Direct Modelling of Regression Effects for Transition Probabilities in Multistate Models

Abstract.  A simple and standard approach for analysing multistate model data is to model all transition intensities and then compute a summary measure such as the transition probabilities based on this. This approach is relatively simple to implement but it is difficult to see what the covariate effects are on the scale of interest. In this paper, we consider an alternative approach that directly models the covariate effects on transition probabilities in multistate models. Our new approach is based on binomial modelling and inverse probability of censoring weighting techniques and is very simple to implement by standard software. We show how to do flexible regression models with possibly time-varying covariate effects.     

Flexible semi-parametric regression of state occupational probabilities in a multistate model with right-censored data

Inference for the state occupation probabilities, given a set of baseline covariates, is an important problem in survival analysis and time to event multistate data. We introduce an inverse censoring probability re-weighted semi-parametric single index model based approach to estimate conditional state occupation probabilities of a given individual in a multistate model under right-censoring. Besides obtaining a temporal regression function, we also test the potential time varying effect of a baseline covariate on future state occupation. We show that the proposed technique has desirable finite sample performances and its performance is competitive when compared with three other existing approaches. We illustrate the proposed methodology using two different data sets. First, we re-examine a well-known data set dealing with leukemia patients undergoing bone marrow transplant with various state transitions. Our second illustration is based on data from a study involving functional status of a set of spinal cord injured patients undergoing a rehabilitation program.     

Inference for outcome probabilities in multi-state models

In bone marrow transplantation studies, patients are followed over time and a number of events may be observed. These include both ultimate events like death and relapse and transient events like graft versus host disease and graft recovery. Such studies, therefore, lend themselves for using an analytic approach based on multi-state models. We will give a review of such methods with emphasis on regression models for both transition intensities and transition- and state occupation probabilities. Both semi-parametric models, like the Cox regression model, and parametric models based on piecewise constant intensities will be discussed.     

Inference for current leukemia free survival

Donor lymphocyte infusion (DLI) for patients who relapse following an allogeneic stem cell transplant has proved remarkably durable. Because of the potential for second remissions with DLI, the current leukemia free survival (CLFS), which is the probability that a patient has not failed the entire course of the treatment, is becoming of interest to clinical investigators. Based on either a multistate Markov model or a linear combination of Kaplan–Meier estimators, we explore regression models for the CLFS. We focus on the two sample problem and we develop confidence bands for the CLFS or for differences in CLFS as well as a Kolmogorov type hypothesis test using a re-sampling technique. We also examine the use of pseudo-values to make inference on the direct effects of covariates on the CLFS function and we develop a score test for the equality of two CLFS. We illustrate these inference methods on a bone marrow transplant dataset.     

On pseudo-values for regression analysis in competing risks models

For regression on state and transition probabilities in multi-state models Andersen et al. (Biometrika 90:15–27, 2003) propose a technique based on jackknife pseudo-values. In this article we analyze the pseudo-values suggested for competing risks models and prove some conjectures regarding their asymptotics (Klein and Andersen, Biometrics 61:223–229, 2005). The key is a second order von Mises expansion of the Aalen-Johansen estimator which yields an appropriate representation of the pseudo-values. The method is illustrated with data from a clinical study on total joint replacement. In the application we consider for comparison the estimates obtained with the Fine and Gray approach (J Am Stat Assoc 94:496–509, 1999) and also time-dependent solutions of pseudo-value regression equations.     

Estimating stage occupation probabilities in non-Markov models

We study non-Markov multistage models under dependent censoring regarding estimation of stage occupation probabilities. The individual transition and censoring mechanisms are linked together through covariate processes that affect both the transition intensities and the censoring hazard for the corresponding subjects. In order to adjust for the dependent censoring, an additive hazard regression model is applied to the censoring times, and all observed counting and “at risk” processes are subsequently given an inverse probability of censoring weighted form. We examine the bias of the Datta–Satten and Aalen–Johansen estimators of stage occupation probability, and also consider the variability of these estimators by studying their estimated standard errors and mean squared errors. Results from different simulation studies of frailty models indicate that the Datta–Satten estimator is approximately unbiased, whereas the Aalen–Johansen estimator either under- or overestimates the stage occupation probability due to the dependent nature of the censoring process. However, in our simulations, the mean squared error of the latter estimator tends to be slightly smaller than that of the former estimator. Studies on development of nephropathy among diabetics and on blood platelet recovery among bone marrow transplant patients are used as demonstrations on how the two estimation methods work in practice. Our analyses show that the Datta–Satten estimator performs well in estimating stage occupation probability, but that the censoring mechanism has to be quite selective before a deviation from the Aalen-Johansen estimator is of practical importance. N. Gunnes—Supported by a grant from the Norwegian Cancer Society.     

