Direct Modelling of Regression Effects for Transition Probabilities in Multistate Models

Estimands and Cumulative Incidence Function Regression in Clinical Trials: Some New Results on Interpretability and Robustness

Regression analyses based on transformations of cumulative incidence functions are often adopted when modeling and testing for treatment effects in clinical trial settings involving competing and semi-competing risks. Common frameworks include the Fine-Gray model and models based on direct binomial regression. Using large sample theory we derive the limiting values of treatment effect estimators based on such models when the data are generated according to multiplicative intensity-based models, and show that the estimand is sensitive to several process features. The rejection rates of hypothesis tests based on cumulative incidence function regression models are also examined for null hypotheses of different types, based on which a robustness property is established. In such settings supportive secondary analyses of treatment effects are essential to ensure a full understanding of the nature of treatment effects. An application to a palliative study of individuals with breast cancer metastatic to bone is provided for illustration.

Inference for transition probabilities in non-Markov multi-state models

Multi-state models are frequently used when data come from subjects observed over time and where focus is on the occurrence of events that the subjects may experience. A convenient modeling assumption is that the multi-state stochastic process is Markovian, in which case a number of methods are available when doing inference for both transition intensities and transition probabilities. The Markov assumption, however, is quite strict and may not fit actual data in a satisfactory way. Therefore, inference methods for non-Markov models are needed. In this paper, we review methods for estimating transition probabilities in such models and suggest ways of doing regression analysis based on pseudo observations. In particular, we will compare methods using land-marking with methods using plug-in. The methods are illustrated using simulations and practical examples from medical research.

A multistate competing risks framework for preconception prediction of pregnancy outcomes

Background

Preconception pregnancy risk profiles—characterizing the likelihood that a pregnancy attempt results in a full-term birth, preterm birth, clinical pregnancy loss, or failure to conceive—can provide critical information during the early stages of a pregnancy attempt, when obstetricians are best positioned to intervene to improve the chances of successful conception and full-term live birth. Yet the task of constructing and validating risk assessment tools for this earlier intervention window is complicated by several statistical features: the final outcome of the pregnancy attempt is multinomial in nature, and it summarizes the results of two intermediate stages, conception and gestation, whose outcomes are subject to competing risks, measured on different time scales, and governed by different biological processes. In light of this complexity, existing pregnancy risk assessment tools largely focus on predicting a single adverse pregnancy outcome, and make these predictions at some later, post-conception time point.

Methods

We reframe the individual pregnancy attempt as a multistate model comprised of two nested multinomial prediction tasks: one corresponding to conception and the other to the subsequent outcome of that pregnancy. We discuss the estimation of this model in the presence of multiple stages of outcome missingness and then introduce an inverse-probability-weighted Hypervolume Under the Manifold statistic to validate the resulting multivariate risk scores. Finally, we use data from the Effects of Aspirin in Gestation and Reproduction (EAGeR) trial to illustrate how this multistate competing risks framework might be utilized in practice to construct and validate a preconception pregnancy risk assessment tool.

Results

In the EAGeR study population, the resulting risk profiles are able to meaningfully discriminate between the four pregnancy attempt outcomes of interest and represent a significant improvement over classification by random chance.

Conclusions

As illustrated in our analysis of the EAGeR data, our proposed prediction framework expands the pregnancy risk assessment task in two key ways—by considering a broader array of pregnancy outcomes and by providing the predictions at an earlier, preconception intervention window—providing obstetricians and their patients with more information and opportunities to successfully guide pregnancy attempts.

Work stress and loss of years lived without chronic disease: an 18-year follow-up of 1.5 million employees in Denmark

We aimed to examine the association between exposure to work stress and chronic disease incidence and loss of chronic disease-free life years in the Danish workforce. The study population included 1,592,491 employees, aged 30-59 in 2000 and without prevalent chronic diseases. We assessed work stress as the combination of job strain and effort-reward imbalance using job exposure matrices. We used Cox regressions to estimate risk of incident hospital-diagnoses or death of chronic diseases (i.e., type 2 diabetes, coronary heart disease, stroke, cancer, asthma, chronic obstructive pulmonary disease, heart failure, and dementia) during 18 years of follow-up and calculated corresponding chronic disease-free life expectancy from age 30 to age 75. Individuals working in occupations with high prevalence of work stress had a higher risk of incident chronic disease compared to those in occupations with low prevalence of work stress (women: HR 1.04 (95% CI 1.02-1.05), men: HR 1.12 (95% CI 1.11-1.14)). The corresponding loss in chronic disease-free life expectancy was 0.25 (95% CI - 0.10 to 0.60) and 0.84 (95% CI 0.56-1.11) years in women and men, respectively. Additional adjustment for health behaviours attenuated these associations among men. We conclude that men working in high-stress occupations have a small loss of years lived without chronic disease compared to men working in low-stress occupations. This finding appeared to be partially attributable to harmful health behaviours. In women, high work stress indicated a very small and statistically non-significant loss of years lived without chronic disease.

