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.     

Mixture models for cancer survival analysis: application to population-based data with covariates

The interest in estimating the probability of cure has been increasing in cancer survival analysis as the curability of many cancer diseases is becoming a reality. Mixture survival models provide a way of modelling time to death when cure is possible, simultaneously estimating death hazard of fatal cases and the proportion of cured case. In this paper we propose an application of a parametric mixture model to relative survival rates of colon cancer patients from the Finnish population-based cancer registry, and including major survival determinants as explicative covariates. Disentangling survival into two different components greatly facilitates the analysis and the interpretation of the role of prognostic factors on survival patterns. For example, age plays a different role in determining, from one side, the probability of cure, and, from the other side, the life expectancy of fatal cases. The results support the hypothesis that observed survival trends are really due to a real prognostic gain for more recently diagnosed patients.     

Prediction of the long-term survival in breast cancer patients according to the present oncological status

After a breast cancer diagnosis, single or multiple events can occur during follow-up (recurrence, metastasis, and death). An analysis of long-term survival should take into account not only the initial characteristics of the patient, but also her oncological status (that is, her history) after surgery. For this purpose, we used a technique proposed by Klein, Keiding and Copelan (1994), to predict the probability of a patient being alive 20 years after surgery for a breast cancer, based on data concerning her oncological status at time t. The first step of the model was to estimate the hazard function for each event of interest (recurrence, metastasis, and death) in a Cox model including initial patient characteristics (age, tumour size, number of involved axillary lymph nodes and the Scarff, Bloom and Richardson (SBR) histo-prognostic grade) and time-dependent covariates representing the occurrence of intermediate events (recurrence and metastasis). The second step was to use these estimations to calculate the conditional probability of being alive 20-t years later for a patient, given her oncological status at time t (t<10 years). In this second step, the method presented by Klein, Keiding and Copelan was extended to include non-proportional hazards. This model has been applied to a population of 3180 patients operated on for a breast cancer at the Institut Gustave Roussy between 1 January 1954 and 31 December 1983. At the time of surgery, the probability of survival at 20 years is 0.78 for all patients. Ten years after surgery, if no recurrence or metastasis are observed, the probability of survival at 20 years will rise to 0.89. If only a recurrence is observed, the probability of a patient being alive at 20 years will drop to 0.72. If a metastasis and no recurrence is observed, the probability of survival at 20 years will be only 0.18. If both recurrence and metastasis are observed the probability of survival at 20 years will be equal to 0.09. In conclusion, the model used dynamically appraises the prognosis and represents a new approach for studying the outcome of breast cancer patients having undergone surgery.     

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.     

A finite mixture survival model to characterize risk groups of neuroblastoma

Neuroblastoma is a childhood cancer with patients experiencing heterogeneous survival outcomes despite aggressive treatment. Disease outcomes range from early death to spontaneous regression of the tumor followed by cure. Owing to this heterogeneity, it is of interest to identify patients with similar types of neuroblastoma so that specific types of treatment can be developed. Oncologists are especially interested in identifying patients who will be cured so that the minimum amount of a potentially toxic treatment can be given to this group of patients. We analyze a large cohort of neuroblastoma patients and develop a finite mixture model that uses covariates to predict the probability of being in a cure group or other (one or more) risk groups. A prediction method is developed that uses the estimated probabilities to assign a patient to different risk groups. The robustness of the model and the prediction method is examined via simulation by looking at misclassification rates under misspecified models. Published in 2009 by John Wiley & Sons, Ltd.     

Unbiased assessment of treatment effects on disease recurrence and survival in clinical trials

Recently there has been considerable discussion concerning the analysis of data and presentation of results from clinical trials in breast cancer. A problem stems from there being several events of interest, notably disease recurrence and death. Current methods based on 'so-called' disease-free survival are criticized on the grounds that they confuse events, such as disease recurrence and death, in the treatment comparison and consequently could be very misleading. Alternative methods of analysis, based on separate measures of treatment effect are presented and illustrated.     

