The k-sample problem in a multi-state model and testing transition probability matrices

The choice of multi-state models is natural in analysis of survival data, e.g., when the subjects in a study pass through different states like ‘healthy’, ‘in a state of remission’, ‘relapse’ or ‘dead’ in a health related quality of life study. Competing risks is another common instance of the use of multi-state models. Statistical inference for such event history data can be carried out by assuming a stochastic process model. Under such a setting, comparison of the event history data generated by two different treatments calls for testing equality of the corresponding transition probability matrices. The present paper proposes solution to this class of problems by assuming a non-homogeneous Markov process to describe the transitions among the health states. A class of test statistics are derived for comparison of \(k\) treatments by using a ‘weight process’. This class, in particular, yields generalisations of the log-rank, Gehan, Peto–Peto and Harrington–Fleming tests. For an intrinsic comparison of the treatments, the ‘leave-one-out’ jackknife method is employed for identifying influential observations. The proposed methods are then used to develop the Kolmogorov–Smirnov type supremum tests corresponding to the various extended tests. To demonstrate the usefulness of the test procedures developed, a simulation study was carried out and an application to the Trial V data provided by International Breast Cancer Study Group is discussed.     

Estimation in a model for a semi-Markov process with covariates under right-censoring

A semi-Markov model with covariates is proposed for a multi-state process with a finite number of states such that the transition probabilities between the states and the distribution functions of the duration times between the occurrence of two states depend on a discrete covariate. The hazard rates for the time elapsed between two successive states depend on the covariate through a proportional hazards model involving a set of regression parameters, while the transition probabilities depend on the covariate in an unspecified way. We propose estimators for these parameters and for the cumulative hazard functions of the sojourn times. A difficulty comes from the fact that when a sojourn time in a state is right-censored, the next state is unknown. We prove that our estimators are consistent and asymptotically Gaussian under the model constraints.     

The Dynamic Model for Cancer Relapse Based on Two-Stage Model of Carcinogenesis

In this article, the time from the start of chemotherapy randomization until cancer relapse is of primary interest. Here, cancer relapse refers to the appearance of the first observable malignant clone after therapy. A dynamic model for cancer relapse after chemotherapy is developed. The model differs from the traditional cure rate models in that it takes into consideration the growth kinetics of malignant tumors using a two-stage carcinogenesis model. The survival and hazard functions for cancer relapse time are derived, and a simulation study is performed to validate the underlying model.     

The star-shaped Λ-coalescent and Fleming–Viot process

The star-shaped Λ-coalescent and corresponding Λ-Fleming–Viot process, where the Λ measure has a single atom at unity, are studied in this article. The transition functions and stationary distribution of the Λ-Fleming–Viot process are derived in a two-type model with mutation. The distribution of the number of non-mutant lines back in time in the star-shaped Λ-coalescent is found. Extensions are made to a model with d types, either with parent-independent mutation or general Markov mutation, and an infinitely-many-types model, when d → ∞. An eigenfunction expansion for the transition functions is found, which has polynomial right eigenfunctions and left eigenfunctions described by hyperfunctions. A further star-shaped model with general frequency-dependent change is considered and the stationary distribution in the Fleming–Viot process derived. This model includes a star-shaped Λ-Fleming–Viot process with mutation and selection. In a general Λ-coalescent explicit formulae for the transition functions and stationary distribution, when there is mutation, are unknown. However, in this article, explicit formulae are derived in the star-shaped coalescent.     

Numerical evaluation of observed sojourn time distributions for a single ion channel incorporating time interval omission

The dynamical aspects of single ion channel gating can be modelled by a semi-Markov process. There is aggregation of states, corresponding to the receptor channel being open or closed, and there is time interval omission, brief sojourns in either the open or closed classes of states not being detected. This paper is concerned with the computation of the probability density functions of observed open (closed) sojourn-times incorporating time interval omission. A system of Volterra integral equations is derived, whose solution governs the required density function. Numerical procedures, using iterative and multistep methods, are described for solving these equations. Examples are given, and in the special case of Markov models results are compared with those obtained by alternative methods. Probabilistic interpretations are given for the iterative methods, which also give lower bounds for the solutions.     

