MKVPCI: a computer program for Markov models with piecewise constant intensities and covariates

A discrete approximation method for modeling interval-censored multistate data

Many longitudinal studies are designed to monitor participants for major events related to the progression of diseases. Data arising from such longitudinal studies are usually subject to interval censoring since the events are only known to occur between two monitoring visits. In this work, we propose a new method to handle interval-censored multistate data within a proportional hazards model framework where the hazard rate of events is modeled by a nonparametric function of time and the covariates affect the hazard rate proportionally. The main idea of this method is to simplify the likelihood functions of a discrete-time multistate model through an approximation and the application of data augmentation techniques, where the assumed presence of censored information facilitates a simpler parameterization. Then the expectation-maximization algorithm is used to estimate the parameters in the model. The performance of the proposed method is evaluated by numerical studies. Finally, the method is employed to analyze a dataset on tracking the advancement of coronary allograft vasculopathy following heart transplantation.

Estimation of covariate effects on net survivals in the relative survival progressive illness-death model

Recently, there has been a lot of development in relative survival field. In the absence of data on the cause of death, the research has tended to focus on the estimation of survival probability of a cancer (as a disease of interest). In many cancers, one nonfatal event that decreases the survival probability can occur. There are a few methods that assess the role of prognostic factors for multiple types of clinical events while dealing with uncertainty about the cause of death. However, these methods require proportional hazard or Markov assumptions. In practice, one or both of these assumptions might be violated. Violation of the proportional hazard assumption can lead to estimates that are biased, and difficult to interpret and violation of Markov assumption results in inconsistent estimators. In this work, we propose a semi-parametric approach to estimate the possibly time-varying regression coefficients in the likely non-Markov relative survival progressive illness-death model. The performance of the proposed estimator is investigated through simulations. We illustrate our approach using data from a study on rectal cancer resected for cure conducted in two French population-based digestive cancer registries.

Metastatic Potential and Survival of Duodenal and Pancreatic Tumors in Multiple Endocrine Neoplasia Type 1: A GTE and AFCE Cohort Study (Groupe d'étude des Tumeurs Endocrines and Association Francophone de Chirurgie Endocrinienne)

OBJECTIVE

To assess the distant metastatic potential of duodeno-pancreatic neuroendocrine tumors (DP-NETs) in patients with MEN1, according to functional status and size.

SUMMARY BACKGROUND DATA

DP-NETs, with their numerous lesions and endocrine secretion-related symptoms, continue to be a medical challenge; unfortunately they can become aggressive tumors associated with distant metastasis, shortening survival. The survival of patients with large nonfunctional DP-NETs is known to be poor, but the overall contribution of DP-NETs to metastatic spread is poorly known.

METHODS

The study population included patients with DP-NETs diagnosed after 1990 and followed in the MEN1 cohort of the Groupe d'étude des Tumeurs Endocrines (GTE). A multistate Markov piecewise constant intensities model was applied to separate the effects of prognostic factors on 1) metastasis, and 2) metastasis-free death or 3) death after appearance of metastases.

RESULTS

Among the 603 patients included, 39 had metastasis at diagnosis of DP-NET, 50 developed metastases during follow-up, and 69 died. The Markov model showed that Zollinger-Ellison-related tumors (regardless of tumor size and thymic tumor pejorative impact), large tumors over 2 cm, and age over 40 years were independently associated with an increased risk of metastases. Men, patients over 40 years old and patients with tumors larger than 2 cm, also had an increased risk of death once metastasis appeared.

CONCLUSIONS

DP-NETs of 2 cm in size or more, regardless of the associated secretion, should be removed to prevent metastasis and increase survival. Surgery for gastrinoma remains debatable.