Expected Labor Market Affiliation: A New Method Illustrated by Estimating the Impact of Perceived Stress on Time in Work, Sickness Absence and Unemployment of 37,605 Danish Employees

As detailed data on labor market affiliation become more accessible, new approaches are needed to address the complex patterns of labor market affiliation. We introduce the expected labor market affiliation (ELMA) method by estimating the time-restricted impact of perceived stress on labor market affiliation in a large sample of Danish employees. Data from two national surveys were linked with a national register. A multi-state proportional hazards model was used to calculate ELMA estimates, i.e., the number of days in work, sickness absence, and unemployment during a 4-year follow-up period, stratified by gender and age. Among employees reporting frequent work-related stress, the expected number of working days decreased with age, ranging from 103 days lost among older women to 37 days lost among younger and middle-aged men. Young and middle-aged women reporting frequent work- and personal life-related stress lost 62 and 81 working days, respectively, and had more days of sickness absence (34 days and 42 days). In conclusion, we showed that perceived stress affects the labor market affiliation. The ELMA estimates provide a detailed understanding of the impact of perceived stress on labor market affiliation over time, and may inform policy and practice towards a more healthy and sustainable working life.

Nonparametric analysis of nonhomogeneous multistate processes with clustered observations

Frequently, clinical trials and observational studies involve complex event history data with multiple events. When the observations are independent, the analysis of such studies can be based on standard methods for multistate models. However, the independence assumption is often violated, such as in multicenter studies, which makes standard methods improper. This work addresses the issue of nonparametric estimation and two‐sample testing for the population‐averaged transition and state occupation probabilities under general multistate models with cluster‐correlated, right‐censored, and/or left‐truncated observations. The proposed methods do not impose assumptions regarding the within‐cluster dependence, allow for informative cluster size, and are applicable to both Markov and non‐Markov processes. Using empirical process theory, the estimators are shown to be uniformly consistent and to converge weakly to tight Gaussian processes. Closed‐form variance estimators are derived, rigorous methodology for the calculation of simultaneous confidence bands is proposed, and the asymptotic properties of the nonparametric tests are established. Furthermore, I provide theoretical arguments for the validity of the nonparametric cluster bootstrap, which can be readily implemented in practice regardless of how complex the underlying multistate model is. Simulation studies show that the performance of the proposed methods is good, and that methods that ignore the within‐cluster dependence can lead to invalid inferences. Finally, the methods are illustrated using data from a multicenter randomized controlled trial.

Semiparametric temporal process regression of survival-out-of-hospital

The recurrent/terminal event data structure has undergone considerable methodological development in the last 10-15 years. An example of the data structure that has arisen with increasing frequency involves the recurrent event being hospitalization and the terminal event being death. We consider the response Survival-Out-of-Hospital, defined as a temporal process (indicator function) taking the value 1 when the subject is currently alive and not hospitalized, and 0 otherwise. Survival-Out-of-Hospital is a useful alternative strategy for the analysis of hospitalization/survival in the chronic disease setting, with the response variate representing a refinement to survival time through the incorporation of an objective quality-of-life component. The semiparametric model we consider assumes multiplicative covariate effects and leaves unspecified the baseline probability of being alive-and-out-of-hospital. Using zero-mean estimating equations, the proposed regression parameter estimator can be computed without estimating the unspecified baseline probability process, although baseline probabilities can subsequently be estimated for any time point within the support of the censoring distribution. We demonstrate that the regression parameter estimator is asymptotically normal, and that the baseline probability function estimator converges to a Gaussian process. Simulation studies are performed to show that our estimating procedures have satisfactory finite sample performances. The proposed methods are applied to the Dialysis Outcomes and Practice Patterns Study (DOPPS), an international end-stage renal disease study.

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.

Direct modeling of regression effects for transition probabilities in the progressive illness-death model

In this work, we present direct regression analysis for the transition probabilities in the possibly non-Markov progressive illness-death model. The method is based on binomial regression, where the response is the indicator of the occupancy for the given state along time. Randomly weighted score equations that are able to remove the bias due to censoring are introduced. By solving these equations, one can estimate the possibly time-varying regression coefficients, which have an immediate interpretation as covariate effects on the transition probabilities. The performance of the proposed estimator is investigated through simulations. We apply the method to data from the Registry of Systematic Lupus Erythematosus RELESSER, a multicenter registry created by the Spanish Society of Rheumatology. Specifically, we investigate the effect of age at Lupus diagnosis, sex, and ethnicity on the probability of damage and death along time. Copyright © 2017 John Wiley & Sons, Ltd.