Pseudo-observations for competing risks with covariate dependent censoring

Regression analysis for competing risks data can be based on generalized estimating equations. For the case with right censored data, pseudo-values were proposed to solve the estimating equations. In this article we investigate robustness of the pseudo-values against violation of the assumption that the probability of not being lost to follow-up (un-censored) is independent of the covariates. Modified pseudo-values are proposed which rely on a correctly specified regression model for the censoring times. Bias and efficiency of these methods are compared in a simulation study. Further illustration of the differences is obtained in an application to bone marrow transplantation data and a corresponding sensitivity analysis.     

Analysis of band‐recovery data in a multistate capture‐recapture framework

Dead recoveries of marked animals are commonly used to estimate survival probabilities. Band‐recovery models can be parameterized either by r (the probability of recovering a band conditional on death of the animal) or by f (the probability that an animal will be killed, retrieved, and have its band reported). The T parametrization can be implemented in a capture‐recapture framework with two states (alive and newly dead), mortality being the transition probability between the two states. The authors show here that the f parametrization can also be implemented in a multistate framework by imposing simple constraints on some parameters. They illustrate it using data on the mallard and the snow goose. However, they mention that because it does not entirely separate the individual survival and encounter processes, the f parametrization must be used with care on reduced models, or in the presence of estimates at the boundary of the parameter space. As they show, a multistate framework allows the use of powerful software for model fitting or testing the goodness‐of‐fit of models; it also affords the implementation of complex models such as those based on mixture of information or uncertain states     

Targeted estimation of state occupation probabilities for the non-Markov illness-death model

We use semi-parametric efficiency theory to derive a class of estimators for the state occupation probabilities of the continuous-time irreversible illness-death model. We consider both the setting with and without additional baseline information available, where we impose no specific functional form on the intensity functions of the model. We show that any estimator in the class is asymptotically linear under suitable assumptions about the estimators of the intensity functions. In particular, the assumptions are weak enough to allow the use of data-adaptive methods, which is important for making the identifying assumption of coarsening at random plausible in realistic settings. We suggest a flexible method for estimating the transition intensity functions of the illness-death model based on penalized Poisson regression. We apply this method to estimate the nuisance parameters of an illness-death model in a simulation study and a real-world application.     

The time‐dependent “cure‐death” model investigating two equally important endpoints simultaneously in trials treating high‐risk patients with resistant pathogens

A variety of primary endpoints are used in clinical trials treating patients with severe infectious diseases, and existing guidelines do not provide a consistent recommendation. We propose to study simultaneously two primary endpoints, cure and death, in a comprehensive multistate cure‐death model as starting point for a treatment comparison. This technique enables us to study the temporal dynamic of the patient‐relevant probability to be cured and alive. We describe and compare traditional and innovative methods suitable for a treatment comparison based on this model. Traditional analyses using risk differences focus on one prespecified timepoint only. A restricted logrank‐based test of treatment effect is sensitive to ordered categories of responses and integrates information on duration of response. The pseudo‐value regression provides a direct regression model for examination of treatment effect via difference in transition probabilities. Applied to a topical real data example and simulation scenarios, we demonstrate advantages and limitations and provide an insight into how these methods can handle different kinds of treatment imbalances. The cure‐death model provides a suitable framework to gain a better understanding of how a new treatment influences the time‐dynamic cure and death process. This might help the future planning of randomised clinical trials, sample size calculations, and data analyses.     