Anti-tumour activity of everolimus and sunitinib in neuroendocrine neoplasms

Comparisons between everolimus and sunitinib regarding their efficacy and safety in neuroendocrine neoplasms (NENs) are scarce. We retrospectively analysed the clinicopathological characteristics and outcomes in 92 patients with well-differentiated (WD) NEN of different origin (57 pancreatic NENs (PanNENs)), treated with molecular targeted therapy (MTT) with everolimus or sunitinib, first- (73:19) or second-line (sequential; 12:22) for progressive disease. Disease control rates (DCR: partial response or stable disease) at first-line were higher in all patients treated with everolimus than sunitinib (64/73 vs 12/19, P = 0.012). In PanNENs, DCR at first-line everolimus was 36/42 versus 9/15 with sunitinib (P = 0.062). Progression-free survival (PFS) at first-line everolimus was longer than sunitinib (31 months (95% CI: 23.1-38.9) vs 9 months (95% CI: 0-18.5); log-rank P < 0.0001) in the whole cohort and the subset of PanNENs (log-rank P < 0.0001). Median PFS at second-line MTT was 12 months with everolimus (95% CI: 4.1-19.9) vs 13 months with sunitinib (95% CI: 9.3-16.7; log-rank P = 0.951). Treatment with sunitinib (HR: 3.47; 95% CI: 1.5-8.3; P value: 0.005), KI67 >20% (HR: 6.38; 95% CI: 1.3-31.3; P = 0.022) and prior chemotherapy (HR: 2.71; 95% CI: 1.2-6.3; P = 0.021) were negative predictors for PFS at first line in multivariable and also confirmed at multi-state modelling analyses. Side effect (SE) analysis indicated events of serious toxicities (Grades 3 and 4: n = 13/85 for everolimus and n = 4/41 for sunitinib). Discontinuation rate due to SEs was 20/85 for everolimus versus 4/41 for sunitinib (P = 0.065). No additive toxicity of second-line MTT was confirmed. Based on these findings, and until reliable predictors of response become available, everolimus may be preferable to sunitinib when initiating MTT in progressive NENs.

Multistate recursively imputed survival trees for time-to-event data analysis: an application to AIDS and mortality post-HIV infection data

BACKGROUND

This study aimed to introduce recursively imputed survival trees into multistate survival models (MSRIST) to analyze these types of data and to identify the prognostic factors influencing the disease progression in patients with intermediate events. The proposed method is fully nonparametric and can be used for estimating transition probabilities.

METHODS

A general algorithm was provided for analyzing multi-state data with a focus on the illness-death and progressive multi-state models. The model considered both beyond Markov and Non-Markov settings. We also proposed a multi-state random survival method (MSRSF) and compared their performance with the classical multi-state Cox model. We applied the proposed method to a dataset related to HIV/AIDS patients based on a retrospective cohort study extracted in Tehran from April 2004 to March 2014 consist of 2473 HIV-infected patients.

RESULTS

The results showed that MSRIST outperformed the classical multistate method using Cox Model and MSRSF in terms of integrated Brier score and concordance index over 500 repetitions. We also identified a set of important risk factors as well as their interactions on different states of HIV and AIDS progression.

CONCLUSIONS

There are different strategies for modelling the intermediate event. We adapted two newly developed data mining technique (RSF and RIST) for multistate models (MSRSF and MSRIST) to identify important risk factors in different stages of the diseases. The methods can capture any complex relationship between variables and can be used as a useful tool for identifying important risk factors in different states of this disease.

Multi-state relative survival modelling of colorectal cancer progression and mortality

Accurate identification of factors associated with progression of colorectal cancer remains a challenge. In particular, it is unclear which statistical methods are most suitable to separate the effects of putative prognostic factors on cancer progression vs cancer-specific and other cause mortality. To address these challenges, we analyzed 10 year follow-up data for patients who underwent curative surgery for colorectal cancer in 1985-2000. Separate analyses were performed in two French cancer registries. Results of three multivariable models were compared: Cox model with recurrence as a time-dependent variable, and two multi-state models, which separated prognostic factor effects on recurrence vs death, with or without recurrence. Conventional multi-state model analyzed all-cause mortality while new relative survival multi-state model focused on cancer-specific mortality. Among the 2517 and 2677 patients in the two registries, about 50% died without a recurrence, and 28% had a recurrence, of whom almost 90% died. In both multi-state models men had significantly increased risk of cancer recurrence in both registries (HR=0.79; 95% CI: 0.68-0.92 and HR=0.83; 95% CI: 0.71-0.96). However, the two multi-state models identified different prognostic factors for mortality without recurrence. In contrast to the conventional model, in the relative survival analyses gender had no independent association with cancer-specific mortality whereas patients diagnosed with stage III cancer had significantly higher risks in both registries (HR=1.67; 95% CI: 1.27-2.22 and HR=2.38; 95% CI: 1.29-3.27). In conclusion, relative survival multi-state model revealed that different factors may be associated with cancer recurrence vs cancer-specific mortality either after or without a recurrence.