Analysing the effects of anaemia on local recurrence of head and neck cancer when covariate values are missing

We present a case study in the analysis of the prognostic effects of anaemia and other covariates on the local recurrence of head and neck cancer in patients who have been treated with radiation therapy. Because it is believed that a large fraction of the patients are cured by the therapy, we use a failure time mixture model for the outcomes, which simultaneously models both the relationship of the covariates to cure and the relationship of the covariates to local recurrence times for subjects who are not cured. A problematic feature of the data is that two covariates of interest having missing values, so that only 75 per cent of the subjects have complete data. We handle the missing-data problem by jointly modelling the covariates and the outcomes, and then fitting the model to all of the data, including the incomplete cases. We compare our approach to two traditional methods for handling missingness, that is, complete-case analysis and the use of an indicator variable for missingness. The comparison with complete-case analysis demonstrates gains in efficiency for joint modelling as well as sensitivity of some results to the method used to handle missing data. The use of an indicator variable yields results that are very similar to those from joint modelling for our data. We also compare the results obtained for the mixture model with results obtained for a standard (non-mixture) survival model. It is seen that the mixture model separates out effects in a way that is not possible with a standard survival model. In particular, conditional on other covariates, we find strong evidence of an association between anaemia and cure, whereas the evidence of an association between anaemia and time to local recurrence for patients who are not cured is weaker.     

Semiparametric mixture cure model analysis with competing risks data: Application to vascular access thrombosis data

The article is motivated by a nephrology study in Taiwan, which enrolled hemodialysis patients who suffered from vascular access thrombosis. After treatment, some patients were cured of thrombosis, while some may experience recurrence of either type (acute or nonacute) of vascular access thrombosis. Our major interest is to estimate the cumulative incidence probability of time to the first recurrence of acute thrombosis after therapy. Since the occurrence of one type of vascular access thrombosis precludes occurrence of the other type, patients are subject to competing risks. To account for the presence of competing risks and cured patients, we develop a mixture model approach to the regression analysis of competing-risks data with a cure fraction. We make inference about the effects of factors on both the cure rate and cumulative incidence function (CIF) for a failure of interest, which are separately specified in the logistic regression model and semiparametric regression model with time-varying and time-invariant effects. Based on two-stage method, we develop novel estimation equations using the inverse probability censoring weight techniques. The asymptotic properties of the estimators are rigorously studied and the plug-in variance estimators can be obtained for constructing interval estimators. We also propose a lack-of-fit test for assessing the adequacy of the proposed model and several tests for time-varying effects. The simulation studies and vascular access thrombosis data analysis are conducted to illustrate the proposed method.     

Multiple imputation of missing covariates for the Cox proportional hazards cure model

We explore several approaches for imputing partially observed covariates when the outcome of interest is a censored event time and when there is an underlying subset of the population that will never experience the event of interest. We call these subjects ‘cured’, and we consider the case where the data are modeled using a Cox proportional hazards (CPH) mixture cure model. We study covariate imputation approaches using fully conditional specification. We derive the exact conditional distribution and suggest a sampling scheme for imputing partially observed covariates in the CPH cure model setting. We also propose several approximations to the exact distribution that are simpler and more convenient to use for imputation. A simulation study demonstrates that the proposed imputation approaches outperform existing imputation approaches for survival data without a cure fraction in terms of bias in estimating CPH cure model parameters. We apply our multiple imputation techniques to a study of patients with head and neck cancer. Copyright © 2016 John Wiley & Sons, Ltd.     

A semi-parametric accelerated failure time cure model

A cure model is a useful approach for analysing failure time data in which some subjects could eventually experience, and others never experience, the event of interest. A cure model has two components: incidence which indicates whether the event could eventually occur and latency which denotes when the event will occur given the subject is susceptible to the event. In this paper, we propose a semi-parametric cure model in which covariates can affect both the incidence and the latency. A logistic regression model is proposed for the incidence, and the latency is determined by an accelerated failure time regression model with unspecified error distribution. An EM algorithm is developed to fit the model. The procedure is applied to a data set of tonsil cancer patients treated with radiation therapy.     

A joint frailty model to estimate the recurrence process and the disease‐specific mortality process without needing the cause of death

In chronic diseases, such as cancer, recurrent events (such as relapses) are commonly observed; these could be interrupted by death. With such data, a joint analysis of recurrence and mortality processes is usually conducted with a frailty parameter shared by both processes. We examined a joint modeling of these processes considering death under two aspects: ‘death due to the disease under study' and ‘death due to other causes', which enables estimating the disease‐specific mortality hazard. The excess hazard model was used to overcome the difficulties in determining the causes of deaths (unavailability or unreliability); this model allows estimating the disease‐specific mortality hazard without needing the cause of death but using the mortality hazards observed in the general population. We propose an approach to model jointly recurrence and disease‐specific mortality processes within a parametric framework. A correlation between the two processes is taken into account through a shared frailty parameter. This approach allows estimating unbiased covariate effects on the hazards of recurrence and disease‐specific mortality. The performance of the approach was evaluated by simulations with different scenarios. The method is illustrated by an analysis of a population‐based dataset on colon cancer with observations of colon cancer recurrences and deaths. The benefits of the new approach are highlighted by comparison with the ‘classical' joint model of recurrence and overall mortality. Moreover, we assessed the goodness of fit of the proposed model. Comparisons between the conditional hazard and the marginal hazard of the disease‐specific mortality are shown, and differences in interpretation are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Anomic aspects of recovery from cancer

