Annals of Applied Statistics

Longitudinal Mixed Membership trajectory models for disability survey data

Daniel Manrique-Vallier

Full-text: Open access


We develop methods for analyzing discrete multivariate longitudinal data and apply them to functional disability data on the U.S. elderly population from the National Long Term Care Survey (NLTCS), 1982–2004. Our models build on a Mixed Membership framework, in which individuals are allowed multiple membership on a set of extreme profiles characterized by time-dependent trajectories of progression into disability. We also develop an extension that allows us to incorporate birth-cohort effects, in order to assess inter-generational changes. Applying these methods, we find that most individuals follow trajectories that imply a late onset of disability, and that younger cohorts tend to develop disabilities at a later stage in life compared to their elders.

Article information

Ann. Appl. Stat., Volume 8, Number 4 (2014), 2268-2291.

First available in Project Euclid: 19 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

NLTCS Mixed Membership trajectories multivariate analysis MCMC cohort analysis


Manrique-Vallier, Daniel. Longitudinal Mixed Membership trajectory models for disability survey data. Ann. Appl. Stat. 8 (2014), no. 4, 2268--2291. doi:10.1214/14-AOAS769.

Export citation


  • Airoldi, E. M., Fienberg, S. E., Joutard, C. and Love, T. M. (2007). Discovering latent patterns with hierarchical Bayesian mixed-membership models. In Data Mining Patterns: New Methods and Applications (P. Poncelet, F. Masseglia and M. Teisseire, eds.) 240–275. Idea Group Inc., Hershey, PA.
  • Airoldi, E. M., Blei, D. M., Fienberg, S. E. and Xing, E. P. (2008). Mixed membership stochastic blockmodels. J. Mach. Learn. Res. 9 1981–2014.
  • Airoldi, E. M., Erosheva, E. A., Fienberg, S. E., Joutard, C., Love, T. and Shringarpure, S. (2010). Reconceptualizing the classification of PNAS articles. Proc. Natl. Acad. Sci. USA 107 20899–20904.
  • Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory 267–281. Akademiai Kiado, Budapest.
  • Bertolet, M. (2008). To weight or not to weight? Incorporating sampling designs into model-based analyses. Ph.D. thesis, Dept. Statistics, Carnegie Mellon Univ., Pittsburgh, PA.
  • Bhattacharya, A. and Dunson, D. B. (2012). Simplex factor models for multivariate unordered categorical data. J. Amer. Statist. Assoc. 107 362–377.
  • Clark, R. F. (1998). An introduction to the National Long-Term Care Survey. Office of Disability, Aging, and Long-Term Care Policy with the U.S. Dept. Health and Human Services. Available at
  • Connor, J. T. (2006). Multivariate mixture models to describe longitudinal patterns of frailty in American seniors. Ph.D. thesis, Dept. Statistics & H. John Heinz III School of Public Policy & Management, Carnegie Mellon Univ., Pittsburgh, PA.
  • Connor, J. T., Fienberg, S. E., Erosheva, E. A. and White, T. (2006). Towards a restructuring of the National Long Term Care Survey: A longitudinal perspective. Prepared for presentation at an Expert Panel Meeting on the National Long Term Care Survey, Committee on National Statistics, National Research Council.
  • Corder, L. S. and Manton, K. G. (1991). National surveys and the health and functioning of the elderly: The effects of design and content. J. Amer. Statist. Assoc. 86 513–525.
  • Erosheva, E. A. (2002). Grade of membership and latent structures with application to disability survey data. Ph.D. thesis, Dept. Statistics, Carnegie Mellon Univ., Pittsburgh, PA.
  • Erosheva, E. A. and Fienberg, S. E. (2005). Bayesian mixed membership models for soft clustering and classification. In Classification—The Ubiquitous Challenge (C. Weihs and W. Gaul, eds.) 11–26. Springer, Berlin.
  • Erosheva, E. A., Fienberg, S. E. and Joutard, C. (2007). Describing disability through individual-level mixture models for multivariate binary data. Ann. Appl. Stat. 1 502–537.
  • Erosheva, E. A., Fienberg, S. E. and Lafferty, J. D. (2004). Mixed-membership models of scientific publications. Proc. Natl. Acad. Sci. USA 101 5220–5227.