Temporal prediction of future state occupation in a multistate model from high-dimensional baseline covariates via pseudo-value regression

In many complex diseases such as cancer, a patient undergoes various disease stages before reaching a terminal state (say disease free or death). This fits a multistate model framework where a prognosis may be equivalent to predicting the state occupation at a future time t. With the advent of high-throughput genomic and proteomic assays, a clinician may intent to use such high-dimensional covariates in making better prediction of state occupation. In this article, we offer a practical solution to this problem by combining a useful technique, called pseudo-value (PV) regression, with a latent factor or a penalized regression method such as the partial least squares (PLS) or the least absolute shrinkage and selection operator (LASSO), or their variants. We explore the predictive performances of these combinations in various high-dimensional settings via extensive simulation studies. Overall, this strategy works fairly well provided the models are tuned properly. Overall, the PLS turns out to be slightly better than LASSO in most settings investigated by us, for the purpose of temporal prediction of future state occupation. We illustrate the utility of these PV-based high-dimensional regression methods using a lung cancer data set where we use the patients’ baseline gene expression values.     

Lower probability models for uncertainty and nondeterministic processes

An extension to the class of conventional numerical probability models for nondeterministic phenomena has been identified by Dempster and Shafer in the class of belief functions. We were originally stimulated by this work, but have since come to believe that the bewildering diversity of uncertainty and chance phenomena cannot be encompassed within either the conventional theory of probability, its relatively minor modifications (e.g., not requiring countable additivity), or the theory of belief functions. In consequence, we have been examining the properties of, and prospects for, the generalization of belief functions that is known as upper and lower, or interval-valued, probability. After commenting on what we deem to be problematic elements of common personalist/subjectivist/Bayesian positions that employ either finitely or countably additive probability to represent strength of belief and that are intended to be normative for rational behavior, we sketch some of the ways in which the set of lower envelopes, a subset of the set of lower probabilities that contains the belief functions, enables us to preserve the core of Bayesian reasoning while admitting a more realistic (e.g., in its reduced insistence upon an underlying precision in our beliefs) class of probability-like models. Particular advantages of lower envelopes are identified in the area of the aggregation of beliefs.

The focus of our own research is in the area of objective probabilistic reasoning about time series generated by physical or other empirical (e.g., societal) processes. As it is not the province of a general mathematical methodology such as probability theory to a priori rule out of existence empirical phenomena, we are concerned by the contraint imposed by conventional probability theory that an empirical process of bounded random variables that is believed to have a time- invariant generating mechanism must then exhibot long-run stable time averages. We have shown that lower probability models that allow for unstable time averages can only lie in the class of undominated lower probabilities, a subset of lower probability models disjoint from the lower envelopes and having the weakest relationship to conventional probability measures. Our research has been devoted to exploring and developing the theory of undominated lower probabilities so that it can be applied to model and understand nondeterministic phenomena, and we have also been interested in identifying actual physical processes (e.g., flicker noises) that exhibit behavior requiring such novel models.     

Using number of failed contact attempts to adjust for non-ignorable non-response

Summary.  We present a general method of adjustment for non-ignorable non-response in studies where one or more further attempts are made to contact initial non-responders. A logistic regression model relates the probability of response at each contact attempt to covariates and outcomes of interest. We assume that the effect of these covariates and outcomes on the probability of response is the same at all contact attempts. Knowledge of the number of contact attempts enables estimation of the model by using only information from the respondents and the number of non-responders. Three approaches for fitting the response models and estimating parameters of substantive interest and their standard errors are compared: a modified conditional likelihood method in which the fitted inverse probabilities of response are used in weighted analyses for the outcomes of interest, an EM procedure with the Louis formula and a Bayesian approach using Markov chain Monte Carlo methods. We further propose the creation of several sets of weights to incorporate uncertainty in the probability weights in subsequent analyses. Our methods are applied as a sensitivity analysis to a postal survey of symptoms in Persian Gulf War veterans and other servicemen.     

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.     

