General tests of the Markov property in multi-state models.

Multi-state models for event history analysis most commonly assume the process is Markov. This article considers tests of the Markov assumption that are applicable to general multi-state models. Two approaches using existing methodology are considered; a simple method based on including time of entry into each state as a covariate in Cox models for the transition intensities and a method involving detecting a shared frailty through a stratified Commenges-Andersen test. In addition, using the principle that under a Markov process the future rate of transitions of the process at times $t > s$ should not be influenced by the state occupied at time $s$, a new class of general tests is developed by considering summaries from families of log-rank statistics where patients are grouped by the state occupied at varying initial time $s$. An extended form of the test applicable to models that are Markov conditional on observed covariates is also derived. The null distribution of the proposed test statistics are approximated by using wild bootstrap sampling. The approaches are compared in simulation and applied to a dataset on sleeping behavior. The most powerful test depends on the particular departure from a Markov process, although the Cox-based method maintained good power in a wide range of scenarios. The proposed class of log-rank statistic based tests are most useful in situations where the non-Markov behavior does not persist, or is not uniform in nature across patient time.

Quantifying the association between progression-free survival and overall survival in oncology trials using Kendall's τ.

This paper considers methods for estimating the association between progression-free and overall survival in oncology trials. Copula-based, nonparametric, and illness-death model-based methods are reviewed. In addition, the approach based on an underlying illness-death model is generalized to allow general parametric models. The performance of these methods, in terms of bias and efficiency, is investigated through simulation and also illustrated using data from a clinical trial of treatments for colon cancer. The simulations suggest that the illness-death model-based method provides good estimates of Kendall's τ across several scenarios. In some situations, copula-based methods perform well but their performance is sensitive to the choice of copula. The Clayton copula is most appropriate in scenarios, which might realistically reflect an oncology trial, but the use of copula models in practice is questionable.

Sample size re-estimation in paired comparative diagnostic accuracy studies with a binary response.

BACKGROUND: The sample size required to power a study to a nominal level in a paired comparative diagnostic accuracy study, i.e. studies in which the diagnostic accuracy of two testing procedures is compared relative to a gold standard, depends on the conditional dependence between the two tests - the lower the dependence the greater the sample size required. A priori, we usually do not know the dependence between the two tests and thus cannot determine the exact sample size required. One option is to use the implied sample size for the maximal negative dependence, giving the largest possible sample size. However, this is potentially wasteful of resources and unnecessarily burdensome on study participants as the study is likely to be overpowered. A more accurate estimate of the sample size can be determined at a planned interim analysis point where the sample size is re-estimated. METHODS: This paper discusses a sample size estimation and re-estimation method based on the maximum likelihood estimates, under an implied multinomial model, of the observed values of conditional dependence between the two tests and, if required, prevalence, at a planned interim. The method is illustrated by comparing the accuracy of two procedures for the detection of pancreatic cancer, one procedure using the standard battery of tests, and the other using the standard battery with the addition of a PET/CT scan all relative to the gold standard of a cell biopsy. Simulation of the proposed method illustrates its robustness under various conditions. RESULTS: The results show that the type I error rate of the overall experiment is stable using our suggested method and that the type II error rate is close to or above nominal. Furthermore, the instances in which the type II error rate is above nominal are in the situations where the lowest sample size is required, meaning a lower impact on the actual number of participants recruited. CONCLUSION: We recommend multinomial model maximum likelihood estimation of the conditional dependence between paired diagnostic accuracy tests at an interim to reduce the number of participants required to power the study to at least the nominal level. TRIAL REGISTRATION: ISRCTN ISRCTN73852054 . Registered 9th of January 2015. Retrospectively registered.

Estimation of time-shift models with application to survival calibration in health technology assessment.

The incremental life expectancy, defined as the difference in mean survival times between two treatment groups, is a crucial quantity of interest in cost-effectiveness analyses. Usually, this quantity is very difficult to estimate from censored survival data with a limited follow-up period. The paper develops estimation procedures for a time-shift survival model that, provided model assumptions are met, gives a reliable estimate of incremental life expectancy without extrapolation beyond the study period. Methods for inference are developed both for individual patient data and when only published Kaplan-Meier curves are available. Through simulation, the estimators are shown to be close to unbiased and constructed confidence intervals are shown to have close to nominal coverage for small to moderate sample sizes. Copyright © 2016 John Wiley & Sons, Ltd.

Item response theory and structural equation modelling for ordinal data: Describing the relationship between KIDSCREEN and Life-H.

Both item response theory and structural equation models are useful in the analysis of ordered categorical responses from health assessment questionnaires. We highlight the advantages and disadvantages of the item response theory and structural equation modelling approaches to modelling ordinal data, from within a community health setting. Using data from the SPARCLE project focussing on children with cerebral palsy, this paper investigates the relationship between two ordinal rating scales, the KIDSCREEN, which measures quality-of-life, and Life-H, which measures participation. Practical issues relating to fitting models, such as non-positive definite observed or fitted correlation matrices, and approaches to assessing model fit are discussed. item response theory models allow properties such as the conditional independence of particular domains of a measurement instrument to be assessed. When, as with the SPARCLE data, the latent traits are multidimensional, structural equation models generally provide a much more convenient modelling framework.

Transition probability estimates for non-Markov multi-state models.