A cancer diagnosis is universally regarded as a traumatic event. Many equate it with a ‘sentence of death’. But an increasing number of cancer patients are being cured.An exploratory study involving intensive interviews with recovered cancer patients indicated that the positive experience of being cured is often mixed with negative elements, including: (1) the withdrawal of the intensified social support which accompanied the diagnosis and early treatment; (2) ambivalence about the discontinuation of treatment; (3) anxiety about recurrence of the disease; (4) adjustment to permanent disabilities resulting from the disease or its treatment; (5) the need to resume life-oriented modes of thought after a successful adjustment to the idea of death; (6) anger at perceived inadequacies in the handling of treatment; and (7) confusion about feelings of depression when the objective situation has improved.Durkheim's concept of anomie originally referred to a societal condition engendered by either positive or negative change. Srole and others adopted the term ‘anomia’ to refer to the social-psychological correlate of this condition, i.e. anomie as experienced by the individual. The present research suggests that the concept of anomia, and specifically, the anomia of good fortune, may be useful in studying the rehabilitation of cancer patients.     

Relative survival multistate Markov model

Prognostic studies often have to deal with two important challenges: (i) separating effects of predictions on different 'competing' events and (ii) uncertainty about cause of death. Multistate Markov models permit multivariable analyses of competing risks of, for example, mortality versus disease recurrence. On the other hand, relative survival methods help estimate disease-specific mortality risks even in the absence of data on causes of death. In this paper, we propose a new Markov relative survival (MRS) model that attempts to combine these two methodologies. Our MRS model extends the existing multistate Markov piecewise constant intensities model to relative survival modeling. The intensity of transitions leading to death in the MRS model is modeled as the sum of an estimable excess hazard of mortality from the disease of interest and an 'offset' defined as the expected hazard of all-cause 'natural' mortality obtained from relevant life-tables. We evaluate the new MRS model through simulations, with a design based on registry-based prognostic studies of colon cancer. Simulation results show almost unbiased estimates of prognostic factor effects for the MRS model. We also applied the new MRS model to reassess the role of prognostic factors for mortality in a study of colorectal cancer. The MRS model considerably reduces the bias observed with the conventional Markov model that does not permit accounting for unknown causes of death, especially if the 'true' effects of a prognostic factor on the two types of mortality differ substantially.     

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.     

Practical recommendations for reporting Fine‐Gray model analyses for competing risk data

In survival analysis, a competing risk is an event whose occurrence precludes the occurrence of the primary event of interest. Outcomes in medical research are frequently subject to competing risks. In survival analysis, there are 2 key questions that can be addressed using competing risk regression models: first, which covariates affect the rate at which events occur, and second, which covariates affect the probability of an event occurring over time. The cause‐specific hazard model estimates the effect of covariates on the rate at which events occur in subjects who are currently event‐free. Subdistribution hazard ratios obtained from the Fine‐Gray model describe the relative effect of covariates on the subdistribution hazard function. Hence, the covariates in this model can also be interpreted as having an effect on the cumulative incidence function or on the probability of events occurring over time. We conducted a review of the use and interpretation of the Fine‐Gray subdistribution hazard model in articles published in the medical literature in 2015. We found that many authors provided an unclear or incorrect interpretation of the regression coefficients associated with this model. An incorrect and inconsistent interpretation of regression coefficients may lead to confusion when comparing results across different studies. Furthermore, an incorrect interpretation of estimated regression coefficients can result in an incorrect understanding about the magnitude of the association between exposure and the incidence of the outcome. The objective of this article is to clarify how these regression coefficients should be reported and to propose suggestions for interpreting these coefficients.     