  • Ferrucci, L., Guralnik, J. M., Simonsick, E., Salive, M. E., Corti, C. and Langlois, J. (1996). Progressive versus catastrophic disability: A longitudinal view of the disablement process. The Journals of Gerontology: Series A 51 M123.
  • Goodman, L. A. (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika 61 215–231.
  • Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed. Springer, New York.
  • Jasra, A., Holmes, C. C. and Stephens, D. A. (2005). Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling. Statist. Sci. 20 50–67.
  • Kreuter, F. and Muthén, B. (2008). Analyzing criminal trajectory profiles: Bridging multilevel and group-based approaches using growth mixture modeling. J. Quant. Criminol. 24 1–31.
  • Kurland, B. F. and Heagerty, P. J. (2005). Directly parameterized regression conditioning on being alive: Analysis of longitudinal data truncated by deaths. Biostatistics 6 241–258.
  • Kurland, B. F., Johnson, L. L., Egleston, B. L. and Diehr, P. H. (2009). Longitudinal data with follow-up truncated by death: Match the analysis method to research aims. Statist. Sci. 24 211–222.
  • Manrique-Vallier, D. (2014). Mixed membership trajectory models. In Handbook on Mixed Membership Models (E. M. Airoldi, D. M. Blei, E. A. Erosheva and S. E. Fienberg, eds.). Chapman & Hall/CRC, London.
  • Manrique-Vallier, D. (2014). Supplement to “Longitudinal Mixed Membership trajectory models for disability survey data.” DOI:10.1214/14-AOAS769SUPP.
  • Manrique-Vallier, D. and Fienberg, S. E. (2008). Population size estimation using individual level mixture models. Biom. J. 50 1051–1063.
  • Manton, K. G. (2008). Recent declines in chronic disability in the elderly U.S. population: Risk factors and future dynamics. Annu. Rev. Public Health 29 91–113.
  • Manton, K. G., Corder, L. and Stallard, E. (1997). Chronic disability trends in elderly United States populations: 1982–1994. Proc. Natl. Acad. Sci. USA 94 2593–2598.
  • Manton, K. G., Gu, X. L. and Lamb, V. L. (2006). Change in chronic disability from 1982 to 2004/2005 as measured by long-term changes in function and health in the US elderly population. Proc. Natl. Acad. Sci. USA 103 18374.
  • Manton, K. G., Lamb, V. L. and Gu, X. (2007). Medicare cost effects of recent US disability trends in the elderly future implications. J. Aging Health 19 359–381.
  • Manton, K. G., Stallard, E. and Woodbury, M. A. (1991). A multivariate event history model based upon fuzzy states: Estimation from longitudinal surveys with informative nonresponse. J. Off. Stat. 7 261–293.
  • Nagin, D. S. (1999). Analyzing developmental trajectories: A semiparametric, group-based approach. Psychol. Methods 4 139–157.
  • Raftery, A. E., Newton, M. A., Satagopan, J. M. and Krivitsky, P. N. (2007). Estimating the integrated likelihood via posterior simulation using the harmonic mean identity. In Bayesian Statistics (J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith and M. West, eds.). Oxford Sci. Publ. 8 371–416. Oxford Univ. Press, Oxford.
  • Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461–464.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 583–639.
  • Stallard, E. (2005). Trajectories of morbidity, disability, and mortality among the US elderly population: Evidence from the 1984–1999 NLTCS. N. Amer. Actuar. J. 11 16–53.
  • Tanner, M. A. (1996). Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions, 3rd ed. Springer, New York.
  • White, T. A. and Erosheva, E. A. (2013). Using group-based latent class transition models to analyze chronic disability data from the National Long-Term Care Survey 1984–2004. Stat. Med. 32 3569–3589.
  • Woodbury, M. A., Clive, J. and Garson, A. Jr. (1978). Mathematical typology: A grade of membership technique for obtaining disease definition. Comput. Biomed. Res. 11 277–98.
  • Xing, E. P., Fu, W. and Song, L. (2010). A state-space mixed membership blockmodel for dynamic network tomography. Ann. Appl. Stat. 4 535–566.

Supplemental materials

  • Supplementary material: Supplement to “Longitudinal Mixed Membership trajectory models for disability survey data”. Estimation using TGoM models with piecewise constant trajectories and tables with posterior estimates for all the fitted models.