Interval-censored time-to-event and competing risk with death: is the illness-death model more accurate than the Cox model?

BACKGROUND

In survival analyses of longitudinal data, death is often a competing event for the disease of interest, and the time-to-disease onset is interval-censored when the diagnosis is made at intermittent follow-up visits. As a result, the disease status at death is unknown for subjects disease-free at the last visit before death. Standard survival analysis consists in right-censoring the time-to-disease onset at that visit, which may induce an underestimation of the disease incidence. By contrast, an illness-death model for interval-censored data accounts for the probability of developing the disease between that visit and death, and provides a better incidence estimate. However, the two approaches have never been compared for estimating the effect of exposure on disease risk.

METHODS

This paper compares through simulations the accuracy of the effect estimates from a semi-parametric illness-death model for interval-censored data and the standard Cox model. The approaches are also compared for estimating the effects of selected risk factors on the risk of dementia, using the French elderly PAQUID cohort data.

RESULTS

The illness-death model provided a more accurate effect estimate of exposures that also affected mortality. The direction and magnitude of the bias from the Cox model depended on the effects of the exposure on disease and death. The application to the PAQUID cohort confirmed the simulation results.

CONCLUSION

If follow-up intervals are wide and the exposure has an impact on death, then the illness-death model for interval-censored data should be preferred to the standard Cox regression analysis.

A decision tree to help determine the best timing and antiretroviral strategy in HIV-infected patients

Optimal antiretroviral strategies for HIV-infected patients still need to be established. To this end a decision tree including different antiretroviral strategies that could be adopted for HIV-infected patients was built. A 10-year follow-up was simulated by using transitional probabilities estimated from a large cohort using a time-homogeneous Markov model. The desired outcome was for patients to maintain a CD4 cell count of >500 cells/mm3 without experiencing AIDS or death. For patients with a baseline HIV viral load ⩾5 log10 copies/ml, boosted protease inhibitor-based immediate highly active antiretroviral therapy (HAART) allowed them to spend 12% more time with CD4 ⩾500/mm3 than did delayed HAART (6·40 vs. 5·69 and 5·57 vs. 4·90 years for baseline CD4 ⩾500 and 350–499/mm3, respectively). In patients with a baseline HIV viral load ⩽3·5 log10 copies/ml, delayed HAART performed better than immediate HAART (6·43 vs. 6·26 and 5·95 vs. 5·18 for baseline CD4 ⩾500 and 350–499/mm3, respectively). Immediate HAART is beneficial in patients with a baseline HIV viral load ⩾5 log10 copies/ml, whereas deferred HAART appears to be the best option for patients with CD4 ⩾350/mm3 and baseline HIV viral load <3·5 log10 copies/ml.

The mstate package for estimation and prediction in non- and semi-parametric multi-state and competing risks models

In recent years, multi-state models have been studied widely in survival analysis. Despite their clear advantages, their use in biomedical and other applications has been rather limited so far. An important reason for this is the lack of flexible and user-friendly software for multi-state models. This paper introduces a package in R, called 'mstate', for each of the steps of the analysis of multi-state models. It can be applied to non- and semi-parametric models. The package contains functions to facilitate data preparation and flexible estimation of different types of covariate effects in the context of Cox regression models, functions to estimate patient-specific transition intensities, dynamic prediction probabilities and their associated standard errors (both Greenwood and Aalen-type). Competing risks models can also be analyzed by means of mstate, as they are a special type of multi-state models. The package is available from the R homepage http://cran.r-project.org. We give a self-contained account of the underlying mathematical theory, including a new asymptotic result for the cumulative hazard function and new recursive formulas for the calculation of the estimated standard errors of the estimated transition probabilities, and we illustrate the use of the key functions of the mstate package by the analysis of a reversible multi-state model describing survival of liver cirrhosis 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.

The disablement process: Factors associated with progression of disability and recovery in French elderly people

PURPOSE

To study the factors associated with progression, recovery and death from different grades of disability in elderly people.

