Multi-State Models & Counting Processes for Disease Progression

In chronic diseases, the disease course and associated clinical event histories for the patient population vary widely. To improve understanding of the prognosis of patients and enable health-care providers to assess and manage resources, we wish to jointly model disease progression, mortality and their relation with patient characteristics. In the work carried at MOX we focus on studying how episodes of hospitalisation for disease-related events, obtained from administrative data, can be used as a surrogate for disease status.


Methodologies & Results

A multi-state model is a stochastic process in which subjects occupy one of a set of discrete states at any time, since they can describe several possible events for a single individual, or the dependence between several individuals. Multi-state models are convenient for describing longitudinal data and/or repeated events. The events are the transitions between the states. In medical applications, the states may represent healthy, different severities of disease, or periods in hospital, and transition rates between states may be modelled in terms of covariates. This class of models allows for an extremely flexible approach that can model almost any kind of longitudinal failure time data. This is particularly relevant for modeling different events, which have an event-related dependence, like occurrence of disease changing the risk of death.

Another natural way to approach the same problem is to consider each patient’s hospitalizations as trajectories of a non-homogeneous counting process. The inter-times between hospitalizations are modelled as independent, parametric, not necessarily identically distributed random variables (for example, Weibull distributions with interval dependent parameters).
To catch the heterogeneity of the observed population the presence of K latent groups of patients behaving differently in terms of disease progression may be assumed. This leads to a Weibull mixture model for each inter-hospitalization time and claims for suitable methods of hazard reconstruction and classification. Frailties are used to account for the natural overdispersion observed in the data.



2013 – 2016 (completed) –

Research on Health and Education systems Assessment using administrative Data

FARB – Finanziamento di Ateneo per la Ricerca di Base

2012 – 2015 (completed)

Logo Niguarda     logo_regione_lombardia
Utilization of Regional Health Source Databases for Evaluating Epidemiology, short- and medium-term outcome and process indicators in patients hospitalized for heart failure

Progetto di Ricerca Finalizzata di Regione Lombardia – HFData-RF-2009-1483329


Related Publications & Pre Prints

Gasperoni, F.; Ieva, F.; Paganoni, A.M.; Jackson, C.; Sharples, L. (2019) Evaluating the effect of healthcare providers on the clinical path of Heart Failure patients through a novel semi-Markov multi-state model.
Submitted  [MOX report]

Gasperoni, F., Ieva, F. Paganoni, A.M., Jackson C., Sharples L.D.  (2019, online first 2018) Nonparametric frailty Cox models for hierarchical time-to-event data.  Biostatistics. doi: 10.1093/biostatistics/kxy071

Paulon, G., De Iorio, M, Guglielmi, A., Ieva, F. (2018) Joint modelling of recurrent events and survival: a Bayesian nonparametric approach. Biostatistics. To appear   doi: 10.1093/biostatistics/kxy026  [MOX report]

Grossetti, F., Ieva, F., Paganoni, A.M. (2018, online first May 2017) A Multi-state Approach to Patients Affected by Chronic Heart Failure The Value Added by Administrative Data. Health Care Management of Science, 21(2): 281–291.   doi:10.1007/s10729-017-9400-z

Bottle, A. Ventura, C.M., Dharmarajan, K., Aylin, P., Ieva, F., Paganoni, A.M. (2018, online first 2017) Regional variation in hospitalisation and mortality in heart failure: comparison of England and Lombardy using multistate modelling. Health Care Management of Science, 21(3): 292–304. doi:10.1007/s10729-017-9410-x

Gasperoni, F., Ieva, F., Barbati, G., Scagnetto, A., Iorio, A., Sinagra, G., Di Lenarda, A. (2017) Multi-state modelling of heart failure care path: A population-based investigation from Italy. PLOS ONE 12(6): e0179176. 

Ieva, F., Paganoni, A.M., Pietrabissa, T. (2017, online first 2016) Dynamic clustering of hazard functions: an application to disease progression in chronic heart failure.  Health Care Management of Science, 20: 353–364.

Ieva, F., Jackson, C.H., Sharples, L.D. (2017, online first 2015) Multi-State modelling of repeated hospitalisation and death in patients with Heart Failure: the use of large  administrative databases in clinical epidemiology. Statistical Methods in Medical Research, 26 (3): 1350-1372

Baraldo, S., Ieva, F., Paganoni, A.M., Vitelli, V. (2013) Generalized functional linear models for recurrent events: an application to re-admission processes in heart failure patients. Scandinavian Journal of Statistics. 40(3): 403-416


R Packages

  • discfrail – Cox Models for Time-to-Event Data with Nonparametric Discrete Group-Specific Frailties
  • msmtools – Building Augmented Data to Run Multi-State Models with ‘msm’



Dr. Viviana Viscardi, MD thesis in Mathematical Engineering (2019)
Parametric proportional hazards models with discrete frailty distributions

Dr. Marta Spreafico, MD thesis in Mathematical Engineering (2018)
Statistical modelling of adherence to drug prescription and its effects on survival in heart failure patients

Dr. Michela Agosti, MD thesis in Mathematical Engineering (2018)
Applications of nonparametric frailty models for the analysis of long term survival in heart failure patients

Dr. Luca Mancini, MD thesis in Mathematical Engineering (2015)
Marked Point Process models for the admnissions of Heart Failured patients

Dr. Chiara Maria Ventura, MD thesis in Mathematical Engineering (2014)
Models for predicting readmissions in heart failure patients: a comparison between Lombardia and England

Dr. Francesca Gasperoni, MD thesis in Mathematical Engineering (2014)
Frailty Multi State Models for the analysis of Heart Failure patients

Dr. Teresa Pietrabissa, MD thesis in Mathematical Engineering (2014)
Hazard reconstruction and clustering for better prognosis of disease progression in heart failure

Dr. Stefano Baraldo, MD thesis in Mathematical Engineering (2009)
Modelli statistici per processi di conteggio. Teoria e applicazioni nella telesorveglianza cardiologica.


