The Annals of Applied Statistics

Bayesian nonparametric hierarchical modeling for multiple membership data in grouped attendance interventions

Terrance D. Savitsky and Susan M. Paddock

Full-text: Open access


We develop a dependent Dirichlet process (DDP) model for repeated measures multiple membership (MM) data. This data structure arises in studies under which an intervention is delivered to each client through a sequence of elements which overlap with those of other clients on different occasions. Our interest concentrates on study designs for which the overlaps of sequences occur for clients who receive an intervention in a shared or grouped fashion whose memberships may change over multiple treatment events. Our motivating application focuses on evaluation of the effectiveness of a group therapy intervention with treatment delivered through a sequence of cognitive behavioral therapy session blocks, called modules. An open-enrollment protocol permits entry of clients at the beginning of any new module in a manner that may produce unique MM sequences across clients. We begin with a model that composes an addition of client and multiple membership module random effect terms, which are assumed independent. Our MM DDP model relaxes the assumption of conditionally independent client and module random effects by specifying a collection of random distributions for the client effect parameters that are indexed by the unique set of module attendances. We demonstrate how this construction facilitates examining heterogeneity in the relative effectiveness of group therapy modules over repeated measurement occasions.

Article information

Ann. Appl. Stat., Volume 7, Number 2 (2013), 1074-1094.

First available in Project Euclid: 27 June 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayesian hierarchical models conditional autoregressive prior Dirichlet process group therapy mental health substance abuse treatment growth curve


Savitsky, Terrance D.; Paddock, Susan M. Bayesian nonparametric hierarchical modeling for multiple membership data in grouped attendance interventions. Ann. Appl. Stat. 7 (2013), no. 2, 1074--1094. doi:10.1214/12-AOAS620.

Export citation


  • Beck, A., Steer, R. and Brown, G. (1996). Manual for the Beck Depression Inventory-II. Psychological Corporation, San Antonio, TX.
  • Besag, J., York, J. and Mollié, A. (1991). Bayesian image restoration, with two applications in spatial statistics. Ann. Inst. Statist. Math. 43 1–59.
  • Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353–355.
  • Carey, K. (2000). A multilevel modelling approach to analysis of patient costs under managed care. Health Economics 9 435–446.
  • Celeux, G., Forbes, F., Robert, C. P. and Titterington, D. M. (2006). Deviance information criteria for missing data models. Bayesian Anal. 1 651–673 (electronic).
  • Congdon, P. (2005). Bayesian Models for Categorical Data. Wiley, Chichester.
  • Crowe, T. P. and Grenyer, B. F. S. (2008). Is therapist alliance or whole group cohesion more influential in group psychotherapy outcomes? Clin. Psychol. Psychother. 15 239–246.
  • Dahl, D. B., Day, R. and Tsai, J. W. (2008). Distance-based probability distribution on set partitions with applications to protein structure prediction. Technical report.
  • Dawid, A. P. (1981). Some matrix-variate distribution theory: Notational considerations and a Bayesian application. Biometrika 68 265–274.
  • De Iorio, M., Müller, P., Rosner, G. L. and MacEachern, S. N. (2004). An ANOVA model for dependent random measures. J. Amer. Statist. Assoc. 99 205–215.
  • Furukawa, T. (2010). Assessment of mood: Guides for clinicians. Journal of Psychosomatic Reseach 68 581–589.
  • Hill, P. W. and Goldstein, H. (1998). Multilevel modeling of educational data with cross-classification and missing identification for units. Journal of Educational and Behavioral Statistics 23 117–128.
  • Hodges, J. S., Carlin, B. P. and Fan, Q. (2003). On the precision of the conditionally autoregressive prior in spatial models. Biometrics 59 317–322.
  • Hoff, P. D. (2011). Separable covariance arrays via the Tucker product, with applications to multivariate relational data. Bayesian Anal. 6 179–196.
  • Jin, X., Carlin, B. P. and Banerjee, S. (2005). Generalized hierarchical multivariate CAR models for areal data. Biometrics 61 950–961.
  • Jones, G. L., Haran, M., Caffo, B. S. and Neath, R. (2006). Fixed-width output analysis for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 101 1537–1547.
  • Lane, P. W. and Nelder, J. A. (1982). Analysis of covariance and standardization as instances of prediction. Biometrics 38 613–621.
  • Langford, I. H., Leyland, A. H., Rasbash, J. and Goldstein, H. (1999). Multilevel modelling of the geographical distributions of diseases. J. R. Stat. Soc. Ser. C. Appl. Stat. 48 253–268.
  • Morgan-Lopez, A. and Fals-Stewart, W. (2006). Analytic complexities associated with group therapy in substance abuse treatment research: Problems, recommendations, and future directions. Experimental and Clinical Psychopharmacology 14 265–273.
  • Neal, R. M. (2000). Markov chain sampling methods for Dirichlet process mixture models. J. Comput. Graph. Statist. 9 249–265.
  • Paddock, S. M. and Savitsky, T. D. (2013). Bayesian hierarchical semiparametric modeling of longitudinal post-treatment outcomes from open-enrollment therapy groups. J. Roy. Statist. Soc. Ser. A 176 795–808.
  • Paddock, S. M., Hunter, S. B., Watkins, K. E. and McCaffrey, D. F. (2011). Analysis of rolling group therapy data using conditionally autoregressive priors. Ann. Appl. Stat. 5 605–627.
  • R Development Core Team (2011). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN-07-0. Available at
  • Rodríguez, A., Dunson, D. B. and Gelfand, A. E. (2008). The nested Dirichlet process. J. Amer. Statist. Assoc. 103 1131–1144.
  • Ryum, T., Hagen, R., Nordahl, H., Vogel, P. and Stiles, T. (2009). Perceived group climate as a predictor of long-term outcome in a randomized controlled trial of cognitive-behavioural group therapy for patients with comorbid psychiatric disorders. Behavioural and Cognitive Psychotherapy 37 497–510.
  • Savitsky, T. and Paddock, S. (2012). Growcurves: Semiparametric hierarchical Bayesian modeling of longitudinal outcomes. R package version 2.15.2. Available at
  • Savitsky, T. and Vannucci, M. (2010). Spiked Dirichlet process priors for Gaussian process models. J. Probab. Stat. 2010 Art. ID 201489, 14.
  • Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statist. Sinica 4 639–650.
  • Smokowski, P., Rose, S. and Bacallao, M. (2001). Damaging experiences in therapeutic groups: How vulnerable consumers become group casualties. Small Group Research 32 223–251.
  • Teh, Y. W., Jordan, M. I., Beal, M. J. and Blei, D. M. (2006). Hierarchical Dirichlet processes. J. Amer. Statist. Assoc. 101 1566–1581.
  • Watkins, K. E., Hunter, S. B., Hepner, K. A., Paddock, S. M., de la Cruz, E., Zhou, A. J. and Gilmore, J. (2011). An effectiveness trial of group cognitive behavioral therapy for patients with persistent depressive symptoms in substance abuse treatment. Archives of General Psychiatry 68 1–8.