Journal of Biostatistics and Epidemiology 2015. 1(3-4):70-79.

Assessing misspecification of individual homogeneity assumption in multi-state models based on asymptotic theory
Ali Zare, Mahmood Mahmoodi, Kazem Mohammad, Hojjat Zeraati, Mostafa Hosseini, Kourosh Holakouie-Naieni


Background & Aim: Multi-state models can help better understand the process of chronic diseases such as cancers.  These models  are influenced  by assumptions  like individual  homogeneity.  This study aimed to investigate the effect of lack of individual homogeneity  assumption  in multi-state models.

Methods & Materials: To investigate the effect of lack of individual homogeneity assumption in multi-state  models,  tracking  model  as well as frailty  factor  with gamma  distribution  were used. Accordingly,  without  any  simulation  and  only  based  on  asymptotic  theory,  the  bias  of  mean transition rate which is among the basic parameters of the multi-state models was studied.

Results: Analysis of the effect of individual homogeneity assumption misspecification revealed that for  different  number  of  follow-ups  as  well  as  censoring  time,  the  mean  transition  rate  and  its variance  were underestimated.  In addition,  if there is a lot of heterogeneity  in reality and if the individual  homogeneous  multi-state  model  is fitted, a significant  bias will exist in the estimated mean transition rate and its variance. The results of this study also showed that the intensity of bias increases with an increase in the degree of heterogeneity.  But with an increase in the number of follow-ups, the intensity of bias decreases, to some extent.

Conclusion: Disregarding individual homogeneity assumption in a heterogeneous population causes bias in the estimation of multi-state model parameters and with an increase in the degree of heterogeneity, the intensity of bias will increase too.


asymptotic theory; frailty,individual homogeneity;gamma distribution; misspecification;multi-state model

Full Text:



Zare A, Mahmoodi M, Mohammad K, Zeraati H, Hosseini M, Naieni KH. Survival analysis of patients with gastric cancer undergoing surgery at the iran cancer institute: a method based on multi-state models. Asian Pac J Cancer Prev 2013;14(11): 6369-73.

Jackson CH. Multi-state models for panel data: Themsm package for R. Journal of Statistical Software 2011; 38(8): 1-28.

Zare A, Mahmoodi M, Mohaammad K,Zeraati H, Hoseini M, Naieni KH. Assessing Markov and time homogeneity assumptions in multi-state models: application in patients with gastric cancer undergoing surgery in the Iran cancer institute. Asian Pac J Cancer Prev 2014; 15(1): 441-7.

Titman AC, Sharples LD. Model diagnostics for multi-state models. Stat Methods Med Res 2010; 19(6): 621-51.

Meira-Machado L, de Una-Alvarez J, Cadarso-Suarez C, Andersen PK. Multi-state models for the analysis of time-to-event data. Stat Methods Med Res 2009; 18(2):195-222.

Andersen PK, Keiding N. Multi-state models for event history analysis. Stat Methods Med Res 2002; 11(2): 91-115.

Foucher Y, Giral M, Soulillou JP, Daures JP. A flexible semi-Markov model for interval-censored data and goodness-of-fit testing. Stat Methods Med Res 2010; 19(2):127-45.

Chen PL, Tien HC. Semi-Markov models for multistate data analysis with periodic observations. Communications in Statistics - Theory and Methods 2004; 33(3): 475-86.

Hougaard P. Multi-state models: a review.Lifetime Data Anal 1999; 5(3): 239-64.

Titman AC, Sharples LD. Semi-Markov models with phase-type sojourn distributions. Biometrics 2010; 66(3): 742-52.

Sharples LD, Taylor GI, Faddy M. A piecewise-homogeneous Markov chain process of lung transplantation. J Epidemiol Biostat 2001; 6(4): 349-55.

Vaupel JW, Manton KG, Stallard E. The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 1979; 16(3): 439-54.

Hougaard P. Modelling Heterogeneity in survival data. Journal of Applied Probability 1991; 28(3): 695-701.