METHOD

The sample included 3198 participants of the PAQUID ('Personnes Agées QUID') cohort, aged 65 and over and community-dwellers at baseline. Subjects were re-interviewed 1, 3, 5, 8 and 10 years after baseline. A five-state Markov model was used to estimate transition intensities between four grades of disability and toward death. We used a hierarchic scale of disability, which combines basic and instrumental activities of daily living and mobility. Several explanatory variables were investigated: medical, personal and environmental factors.

RESULTS

The factors associated with progression and/or no recovery of disability were cardiovascular diseases, stroke and diabetes, low cognition, visual impairment and dyspnoea (for pathologies and impairments), older age, female gender, low educational level (for risk factors), depression (for intra-individual factor) and being married, recent hospitalization and number of drugs (for extra-individual factors). Older age, male gender, tobacco consumption and living in an urban area were associated with mortality.

CONCLUSIONS

These findings confirm the independent contribution of each group of variables in the disablement process and stress their different impact on progression of disability or on recovery from different grades of disability.

Markov modelling of HIV infection evolution in the HAART era

The aim was to investigate the impact of the main prognostic factors on HIV evolution. A multi-state Markov model was applied in a cohort of 2126 patients to estimate impact of these factors on patients' clinical and immunological evolutions. Clinical progression and immunological deterioration shared most of their prognostic factors: male gender, intravenous drug use, weight loss, low haemoglobin level (<110 g/l), CD8 cell count (<500/mm(3)) and HIV viral load (>5 log(10) copies/ml). Highly active retroviral therapy reduced the risks of clinical progression and immune deterioration whatever patients' CD4 cell count. Risk reductions were 41-60% for protease inhibitor-based and 27-68% for non-nucleoside reverse transcriptase inhibitor-based regimens. Three-year transition probabilities showed that only patients with a CD4 cell count >or=350 CD4/mm(3) could in most cases maintain their immunity. This model provides 'real life' transition probabilities from one immunological stage to another, allowing decision analyses that could help determine the beneficial therapeutic strategies for HIV-infected patients.

Multi-state models for the analysis of time-to-event data

The experience of a patient in a survival study may be modelled as a process with two states and one possible transition from an "alive" state to a "dead" state. In some studies, however, the "alive" state may be partitioned into two or more intermediate (transient) states, each of which corresponding to a particular stage of the illness. In such studies, multi-state models can be used to model the movement of patients among the various states. In these models issues, of interest include the estimation of progression rates, assessing the effects of individual risk factors, survival rates or prognostic forecasting. In this article, we review modelling approaches for multi-state models, and we focus on the estimation of quantities such as the transition probabilities and survival probabilities. Differences between these approaches are discussed, focussing on possible advantages and disadvantages for each method. We also review the existing software currently available to fit the various models and present new software developed in the form of an R library to analyse such models. Different approaches and software are illustrated using data from the Stanford heart transplant study and data from a study on breast cancer conducted in Galicia, Spain.

Disease progression and survival in ALS: first multi-state model approach

Although several prognostic factors have been identified in ALS, there remains some discordance concerning the prognostic significance of the age and clinical form at onset. In order to clarify these findings, we have analysed already known prognostic factors using a multi-state model. Two hundred and twenty-two sporadic ALS patients were followed. A simple unidirectional three-states model was used to summarize clinical course of ALS. States 1 and 2 reflected the progression of neurological impairment and state 3 represented the end of follow-up (tracheotomy or death). Gender, diagnostic delay, body mass index (BMI) and slow vital capacity (SVC) were also recorded. A time-inhomogeneous Markov model with piecewise constant transition intensities was used to estimate the effect of the covariates in each transition. The bulbar form at onset was only correlated with a more rapid clinical progression between state 1 and state 2. In contrast, an advanced age at diagnosis affected only survival from state 2. This methodological approach suggests that these two factors have a different prognostic significance: age at onset is related to patient's survival and the clinical form at onset predicts the progression of motoneuronal impairment in different regions.

tdc.msm: an R library for the analysis of multi-state survival data

The aim of this paper is to present an R library, called tdc.msm, developed to analyze multi-state survival data. In this library, the time-dependent regression model and multi-state models are included as two possible approaches for such data. For the multi-state modelling five different models are considered, allowing the user to choose between Markov and semi-Markov property, as well as to use homogeneous or non-homogeneous models. Specifically, the following multi-state models in continuous time were implemented: Cox Markov model; Cox semi-Markov model; homogeneous Markov model; non-homogeneous piecewise model and non-parametric Markov model. This software can be used to fit multi-state models with one initial state (e.g., illness diagnosis), a finite number of intermediate states, representing, for example, a change of treatment, and one absorbing state corresponding to a terminal event of interest. Graphical output includes survival estimates, transition probabilities estimates and smooth log hazard for continuous covariates.

