Gauss–Hermite quadrature. The methods are evaluated through simulation studies representing clinically plausible scenarios of a multicenter trial and IPD meta‐analysis, showing good performance of the estimation method. The flexible parametric mixed effects model is illustrated using a dataset of patients with kidney disease and repeated times to infection and an IPD meta‐analysis of prognostic factor studies in patients with breast cancer. User‐friendly Stata software is provided to implement the methods. Copyright © 2014 John Wiley & Sons, Ltd.     

Semiparametric Bayesian inference on skew–normal joint modeling of multivariate longitudinal and survival data

We propose a semiparametric multivariate skew–normal joint model for multivariate longitudinal and multivariate survival data. One main feature of the posited model is that we relax the commonly used normality assumption for random effects and within‐subject error by using a centered Dirichlet process prior to specify the random effects distribution and using a multivariate skew–normal distribution to specify the within‐subject error distribution and model trajectory functions of longitudinal responses semiparametrically. A Bayesian approach is proposed to simultaneously obtain Bayesian estimates of unknown parameters, random effects and nonparametric functions by combining the Gibbs sampler and the Metropolis–Hastings algorithm. Particularly, a Bayesian local influence approach is developed to assess the effect of minor perturbations to within‐subject measurement error and random effects. Several simulation studies and an example are presented to illustrate the proposed methodologies. Copyright © 2014 John Wiley & Sons, Ltd.     

Direct likelihood inference on the cause‐specific cumulative incidence function: A flexible parametric regression modelling approach

In a competing risks analysis, interest lies in the cause‐specific cumulative incidence function (CIF) that can be calculated by either (1) transforming on the cause‐specific hazard or (2) through its direct relationship with the subdistribution hazard. We expand on current competing risks methodology from within the flexible parametric survival modelling framework (FPM) and focus on approach (2). This models all cause‐specific CIFs simultaneously and is more useful when we look to questions on prognosis. We also extend cure models using a similar approach described by Andersson et al for flexible parametric relative survival models. Using SEER public use colorectal data, we compare and contrast our approach with standard methods such as the Fine & Gray model and show that many useful out‐of‐sample predictions can be made after modelling the cause‐specific CIFs using an FPM approach. Alternative link functions may also be incorporated such as the logit link. Models can also be easily extended for time‐dependent effects.     

Geographical variation in cancer patient survival in Finland: chance,confounding, or effect of treatment?

STUDY OBJECTIVE--The aim was to determine whether survival of cancer patients in Finland varies with their place of residence, and if so, what proportion of the variation might be due to health services rather than to confounding variables. DESIGN--Patients with breast and prostatic cancer diagnosed in Finland between 1970 and 1981 were classified by place of residence (from 21 hospital districts), and area specific 5 year relative survival rates were estimated. SETTING--This was a population based survey of the whole of Finland. PATIENTS--16,754 cases of breast cancer and 9483 cases of prostatic cancer were identified. Of these, 0.5% of breast cancers and 4.1% of prostatic cancers were excluded because diagnosis was based only on necropsy findings or because the only information available was from the death certificate. MEASUREMENTS AND MAIN RESULTS--There was a large variation in rates, ranging from 59% to 76% for breast cancer, and from 30% to 65% for prostatic cancer. However, after accounting for age of patient and extent of disease, the standardised differences for prostatic cancer closely followed random distribution, indicating equal results of treatment in different areas. For breast cancer there was more variation than expected by chance and patients resident in any of the university central hospital districts with modern radiotherapy equipment survived better than other patients. CONCLUSIONS--There is little indication that large variations in crude mortality rates from these two cancers in different parts of Finland are due to inequalities of medical care, though a small effect on breast cancer survival which might be care related was shown.