Bayesian adjusted R2 for the meta-analytic evaluation of surrogate time-to-event endpoints in clinical trials

A two-stage model for evaluating both trial-level and patient-level surrogacy of correlated time-to-event endpoints has been introduced, using patient-level data when multiple clinical trials are available. However, the associated maximum likelihood approach often suffers from numerical problems when different baseline hazards among trials and imperfect estimation of treatment effects are assumed. To address this issue, we propose performing the second-stage, trial-level evaluation of potential surrogates within a Bayesian framework, where we may naturally borrow information across trials while maintaining these realistic assumptions. Posterior distributions on surrogacy measures of interest may then be used to compare measures or make decisions regarding the candidacy of a specific endpoint. We perform a simulation study to investigate differences in estimation performance between traditional maximum likelihood and new Bayesian representations of common meta-analytic surrogacy measures, while assessing sensitivity to data characteristics such as number of trials, trial size, and amount of censoring. Furthermore, we present both frequentist and Bayesian trial-level surrogacy evaluations of time to recurrence for overall survival in two meta-analyses of adjuvant therapy trials in colon cancer. With these results, we recommend Bayesian evaluation as an attractive and numerically stable alternative in the multitrial assessment of potential surrogate endpoints.     

Weighted hurdle regression method for joint modeling of cardiovascular events likelihood and rate in the US dialysis population

We propose a new weighted hurdle regression method for modeling count data, with particular interest in modeling cardiovascular events in patients on dialysis. Cardiovascular disease remains one of the leading causes of hospitalization and death in this population. Our aim is to jointly model the relationship/association between covariates and (i) the probability of cardiovascular events, a binary process, and (ii) the rate of events once the realization is positive—when the ‘hurdle’ is crossed—using a zero‐truncated Poisson distribution. When the observation period or follow‐up time, from the start of dialysis, varies among individuals, the estimated probability of positive cardiovascular events during the study period will be biased. Furthermore, when the model contains covariates, then the estimated relationship between the covariates and the probability of cardiovascular events will also be biased. These challenges are addressed with the proposed weighted hurdle regression method. Estimation for the weighted hurdle regression model is a weighted likelihood approach, where standard maximum likelihood estimation can be utilized. The method is illustrated with data from the United States Renal Data System. Simulation studies show the ability of proposed method to successfully adjust for differential follow‐up times and incorporate the effects of covariates in the weighting. Copyright © 2014 John Wiley & Sons, Ltd.     

A statistic for comparing two correlated markers which are prognostic for time to an event

A proliferation of biologic markers has led to increased interest in methods for comparing the prognostic value of markers in predicting time to recurrence of or death from disease. This paper proposes a non-parametric test statistic for comparing two continuous markers when the outcome of interest is time to an event and the outcome is subject to right censoring. Results from Monte Carlo studies show that the new testing procedure is reliable for practical use. We present an example in which we use the statistic successfully in a colon cancer study to select between two potentially useful markers.     

Analyzing longitudinal data with patients in different disease states during follow‐up and death as final state

This paper considers the analysis of longitudinal data complicated by the fact that during follow‐up patients can be in different disease states, such as remission, relapse or death. If both the response of interest (for example, quality of life (QOL)) and the amount of missing data depend on this disease state, ignoring the disease state will yield biased means. Death as the final state is an additional complication because no measurements after death are taken and often the outcome of interest is undefined after death. We discuss a new approach to model these types of data. In our approach the probability to be in each of the different disease states over time is estimated using multi‐state models. In each different disease state, the conditional mean given the disease state is modeled directly. Generalized estimation equations are used to estimate the parameters of the conditional means, with inverse probability weights to account for unobserved responses. This approach shows the effect of the disease state on the longitudinal response. Furthermore, it yields estimates of the overall mean response over time, either conditionally on being alive or after imputing predefined values for the response after death. Graphical methods to visualize the joint distribution of disease state and response are discussed. As an example, the analysis of a Dutch randomized clinical trial for breast cancer is considered. In this study, the long‐term impact on the QOL for two different chemotherapy schedules was studied with three disease states: alive without relapse, alive after relapse and death. Copyright © 2009 John Wiley & Sons, Ltd.     

49例老年结肠癌并发急性梗阻的临床诊治分析

目的探讨老年结肠癌并发急性梗阻的诊断及治疗方法。方法回顾性分析我院2009年1月至2010年4月收治的49倒蛄肠癌并发急性肠梗阻的患者资料,总结其临床诊断方法及治疗手段。结果49倒患者中,治愈45例（91．84％）,固手术期共死亡4例,死亡原因包括原有心肌疾病复发1例,肺部感染1例,多脏器功能衰竭2例。结论对老年结肠癌并发急性肠梗阻的患者,需要医生进行早期准确诊断,根据患者的实际情况选择合适的治疗方法,提高治愈率,减少并发症的发生,降低死亡率。