Invited Talks

  • 2015/11 CERGAS Bocconi, Milano (Italy).
    Title: “Exploiting the use of large Administrative Databases for Modelling Repeated Hospitalizations and Death in Patients with Heart Failure”
  • 2015/09  Cardiovascular Centre of Trieste (Italy).
    Title: “Multi-State Modelling of Repeated Hospitalization and Death in Patients with Heart Failure: how to exploit the use of Administrative Databases in Clinical Care during and after Discharge”
  • 2014/11   Institute for Integrating Statistics in Decision Sciences (I2SDS), School of Businness, George Washington University, Washington DC (US).
    Title: “Statistical Models for Improving Prognosis of Heart Failure: Hazard Reconstruction, Clustering and Prediction of Disease Progression”
  • 2014/02   SISSA – International School for Advanced Studied, Trieste (Italy).
    Title: “Data mining of administrative healthcare databases: how statistics gets insights on chronic heart failures”



  • IBC 2018-29th International Biometric Conference, Barcelona (SP), July 8-13, 2018
    Talk: Application of advanced statistical methods to clinical administrative databases for analyzing patients’ survival adjusting for pharmacological treatments
  • SIS 2018 – 49th Conference of the Italian Society of Statistics, Palermo (IT), June 19-22, 2018
    Talk: Multi-State model with nonparametric discrete frailty
  • SIS 2018 – 49th Conference of the Italian Society of Statistics, Palermo (IT), June 19-22, 2018
    Poster: Investigating the effect of drugs consumption on survival outcome of Heart Failure patients using joint models: a case study based on regional administrative data
  • ISNPS 2018 – 4th Conference of the International Society for Nonparametric Statistics, Salerno (IT), June 11-15, 2018
    Talk: Cox model with nonparametric frailty for modeling hierarchical survival data
  • SISMEC 2017 intermediate conference, Gargnano (IT), September 13-16, 2017
    Poster: Evaluation of Heart Failure Patterns of Care via Multi State Models: a population based approach based on Administrative Data.
  • ISCB 2017-38th Conference of the International Society of Clinical Biostatistics, Vigo (SP), July 9-13, 2017
    Talk: Multi state modelling for heart failure care path: a population-based study using administrative data
  • ISCB 2017-38th Conference of the International Society of Clinical Biostatistics, Vigo (SP), July 9-13, 2017
    Poster: Built-in clustering technique for time-to-event data through a discrete frailty term
  • SIS 2017 Conference of the Italian Society of Statistics, Firenze (IT), June 28-30, 2017
    Talk: Network Analysis of Comorbidity Patterns in Heart Failure Patients using Administrative Data
  • SIS 2017 Conference of the Italian Society of Statistics, Firenze (IT), June 28-30, 2017
    Talk: Nonparametric shared frailty model for classification of survival data
  • ISCB 2016-37th Conference of the International Society of Clinical Biostatistics, Birmingham (UK), August 21-25, 2016
    Talk: Multi-state models for cardiovascular disease
  • SIS 2016-48th Conference of the Italian Society of Statistics, Salerno (IT), June 8-10, 2016
    Talk: Multi-state models for hospitalizations of heart failure patients in Trieste
  • IMA 2016-8th International Conference on Quantitative Modelling in the Management of Health and Social Care, London (UK), March 21-24, 2016
    Talk: Modelling healthcare path of heart failure patients in the realities of Trieste through a multi-state approach
  • L’evoluzione della ricerca epidemiologica basata sugli archivi sanitari: uno sguardo al futuro, Pavia (IT),  November 24-25, 2016
    Poster: On the use of Administrative Data in Healthcare Planning and Research: a case study on Heart Failure patients in Lombardia from 2000 to 2012
  • Aggiornamenti in cardiologia, ecocardiografia clinica ed EcocolorDoppler vascolare, Trieste (IT), February 5-6, 2016
    Talk (invited): Multi-state models: ospedalizzazioni ripetute e PTDA nello scompenso cardiaco
  • ISCB 2014 – 35th Conference of the International Society of Clinical Biostatistics, Wien (Austria), August 24-28, 2014
    Talk: Statistical models for improving prognosis of chronic cardiovascular diseases: hazard reconstruction and clustering of patients affected by heart failure
  • IBC 2014 – 27th International Biometric Conference, Florence (Italy), July 6-11, 2014
    Talk: Study of Disease Progression via Dynamic Clustering of Hazard Functions in non-homogeneous Counting Processes: an application to Heart Failure
  • Doing Research in Healthcare with Administrative Databases, Milan (IT), June 30th, 2014
    Talk: Healthcare assessment from administrative data: multi state models for hospitalizations and time to death in chronic heart failure
  • MRC Conference on Biostatistics, Cambridge (UK), March 24-26, 2014
    Talk: Multi State modelling of hospitalization process and time to death in patients affected by Heart Failure: how to assess the burden of an extensive pathology from administrative data
  • SCo 2013, Milan (Italy), September 9-11, 2013
    Talk: Multi-state modeling of the hospitalization process of patients affected by Chronic Heart Failure
    Poster: Optical Identification of Subjects at High Risk for developing Breast Cancer
  • ISCB 2013 – 34th annual conference of the International Society of Clinical Biostatistics, Munich (Germany), August 25-29, 2013
    Talk: Multi-state models for the joint prediction of time to hospitalizations and death in heart failure patients