Non-parametric estimation of the transition probabilities in multi-state models is considered for non-Markov processes. Firstly, a generalization of the estimator of Pepe et al., (1991) (Statistics in Medicine) is given for a class of progressive multi-state models based on the difference between Kaplan-Meier estimators. Secondly, a general estimator for progressive or non-progressive models is proposed based upon constructed univariate survival or competing risks processes which retain the Markov property. The properties of the estimators and their associated standard errors are investigated through simulation. The estimators are demonstrated on datasets relating to survival and recurrence in patients with colon cancer and prothrombin levels in liver cirrhosis patients.

A pool-adjacent-violators type algorithm for non-parametric estimation of current status data with dependent censoring.

A likelihood based approach to obtaining non-parametric estimates of the failure time distribution is developed for the copula based model of Wang et al. (Lifetime Data Anal 18:434-445, 2012) for current status data under dependent observation. Maximization of the likelihood involves a generalized pool-adjacent violators algorithm. The estimator coincides with the standard non-parametric maximum likelihood estimate under an independence model. Confidence intervals for the estimator are constructed based on a smoothed bootstrap. It is also shown that the non-parametric failure distribution is only identifiable if the copula linking the observation and failure time distributions is fully-specified. The method is illustrated on a previously analyzed tumorigenicity dataset.

Flexible nonhomogeneous markov models for panel observed data.

Methods for fitting nonhomogeneous Markov models to panel-observed data using direct numerical solution to the Kolmogorov Forward equations are developed. Nonhomogeneous Markov models occur most commonly when baseline transition intensities depend on calendar time, but may also occur with deterministic time-dependent covariates such as age. We propose transition intensities based on B-splines as a smooth alternative to piecewise constant intensities and also as a generalization of time transformation models. An expansion of the system of differential equations allows first derivatives of the likelihood to be obtained, which can be used in a Fisher scoring algorithm for maximum likelihood estimation. The method is evaluated through a small simulation study and demonstrated on data relating to the development of cardiac allograft vasculopathy in posttransplantation patients.

Accounting for bias due to a non-ignorable tracing mechanism in a retrospective breast cancer cohort study.

We consider the analysis of competing risks in a retrospective breast cancer cohort study where tracing of patients is dependent on survival to a pre-specified truncation time. We demonstrate that if ignored, the observed cause-specific hazards will become distorted before the truncation time. Two approaches to account for the tracing bias are considered. First, a likelihood-based method using piecewise constant transition intensities under a Markov assumption. Second, a pseudo-likelihood method using inverse probability of tracing weights. For the breast cancer example, both methods improve the precision of estimates compared with a conventional approach based on excluding patients.

Semi-Markov models with phase-type sojourn distributions.

Continuous-time multistate models are widely used for categorical response data, particularly in the modeling of chronic diseases. However, inference is difficult when the process is only observed at discrete time points, with no information about the times or types of events between observation times, unless a Markov assumption is made. This assumption can be limiting as rates of transition between disease states might instead depend on the time since entry into the current state. Such a formulation results in a semi-Markov model. We show that the computational problems associated with fitting semi-Markov models to panel-observed data can be alleviated by considering a class of semi-Markov models with phase-type sojourn distributions. This allows methods for hidden Markov models to be applied. In addition, extensions to models where observed states are subject to classification error are given. The methodology is demonstrated on a dataset relating to development of bronchiolitis obliterans syndrome in post-lung-transplantation patients.

Model diagnostics for multi-state models.

Multi-state models are a popular method of describing medical processes that can be represented as discrete states or stages. They have particular use when the data are panel-observed, meaning they consist of discrete snapshots of disease status at irregular time points which may be unique to each patient. However, due to the difficulty of inference in more complicated cases, strong assumptions such as the Markov property, patient homogeneity and time homogeneity are applied. It is important that the validity of these assumptions is tested. A review of methods for diagnosing model fit for panel-observed continuous-time Markov and misclassification-type hidden Markov models is given, with illustrative application to a dataset on cardiac allograft vasculopathy progression in post-heart transplant patients.

Computation of the asymptotic null distribution of goodness-of-fit tests for multi-state models.

We develop an improved approximation to the asymptotic null distribution of the goodness-of-fit tests for panel observed multi-state Markov models (Aguirre-Hernandez and Farewell, Stat Med 21:1899-1911, 2002) and hidden Markov models (Titman and Sharples, Stat Med 27:2177-2195, 2008). By considering the joint distribution of the grouped observed transition counts and the maximum likelihood estimate of the parameter vector it is shown that the distribution can be expressed as a weighted sum of independent chi(1)(2) random variables, where the weights are dependent on the true parameters. The performance of this approximation for finite sample sizes and where the weights are calculated using the maximum likelihood estimates of the parameters is considered through simulation. In the scenarios considered, the approximation performs well and is a substantial improvement over the simple chi(2) approximation.

A general goodness-of-fit test for Markov and hidden Markov models.

Markov models are a convenient and useful method of estimating transition rates between levels of a categorical response variable, such as a disease stage, which changes over time. In medical applications the response variable is typically observed at irregular intervals. A Pearson-type goodness-of-fit test for such models was proposed by Aguirre-Hernandez and Farewell (Statist. Med. 2002; 21:1899-1911), but this test is not applicable in the common situation where the process includes an absorbing state, such as death, for which the time of entry is known precisely nor when the data include censored state observations. This paper presents a modification to the Pearson-type test to allow for these cases. An extension of the method, to allow for the class of hidden Markov models where the response variable is subject to misclassification error, is given. The method is applied to data on cardiac allograft vasculopathy in post-heart-transplant patients.