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 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 pseudo-value based high dimensional regression methods using a lung cancer data set where we use the patients' baseline gene expression values.

A flexible semiparametric modeling approach for doubly censored data with an application to prostate cancer

Doubly censored data often arise in medical studies of disease progression involving two related events for which both an originating and a terminating event are interval-censored. Although regression modeling for such doubly censored data may be complicated, we propose a simple semiparametric regression modeling strategy based on jackknife pseudo-observations obtained using nonparametric estimators of the survival function. Inference is carried out via generalized estimating equations. Simulations studies show that the proposed method produces virtually unbiased covariate effect estimates, even for moderate sample sizes. A prostate cancer study example illustrates the practical advantages of the proposed approach.

Penalized estimation for competing risks regression with applications to high-dimensional covariates

High-dimensional regression has become an increasingly important topic for many research fields. For example, biomedical research generates an increasing amount of data to characterize patients' bio-profiles (e.g. from a genomic high-throughput assay). The increasing complexity in the characterization of patients' bio-profiles is added to the complexity related to the prolonged follow-up of patients with the registration of the occurrence of possible adverse events. This information may offer useful insight into disease dynamics and in identifying subset of patients with worse prognosis and better response to the therapy. Although in the last years the number of contributions for coping with high and ultra-high-dimensional data in standard survival analysis have increased (Witten and Tibshirani, 2010. Survival analysis with high-dimensional covariates. Statistical Methods in Medical Research19(1), 29–51), the research regarding competing risks is less developed (Binder and others, 2009. Boosting for high-dimensional time-to-event data with competing risks. Bioinformatics25(7), 890–896). The aim of this work is to consider how to do penalized regression in the presence of competing events. The direct binomial regression model of Scheike and others (2008. Predicting cumulative incidence probability by direct binomial regression. Biometrika95(1), 205–220) is reformulated in a penalized framework to possibly fit a sparse regression model. The developed approach is easily implementable using existing high-performance software to do penalized regression. Results from simulation studies are presented together with an application to genomic data when the endpoint is progression-free survival. An R function is provided to perform regularized competing risks regression according to the binomial model in the package timereg (Scheike and Martinussen, 2006. Dynamic Regression models for survival data. New York: Springer), available through CRAN.

The proportional odds cumulative incidence model for competing risks

We suggest an estimator for the proportional odds cumulative incidence model for competing risks data. The key advantage of this model is that the regression parameters have the simple and useful odds ratio interpretation. The model has been considered by many authors, but it is rarely used in practice due to the lack of reliable estimation procedures. We suggest such procedures and show that their performance improve considerably on existing methods. We also suggest a goodness-of-fit test for the proportional odds assumption. We derive the large sample properties and provide estimators of the asymptotic variance. The method is illustrated by an application in a bone marrow transplant study and the finite-sample properties are assessed by simulations.

Comparison of stopped Cox regression with direct methods such as pseudo-values and binomial regression

By far the most popular model to obtain survival predictions for individual patients is the Cox model. The Cox model does not make any assumptions on the underlying hazard, but it relies heavily on the proportional hazards assumption. The most common ways to circumvent this robustness problem are 1) to categorize patients based on their prognostic risk score and to base predictions on Kaplan-Meier curves for the risk categories, or 2) to include interactions with the covariates and suitable functions of time. Robust estimators of the t(0)-year survival probabilities can also be obtained from a "stopped Cox" regression model, in which all observations are administratively censored at t(0). Other recent approaches to solve this robustness problem, originally proposed in the context of competing risks, are pseudo-values and direct binomial regression, based on unbiased estimating equations. In this paper stopped Cox regression is compared with these direct approaches. This is done by means of a simulation study to assess the biases of the different approaches and an analysis of breast cancer data to get some feeling for the performance in practice. The tentative conclusion is that stopped Cox and direct models agree well if the follow-up is not too long. There are larger differences for long-term follow-up data. There stopped Cox might be more efficient, but less robust.

Varying coefficient subdistribution regression for left-truncated semi-competing risks data

Semi-competing risks data frequently arise in biomedical studies when time to a disease landmark event is subject to dependent censoring by death, the observation of which however is not precluded by the occurrence of the landmark event. In observational studies, the analysis of such data can be further complicated by left truncation. In this work, we study a varying co-efficient subdistribution regression model for left-truncated semi-competing risks data. Our method appropriately accounts for the specifical truncation and censoring features of the data, and moreover has the flexibility to accommodate potentially varying covariate effects. The proposed method can be easily implemented and the resulting estimators are shown to have nice asymptotic properties. We also present inference, such as Kolmogorov-Smirnov type and Cramér Von-Mises type hypothesis testing procedures for the covariate effects. Simulation studies and an application to the Denmark diabetes registry demonstrate good finite-sample performance and practical utility of the proposed method.