Disability evolution in multiple sclerosis: how to deal with missing transition times in the Markov model?

Markov modeling of disability progression in multiple sclerosis requires knowledge of all times of transitions from a given level of disability to the next level, but such data are often missing. We address methodological challenges due to partly missing transition times. To estimate the effects of prognostic factors on the risk of transitions between three consecutive disability levels, two methods were used to deal with missing data. Listwise deletion limited the analysis to subjects with complete data. Multiple imputation of missing data revealed that data were missing at random (MAR mechanism) and imputed the missing transition times from the Weibull model. The results were then compared with the full data set with the actual times established through chart review. Multiple imputation estimates were systematically closer to those from the full data set than the listwise deletion estimates.

A population-based assessment of the prognostic value of the CD19 positive lymphocyte count in B-cell chronic lymphocytic leukemia using Cox and Markov models

No population-based study has assessed the prognostic impact on survival of the CD19 positive lymphocyte count, evaluated by immunophenotyping at diagnosis, in B-cell chronic lymphocytic leukemia (B-CLL). Aiming at addressing this issue, we investigated the clinical outcome of a well-defined population of B-CLL patients. Survival of B-CLL patients, diagnosed between 1990 and 1999 and recorded by the Registry of Hematological Malignancies of the Côte d'Or, was analysed applying Cox's regression model to the 237 included cases and to the 195 Binet stage A patients. To assess simultaneously the predictive value of each parameter on the risk of disease progression and on the risk of death, we completed this analysis by applying a three-states homogeneous Markov model to the whole study population. Analysis of the entire population showed that age (p < 0.001), Binet stage (p = 0.008) and CD19 positive lymphocyte count (p = 0.038) were three independent prognostic factors. However, in stage A patients, only progression into a more advanced stage, analysed as a time-dependent variable, and age had a clear impact on survival (p < 0.001 for both). Markov model revealed that an increased CD19 positive lymphocyte count increased the risk of disease progression in stage A patients (p = 0.002) but did not have direct impact on survival of either stage A patients with stable disease or stage B or C patients. An increased CD19 positive lymphocyte count at diagnosis is a marker of an increased risk of disease progression in stage A patients. Thus, it can be a useful tool for the clinical management of these patients.

Estimated progression rates in three United Kingdom hepatitis C cohorts differed according to method of recruitment

OBJECTIVES

To estimate hepatitis C virus (HCV) progression rates between disease stages prior to cirrhosis, using data from liver biopsies in three observational cohorts. To demonstrate how the method of cohort recruitment can influence the estimation of HCV-progression rates.

STUDY DESIGN AND SETTING

Data came from three United Kingdom observational cohorts, assembled from different referral sources. In total, 987 HCV-infected patients with an estimated (or known) date of infection and at least one histologically scored liver biopsy were eligible for inclusion in the analysis. Liver biopsy scores were used to determine the stage of HCV-related liver disease. A three-state continuous time Markov model was used to estimate covariate-specific average probabilities of progression of disease.

RESULTS

After adjusting for confounders, considerably different rates of disease progression were estimated in the three cohorts. For a group of patients with the same demographics, the estimated 20-year probability of progression to cirrhosis was 12% (95% confidence interval CI = 6-22) in a hospital-based cohort, 6% (95% CI = 3-13) in a posttransfusion cohort, and 23% (95% CI = 14-37) in a cohort recruited from a tertiary referral center.

CONCLUSION

Researchers using estimates of disease progression should be aware that the method of cohort recruitment has considerable influence on the progression rates that are derived.