An integrated model for economic design of chi-square control chart and maintenance planning

Considered process in this article is a two-stage dependent process. Each item in this process has two quality characteristics as x and y while x and y are related to the stage 1 and 2, respectively. Each stage has two operational states as the in-control state and out-of-control state and transition time from the in-control state to the out-of-control state follows a general continues distribution function. The process is monitored using a chi-square control chart. An integrated model that coordinates the decisions related to the economic design of the used control chart and maintenance planning is presented. For the evaluation of the integrated model performance, a stand-alone maintenance model is also presented, and the performance of these two models is compared with each other.     

Fitting a Semi-Parametric Mixture Model for Competing Risks in Survival Data

A model for survival analysis is studied that is relevant for samples which are subject to multiple types of failure. In comparison with a more standard approach, through the appropriate use of hazard functions and transition probabilities, the model allows for a more accurate study of cause-specific failure with regard to both the timing and type of failure. A semiparametric specification of a mixture model is employed that is able to adjust for concomitant variables and allows for the assessment of their effects on the probabilities of eventual causes of failure through a generalized logistic model, and their effects on the corresponding conditional hazard functions by employing the Cox proportional hazards model. A carefully formulated estimation procedure is presented that uses an EM algorithm based on a profile likelihood construction. The methods discussed, which could also be used for reliability analysis, are applied to a prostate cancer data set.     

Multistate Survival Models as Transient Electrical Networks

In multistate survival analysis, the sojourn of a patient through various clinical states is shown to correspond to the diffusion of 1 C of electrical charge through an electrical network. The essential comparison has differentials of probability for the patient to correspond to differentials of charge, and it equates clinical states to electrical nodes. Indeed, if the death state of the patient corresponds to the sink node of the circuit, then the transient current that would be seen on an oscilloscope as the sink output is a plot of the probability density for the survival time of the patient. This electrical circuit analogy is further explored by considering the simplest possible survival model with two clinical states, alive and dead (sink), that incorporates censoring and truncation. The sink output seen on an oscilloscope is a plot of the Kaplan–Meier mass function. Thus, the Kaplan–Meier estimator finds motivation from the dynamics of current flow, as a fundamental physical law, rather than as a nonparametric maximum likelihood estimate (MLE). Generalization to competing risks settings with multiple death states (sinks) leads to cause‐specific Kaplan–Meier submass functions as outputs at sink nodes. With covariates present, the electrical analogy provides for an intuitive understanding of partial likelihood and various baseline hazard estimates often used with the proportional hazards model.     

Analysis of cure rate survival data under proportional odds model

Due to significant progress in cancer treatments and management in survival studies involving time to relapse (or death), we often need survival models with cured fraction to account for the subjects enjoying prolonged survival. Our article presents a new proportional odds survival models with a cured fraction using a special hierarchical structure of the latent factors activating cure. This new model has same important differences with classical proportional odds survival models and existing cure-rate survival models. We demonstrate the implementation of Bayesian data analysis using our model with data from the SEER (Surveillance Epidemiology and End Results) database of the National Cancer Institute. Particularly aimed at survival data with cured fraction, we present a novel Bayes method for model comparisons and assessments, and demonstrate our new tool’s superior performance and advantages over competing tools.     

ADAPTIVE GROUP SEQUENTIAL PROCEDURES FOR MULTIPLE COMPARISON OF TREATMENTS

ABSTRACT

We present a flexible group sequential procedure for comparing several treatments to a control. Though longitudinal data corresponding to a two stage mixed effects model are considered, ranges of application include any process with independent increments. The procedure allows the experimenter to drop the inferior treatments from the trial as soon as they are detected. It control strongly the familywise error rate. We also discuss a new error spending function (ESF) and study the performance of the procedure using various ESFs and time scales. Finally, the procedure is illustrated on a real example and implementation considerations are discussed.     