Progression of liver cirrhosis to HCC: an application of hidden Markov model

Background

Health service databases of administrative type can be a useful tool for the study of progression of a disease, but the data reported in such sources could be affected by misclassifications of some patients' real disease states at the time. Aim of this work was to estimate the transition probabilities through the different degenerative phases of liver cirrhosis using health service databases.

Methods

We employed a hidden Markov model to determine the transition probabilities between two states, and of misclassification. The covariates inserted in the model were sex, age, the presence of comorbidities correlated with alcohol abuse, the presence of diagnosis codes indicating hepatitis C virus infection, and the Charlson Index. The analysis was conducted in patients presumed to have suffered the onset of cirrhosis in 2000, observing the disease evolution and, if applicable, death up to the end of the year 2006.

Results

The incidence of hepatocellular carcinoma (HCC) in cirrhotic patients was 1.5% per year. The probability of developing HCC is higher in males (OR = 2.217) and patients over 65 (OR = 1.547); over 65-year-olds have a greater probability of death both while still suffering from cirrhosis (OR = 2.379) and if they have developed HCC (OR = 1.410). A more severe casemix affects the transition from HCC to death (OR = 1.714). The probability of misclassifying subjects with HCC as exclusively affected by liver cirrhosis is 14.08%.

Conclusions

The hidden Markov model allowing for misclassification is well suited to analyses of health service databases, since it is able to capture bias due to the fact that the quality and accuracy of the available information are not always optimal. The probability of evolution of a cirrhotic subject to HCC depends on sex and age class, while hepatitis C virus infection and comorbidities correlated with alcohol abuse do not seem to have an influence.

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.

Dynamic predicting by landmarking as an alternative for multi-state modeling: an application to acute lymphoid leukemia data

This paper considers the problem of obtaining a dynamic prediction for 5-year failure free survival after bone marrow transplantation in ALL patients using data from the EBMT, the European Group for Blood and Marrow Transplantation. The paper compares the new landmark methodology as developed by the first author and the established multi-state modeling as described in a recent Tutorial in Biostatistics in Statistics in Medicine by the second author and colleagues. As expected the two approaches give similar results. The landmark methodology does not need complex modeling and leads to easy prediction rules. On the other hand, it does not give the insight in the biological processes as obtained for the multi-state model.

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.

A Markov model to evaluate hospital readmission

Background

The analysis of non-fatal recurring events is frequently found in studies on chronic-degenerative diseases. The aim of this paper is to estimate the probability of readmission of patients with Chronic Obstructive Pulmonary Disease (COPD) or with Respiratory Failure (RF).

Methods

The Repeated hospital admissions of a patient are considered as a Markov Chain. The transitions between the states are estimated using the Nelson-Aalen estimator. The analysis was carried out using the Puglia Region hospital patient discharge database for the years 1998–2005. Patients were selected on the basis of first admission between 01/01/2001 and 31/12/2005 with ICD-9-CM code of COPD or RF as principal and/or secondary diagnosis. For those selected two possible transitions were considered in the case they had the second and third admission with an ICD-9-CM code of COPD or RF as principal diagnosis.

Results

The probability of readmission is increased in patients with a diagnosis of RF (OR = 1.618 in the first transition and 1.279 in the second) and also in those with a diagnosis of COPD or RF as the principal diagnosis at first admission (OR = 1.615 in the first transition and 1.193 in the second). The clinical gravity and the ward from which they were discharged did not significantly influence the probability of readmission.

Conclusion

The time to readmission depends on the gravity of the pathology at onset. In patients with a grave clinical picture, either COPD or Respiratory Failure, when treated and controlled after the first admission, they become minor problems and they are indicated among secondary diagnoses in any further admission.

Time-dependent covariates in the proportional subdistribution hazards model for competing risks

Separate Cox analyses of all cause-specific hazards are the standard technique of choice to study the effect of a covariate in competing risks, but a synopsis of these results in terms of cumulative event probabilities is challenging. This difficulty has led to the development of the proportional subdistribution hazards model. If the covariate is known at baseline, the model allows for a summarizing assessment in terms of the cumulative incidence function. black Mathematically, the model also allows for including random time-dependent covariates, but practical implementation has remained unclear due to a certain risk set peculiarity. We use the intimate relationship of discrete covariates and multistate models to naturally treat time-dependent covariates within the subdistribution hazards framework. The methodology then straightforwardly translates to real-valued time-dependent covariates. As with classical survival analysis, including time-dependent covariates does not result in a model for probability functions anymore. Nevertheless, the proposed methodology provides a useful synthesis of separate cause-specific hazards analyses. We illustrate this with hospital infection data, where time-dependent covariates and competing risks are essential to the subject research question.