Markov Modelling of Immunological and Virological States in HIV-1 Infected Patients

The purpose of this study was to evaluate the evolution of HIV infected patients and to bring out some significant factors associated with this pathology. The main criteria revealing the State of illness is viral load measurement (VL). However the CD4 lymphocytes also represent an important marker as these reflect the State of the immune reservoir. Many studies have been carried out in this field and different models have been proposed with a view to a better understanding of this disease. Multi State Markov models defined in terms of CD4 counts, or in terms of viral load, have proved to be very useful tools for modelling HIV disease progression. The model we have developed in this study is based on both the CD4 lymphocytes counts and VL. Markov models are characterized by transition intensities. In this paper we explored several structures in succession. First, we used a homogeneous continuous time Markov process with four states defined by crossed values of CD4 and VL in a given patient at a given time. Then, the effect of certain covariates on the infection process was introduced into the model via the transition intensity functions, as with a Cox regression model. Since the hypothesis of homogeneity may be unrealistic in certain cases, we also considered piecewise homogeneous Markov models. Finally, the effects of covariates and time were combined in a piecewise homogeneous model with a covariate. We applied these methods to data from 1313 HIV-infected patients included in the NADIS cohort.

The contribution of dementia to the disablement process and modifying factors

The objective of this study was to examine the impact of dementia on disability progression and mortality, and to analyze the modifying effect of risk factors and extra-individual factors. A sample of 3,403 participants in the PAQUID study was followed for 10 years. Disability was assessed on a 4-grade scale: no disability, disabled only on the Rosow-Breslau scale, disabled on the Rosow-Breslau scale and on instrumental activities of daily living (IADL) scales, and disabled on the Rosow-Breslau, IADL and activities of daily living (ADL) scales. A Markov model was used to estimate the effect of explanatory variables on disability and mortality. Controlling for age, gender, education, place of residence, medical care and informal support, dementia had a strong significant effect on progression to IADL and then to ADL disability. Dementia did not increase the risk of death, once disability was taken into account, except from the lowest disability grade.

Modeling recurrence in colorectal cancer

OBJECTIVE

To assess the role of recurrence in prognosis of colon cancer, we investigated several methodologic issues, including application of classic survival analysis and Markov model.

STUDY DESIGN AND SETTING

The data were recorded by the Registry of Digestive Tumors of Côte d'Or, France, for 874 patients who had been treated by surgery between 1976 and 1984 and followed for up to 11 years. Survival analyses included the Cox proportional hazards model and its two generalizations that allow recurrence to be taken into account as a time-dependent covariate or as a competing outcome. The Markov model was used to analyze simultaneously recurrence and death.

RESULTS

The competing risks approach is not appropriate because censoring is indisputably informative. The Markov model and the Cox model, with recurrence as a time-dependent covariate, provided similar results, demonstrating the impact of age and gender on recurrence and revealing a reduction in the effect of site and stage on mortality.

CONCLUSION

A Markov multistate model seems to give new insights about the course of digestive cancer progression and into the role of recurrence in this process.

The analysis of asthma control under a Markov assumption with use of covariates

In studies of disease states and their relation to evolution, data on the state are usually obtained at in frequent time points during follow-up. Moreover in many applications, there are measured covariates on each individual under study and interest centres on the relationship between these covariates and the disease evolution. We developed a continuous-time Markov model with use of time-dependent covariates and a Markov model with piecewise constant intensities to model asthma evolution. Methods to estimate the effect of covariates on transition intensities, to test the assumption of time homogeneity and to assess goodness-of-fit are proposed. We apply these methods to asthma control. We consider a three-state model and we discuss in detail the analysis of asthma control evolution.

Inference for multi-state models from interval-censored data

Clinical statuses of subjects are often observed at a finite number of visits. This leads to interval-censored observations of times of transition from one state to another. The likelihood can still easily be written in terms of both transition probabilities and transition intensities. In homogeneous Markov models, transition probabilities can be expressed simply in terms of transition intensities, but this is not the case in more general multi-state models. In addition, inference in homogeneous Markov models is easy because these are parametric models. Non-parametric approaches to non-homogeneous Markov models may follow two paths: one is the completely non-parametric approach and can be seen as a generalisation of the Turnbull approach; the other implies a restriction to smooth intensities models. In particular, the penalized likelihood method has been applied to this problem. This paper gives a review of these topics.