Klein JP, Moeschberger ML. Survival analysis: techniques for censored and truncated data. New York, NY: Springer Science & Business Media; 2003.

Kalbfleisch J, Prentice RL. The Statistical Analysis of Failure Time Data. 2nd ed. Hoboken, NJ: John Wiley & Sons; 2011.

Wienke A. Frailty models in survival analysis. Boca Raton, F: CRC Press; 2010.

Kleinbaum D, Klein M. Survival analysis: a self-learning text. 3rd ed. New York, NY: Springer Science & Business Media; 2011.

Cook RJ, Yi GY, Lee KA, Gladman DD. A conditional Markov model for clustered progressive multistate processes under incomplete observation. Biometrics 2004;60(2): 436-43.

Cook RJ. A mixed model for two-state Markov processes under panel observation. Biometrics 1999; 55(3): 915-20.

Satten GA. Estimating the extent of tracking in interval-censored Chain-Of-Events data. Biometrics 1999; 55(4): 1228-31.

Chen HH, Duffy SW, Tabar L. A mover- stayer mixture of Markov chain models for the assessment of dedifferentiation and tumour progression in breast cancer. Journal of Applied Statistics 2010; 24(3): 265-78.

Cook RJ, Kalbfleisch JD, Yi GY. A generalized mover-stayer model for panel data. Biostatistics 2002; 3(3): 407-20.

Titman AC. Flexible nonhomogeneous Markov models for panel observed data. Biometrics 2011; 67(3): 780-7.

Hsieh HJ, Chen TH, Chang SH. Assessing chronic disease progression using non- homogeneous exponential regression Markov models: an illustration using a selective breast cancer screening in Taiwan. Stat Med 2002; 21(22): 3369-82.

Pérez-Ocón R, Ruiz-Castro JE, Gámiz-Pérez ML. Markov models with lognormal transition rates in the analysis of survival times. Test 2000; 9(2): 353-70.

Pérez-Ocón R, Ruiz-Castro JE, Gámiz-Pérez ML. Non-homogeneous Markov models in the analysis of survival after breast cancer. Journal of the Royal Statistical Society: Series C (Applied Statistics) 2001; 50(1):111-24.

Yen AM, Chen TH. Mixture multi-state markov regression model. Journal of Applied Statistics 2007; 34(1): 11-21

White H. Maximum likelihood estimation of misspecified models. Econometrica, 1982;50(1): 1-25.

Cox DR. Tests of separate families of hypotheses. Proc Fourth Berkeley Symp on Math Statist and Prob 1961; 1: 105-23.

Hwang W, Brookmeyer R. Design of panel studies for disease progression with multiple stages. Lifetime Data Analysis 2003; 9(3):261-74.

de Stavola BL. Sampling designs for short panel data. Econometrica 1986; 54(2): 415-24.

Lehmann EL. Elements of large-sample theory. New York, NY: Springer Science & Business Media; 1999.

Jackson CH, Sharples LD, Thompson SG, Duffy SW, Couto E. Multistate Markov models for disease progression with classification error. The Statistician 2003;52: 193-209.

Jackson CH, Sharples LD. Hidden Markov models for the onset and progression of bronchiolitis obliterans syndrome in lung transplant recipients. Stat Med 2002; 21(1):113-28.

Ng ET, Cook RJ. Modeling two-state disease processes with random effects. Lifetime Data Anal 1997; 3(4): 315-35.

Bijwaard G. Multistate event history analysis with frailty. Demographic Research 2014; 30(58): 1591-620.

Putter H, van Houwelingen HC. Frailties in multi-state models: Are they identifiable? Do we need them? Stat Methods Med Res 2011.


  • There are currently no refbacks.

Creative Commons Attribution-NonCommercial 3.0

This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License which allows users to read, copy, distribute and make derivative works for non-commercial purposes from the material, as long as the author of the original work is cited properly.