Confidence Bands for the Difference of Two Survival Curves Under Proportional Hazards Model

A common approach to testing for differences between the survival rates of two therapies is to use a proportional hazards regression model which allows for an adjustment of the two survival functions for any imbalance in prognostic factors in the comparison. When the relative risk of one treatment to the other is not constant over time the question of which therapy has a survival advantage is difficult to determine from the Cox model. An alternative approach to this problem is to plot the difference between the two predicted survival functions with a confidence band that provides information about when these two treatments differ. Such a band will depend on the covariate values of a given patient. In this paper we show how to construct a confidence band for the difference of two survival functions based on the proportional hazards model. A simulation approach is used to generate the bands. This approach is used to compare the survival probabilities of chemotherapy and allogeneic bone marrow transplants for chronic leukemia.     

A generalization of the compound rayleigh distribution: using a bayesian method on cancer survival times

In this paper the generalized compound Rayleigh model, exhibiting flexible hazard rate, is high¬lighted. This makes it attractive for modelling survival times of patients showing characteristics of a random hazard rate. The Bayes estimators are derived for the parameters of this model and some survival time parameters from a right censored sample. This is done with respect to conjugate and discrete priors on the parameters of this model, under the squared error loss function, Varian's asymmetric linear-exponential (linex) loss function and a weighted linex loss function. The future survival time of a patient is estimated under these loss functions. A Monte Carlo simu¬lation procedure is used where closed form expressions of the estimators cannot be obtained. An example illustrates the proposed estimators for this model.     

Modeling Two-state Disease Processes with Random Effects

Many chronic medical conditions are manifested by alternating sojourns in symptom-free and symptomatic states. In many cases, in addition to their relapsing and remitting nature, these conditions lead to worsening disease patterns over time and may exhibit seasonal trends. We develop a mixed-effect two-state model for such disease processes in which covariate effects are modeled multiplicatively on transition intensities. The transition intensities, in turn, are functions of three time scales: the semi-Markov scale involving the backward recurrence time for the cyclical component, the Markov scale for the time trend component, and a seasonal time scale. Multiplicative bivariate log-normal random effects are introduced to accommodate heterogeneity in disease activity between subjects and to admit a possible negative correlation between the transition intensities. Maximum likelihood estimation is carried out using Gauss-Hermite integration and a standard Newton-Raphson procedure. Tests of homogeneity are presented based on score statistics. An application of the methodology to data from a multi-center clinical trial of chronic bronchitis is provided for illustrative purposes.     

On inference from Markov chain macro-data using transforms

We consider data that are longitudinal, arising from n individuals over m time periods. Each individual moves according to the same homogeneous Markov chain, with s states. If the individual sample paths are observed, so that ‘micro-data’ are available, the transition probability matrix is estimated by maximum likelihood straightforwardly from the transition counts. If only the overall numbers in the various states at each time point are observed, we have ‘macro-data’, and the likelihood function is difficult to compute. In that case a variety of methods has been proposed in the literature. In this paper we propose methods based on generating functions and investigate their performance.     

A parametric model fitting time to first event for overdispersed data: application to time to relapse in multiple sclerosis

In this article, we propose a parametric model for the distribution of time to first event when events are overdispersed and can be properly fitted by a Negative Binomial distribution. This is a very common situation in medical statistics, when the occurrence of events is summarized as a count for each patient and the simple Poisson model is not adequate to account for overdispersion of data. In this situation, studying the time of occurrence of the first event can be of interest. From the Negative Binomial distribution of counts, we derive a new parametric model for time to first event and apply it to fit the distribution of time to first relapse in multiple sclerosis (MS). We develop the regression model with methods for covariate estimation. We show that, as the Negative Binomial model properly fits relapse counts data, this new model matches quite perfectly the distribution of time to first relapse, as tested in two large datasets of MS patients. Finally we compare its performance, when fitting time to first relapse in MS, with other models widely used in survival analysis (the semiparametric Cox model and the parametric exponential, Weibull, log-logistic and log-normal models).     