A frequentist understanding of sets of measures

We present a mathematical theory of objective, frequentist chance phenomena that uses as a model a set of probability measures. In this work, sets of measures are not viewed as a statistical compound hypothesis or as a tool for modeling imprecise subjective behavior. Instead we use sets of measures to model stable (although not stationary in the traditional stochastic sense) physical sources of finite time series data that have highly irregular behavior. Such models give a coarse-grained picture of the phenomena, keeping track of the range of the possible probabilities of the events. We present methods to simulate finite data sequences coming from a source modeled by a set of probability measures, and to estimate the model from finite time series data. The estimation of the set of probability measures is based on the analysis of a set of relative frequencies of events taken along subsequences selected by a collection of rules. In particular, we provide a universal methodology for finding a family of subsequence selection rules that can estimate any set of probability measures with high probability.     

Ordered response models

Summary We discuss regression models for ordered responses, such as ratings of bonds, schooling attainment, or measures of subjective well-being. Commonly used models in this context are the ordered logit and ordered probit regression models. They are based on an underlying latent model with single index function and constant thresholds. We argue that these approaches are overly restrictive and preclude a flexible estimation of the effect of regressors on the discrete outcome probabilities. For example, the signs of the marginal probability effects can only change once when moving from the smallest category to the largest one. We then discuss several alternative models that overcome these limitations. An application illustrates the benefit of these alternatives. We are grateful to an anonymous referee for valuable comments.     

Dynamic inference for non-Markov transition probabilities under random right censoring

The main contribution of this article is the verification of weak convergence of a general non-Markov (NM) state transition probability estimator by Titman, which has not yet been done for any other general NM estimator. A similar theorem is shown for the bootstrap, yielding resampling-based inference methods for statistical functionals. Formulas of the involved covariance functions are presented in detail. Particular applications include the conditional expected length of stay in a specific state, given occupation of another state in the past, and the construction of time-simultaneous confidence bands for the transition probabilities. The expected lengths of stay in a two-sample liver cirrhosis dataset are compared and confidence intervals for their difference are constructed. With borderline significance and in comparison to the placebo group, the treatment group has an elevated expected length of stay in the healthy state given an earlier disease state occupation. In contrast, the Aalen-Johansen (AJ) estimator-based confidence interval, which relies on a Markov assumption, leads to a drastically different conclusion. Also, graphical illustrations of confidence bands for the transition probabilities demonstrate the biasedness of the AJ estimator in this data example. The reliability of these results is assessed in a simulation study.     

Model diagnostics for marginal regression analysis of correlated binary data

We propose several diagnostic methods for checking the adequacy of marginal regression models for analyzing correlated binary data. We use a parametric marginal model based on latent variables and derive the projection (hat) matrix, Cook's distance, various residuals and Mahalanobis distance between the observed binary responses and the estimated probabilities for a cluster. Emphasized are several graphical methods including the simulated Q-Q plot, the half-normal probability plot with a simulated envelope, and the partial residual plot. The methods are illustrated with a real life example.     

Proportional hazards model for competing risks data with missing cause of failure

We consider the semiparametric proportional hazards model for the cause-specific hazard function in analysis of competing risks data with missing cause of failure. The inverse probability weighted equation and augmented inverse probability weighted equation are proposed for estimating the regression parameters in the model, and their theoretical properties are established for inference. Simulation studies demonstrate that the augmented inverse probability weighted estimator is doubly robust and the proposed method is appropriate for practical use. The simulations also compare the proposed estimators with the multiple imputation estimator of Lu and Tsiatis (2001). The application of the proposed method is illustrated using data from a bone marrow transplant study.     

Non-homogeneous Markov models in the analysis of survival after breast cancer

A cohort of 300 women with breast cancer who were submitted for surgery is analysed by using a non-homogeneous Markov process. Three states are onsidered: no relapse, relapse and death. As relapse times change over time, we have extended previous approaches for a time homogeneous model to a non omogeneous multistate process. The trends of the hazard rate functions of transitions between states increase and then decrease, showing that a changepoint can be considered. Piecewise Weibull distributions are introduced as transition intensity functions. Covariates corresponding to treatments are incorporated in the model multiplicatively via these functions. The likelihood function is built for a general model with k changepoints and applied to the data set, the parameters are estimated and life-table and transition probabilities for treatments in different periods of time are given. The survival probability functions for different treatments are plotted and compared with the corresponding function for the homogeneous model. The survival functions for the various cohorts submitted for treatment are fitted to the mpirical survival functions.