Monte Carlo methods for nonparametric survival model determination

In the causal analysis of survival data a time-based response is related to a set of explanatory variables. Definition of the relation between the time and the covariates may become a difficult task, particularly in the preliminary stage, when the information is limited. Through a nonparametric approach, we propose to estimate the survival function allowing to evaluate the relative importance of each potential explanatory variable, in a simple and explanatory fashion. To achieve this aim, each of the explanatory variables is used to partition the observed survival times. The observations are assumed to be partially exchangeable according to such partition. We then consider, conditionally on each partition, a hierarchical nonparametric Bayesian model on the hazard functions. We define and compare different prior distribution for the hazard functions.     

Semiparametric model for semi-competing risks data with application to breast cancer study

For many forms of cancer, patients will receive the initial regimen of treatments, then experience cancer progression and eventually die of the disease. Understanding the disease process in patients with cancer is essential in clinical, epidemiological and translational research. One challenge in analyzing such data is that death dependently censors cancer progression (e.g., recurrence), whereas progression does not censor death. We deal with the informative censoring by first selecting a suitable copula model through an exploratory diagnostic approach and then developing an inference procedure to simultaneously estimate the marginal survival function of cancer relapse and an association parameter in the copula model. We show that the proposed estimators possess consistency and weak convergence. We use simulation studies to evaluate the finite sample performance of the proposed method, and illustrate it through an application to data from a study of early stage breast cancer.     

A comparison of non-homogeneous Markov regression models with application to Alzheimer's disease progression

Markov regression models are useful tools for estimating the impact of risk factors on rates of transition between multiple disease states. Alzheimer's disease (AD) is an example of a multi-state disease process in which great interest lies in identifying risk factors for transition. In this context, non-homogeneous models are required because transition rates change as subjects age. In this report we propose a non-homogeneous Markov regression model that allows for reversible and recurrent disease states, transitions among multiple states between observations, and unequally spaced observation times. We conducted simulation studies to demonstrate performance of estimators for covariate effects from this model and compare performance with alternative models when the underlying non-homogeneous process was correctly specified and under model misspecification. In simulation studies, we found that covariate effects were biased if non-homogeneity of the disease process was not accounted for. However, estimates from non-homogeneous models were robust to misspecification of the form of the non-homogeneity. We used our model to estimate risk factors for transition to mild cognitive impairment (MCI) and AD in a longitudinal study of subjects included in the National Alzheimer's Coordinating Center's Uniform Data Set. Using our model, we found that subjects with MCI affecting multiple cognitive domains were significantly less likely to revert to normal cognition.     

Semiparametric methods for center effect measures based on the ratio of survival functions

The survival function is often of chief interest in epidemiologic studies of time to an event. We develop methods for evaluating center-specific survival outcomes through a ratio of survival functions. The proposed method assumes a center-stratified additive hazards model, which provides a convenient framework for our purposes. Under the proposed methods, the center effects measure is cast as the ratio of subject-specific survival functions under two scenarios: the scenario in which the subject is treated at center \(j\) ; and that wherein the subject is treated at a hypothetical center with survival function equal to the population average. The proposed measure reduces to the ratio of baseline survival functions, but is invariant to the choice of baseline covariate level. We derive the asymptotic properties of the proposed estimators, and assess finite-sample characteristics through simulation. The proposed methods are applied to national kidney transplant data.     

Parameter estimation in regression for long-term survival rate from censored data

In recent years, regression models have been shown to be useful for predicting the long-term survival probabilities of patients in clinical trials. The importance of a regression model is that once the regression parameters are estimated information about the regressed quantity is immediate. A simple estimator is proposed for the regression parameters in a model for the long-term survival rate. The proposed estimator is seen to arise from an estimating function that has the missing information principle underlying its construction. When the covariate takes values in a finite set, the proposed estimating function is equivalent to an ad hoc estimating function proposed in the literature. However, in general, the two estimating functions lead to different estimators of the regression parameter. For discrete covariates, the asymptotic covariance matrix of the proposed estimator is simple to calculate using standard techniques involving the predictable covariation process of martingale transforms. An ad hoc extension to the case of a one-dimensional continuous covariate is proposed. Simplicity and generalizability are two attractive features of the proposed approach. The last mentioned feature is not enjoyed by the other estimator.