The Annals of Applied Statistics

Latent class modeling using matrix covariates with application to identifying early placebo responders based on EEG signals

Bei Jiang, Eva Petkova, Thaddeus Tarpey, and R. Todd Ogden

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Latent class models are widely used to identify unobserved subgroups (i.e., latent classes) based upon one or more manifest variables. The probability of belonging to each subgroup is typically modeled as a function of a set of measured covariates. In this paper, we extend existing latent class models to incorporate matrix covariates. This research is motivated by a randomized placebo-controlled depression clinical trial. One study goal is to identify a subgroup of subjects who experience symptoms improvement early on during antidepressant treatment, which is considered to be an indication of a placebo rather than a true pharmacological response. We want to relate the likelihood of belonging to this subgroup of early responders to baseline electroencephalography (EEG) measurement that takes the form of a matrix. The proposed method is built upon a low-rank Candecomp/Parafac (CP) decomposition of the target coefficient matrix through low-dimensional latent variables, which effectively reduces the model dimensionality. We adopt a Bayesian hierarchical modeling approach to estimate the latent variables, which allows a flexible way to incorporate prior knowledge about covariate effect heterogeneity and offers a data-driven method of regularization. Simulation studies suggest that the proposed method is robust against potentially misspecified rank in the CP decomposition. With the motivating example we show how the proposed method can be applied to extract valuable information from baseline EEG measurements that explains the likelihood of belonging to the early responder subgroup, helping to identify placebo responders and suggesting new targets for the study of placebo response.

Article information

Ann. Appl. Stat., Volume 11, Number 3 (2017), 1513-1536.

Received: February 2016
Revised: March 2017
First available in Project Euclid: 5 October 2017

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Candecomp/Parafac (CP) matrix decomposition Bayesian hierarchical modeling data-driven regularization major depression placebo effect


Jiang, Bei; Petkova, Eva; Tarpey, Thaddeus; Ogden, R. Todd. Latent class modeling using matrix covariates with application to identifying early placebo responders based on EEG signals. Ann. Appl. Stat. 11 (2017), no. 3, 1513--1536. doi:10.1214/17-AOAS1044.

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  • Akaike, H. (1974). A new look at the statistical model identification. IEEE Trans. Automat. Control 19 716–723.
  • Albert, J. H. and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. J. Amer. Statist. Assoc. 88 669–679.
  • Bandeen-Roche, K., Miglioretti, D. L., Zeger, S. L. and Rathouz, P. J. (1997). Latent variable regression for multiple discrete outcomes. J. Amer. Statist. Assoc. 92 1375–1386.
  • Besag, J. and Kooperberg, C. (1995). On conditional and intrinsic autoregressions. Biometrika 82 733–746.
  • Bonate, P. L. and Howard, D. R. (2011). Pharmacokinetics in Drug Development. Advances and Applications 3. Springer, Berlin.
  • Bruder, G. E., Stewart, J. W., Tenke, C. E., McGrath, P. J., Leite, P., Bhattacharya, N. and Quitkin, F. M. (2001). Electroencephalographic and perceptual asymmetry differences between responders and nonresponders to an SSRI antidepressant. Biol. Psychiatry 49 416–425.
  • Bruder, G. E., Sedoruk J. P., Stewart J. W., McGrath, P., Quitkin, F. M. and Tenke, C. E. (2008). EEG alpha measures predict therapeutic response to an SSRI antidepressant: Pre and post treatment findings. Biol. Psychiatry 63 1171.
  • Chib, S. (1995). Marginal likelihood from the Gibbs output. J. Amer. Statist. Assoc. 90 1313–1321.
  • Ciarleglio, A., Petkova, E., Ogden, R. T. and Tarpey, T. (2015). Treatment decisions based on scalar and functional baseline covariates. Biometrics 71 884–894.
  • Clogg, C. C. (1995). Latent class models. In Handbook of Statistical Modeling for the Social and Behavioral Sciences (G. Arminger, C. C. Clogg and M. E. Sobel, eds.) 311–359. Plenum Press, New York.
  • Collins, L. M. and Lanza, S. T. (2013). Latent Class and Latent Transition Analysis: With Applications in The Social, Behavioral, and Health Sciences. Wiley, Hoboken, NJ.
  • Elliott, M. R. (2007). Identifying latent clusters of variability in longitudinal data. Biostatistics 8 756–771.
  • Elliott, M. R., Gallo, J. J., Ten Have, T. R., Bogner, H. R. and Katz, I. R. (2005). Using a Bayesian latent growth curve model to identify trajectories of positive affect and negative events following myocardial infarction. Biostatistics 6 119–143.
  • Garrett, E. S. and Zeger, S. L. (2000). Latent class model diagnosis. Biometrics 56 1055–1067.
  • Geisser, S. and Eddy, W. F. (1979). A predictive approach to model selection. J. Amer. Statist. Assoc. 74 153–160.
  • Gelfand, A. E. and Dey, D. K. (1994). Bayesian model choice: Asymptotics and exact calculations. J. R. Stat. Soc. Ser. B. Stat. Methodol. 56 501–514.
  • Gelman, A., Hwang, J. and Vehtari, A. (2014). Understanding predictive information criteria for Bayesian models. Stat. Comput. 24 997–1016.
  • Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci. 7 457–472.
  • Gormley, I. C. and Murphy, T. B. (2011). Mixture of experts modelling with social science applications. In Mixtures: Estimation and Applications. Wiley Ser. Probab. Stat. 101–121. Wiley, Chichester.
  • Holsboer, F. (2008). How can we realize the promise of personalized antidepressant medicines? Nat. Rev., Neurosci. 9 638–646.
  • Hung, H. and Wang, C.-C. (2013). Matrix variate logistic regression model with application to EEG data. Biostatistics 14 189–202.
  • Jacobs, R. A., Jordan, M. I., Nowlan, S. J. and Hinton, G. E. (1991). Adaptive mixtures of local experts. Neural Comput. 3 79–87.
  • Jiang, B., Elliott, M. R., Sammel, M. D. and Wang, N. (2015). Joint modeling of cross-sectional health outcomes and longitudinal predictors via mixtures of means and variances. Biometrics 71 487–497.
  • Jiang, B., Petkova, E., Tarpey, T. and Ogden, R. T (2017). Supplement to “Latent class modeling using matrix covariates with application to identifying early placebo responders based on EEG signals.” DOI:10.1214/17-AOAS1044SUPP.
  • Jordan, M. I. and Jacobs, R. A. (1994). Hierarchical mixtures of experts and the EM algorithm. Neural Comput. 6 181–214.
  • Joyce, P. R. and Paykel, E. S. (1989). Predictors of drug response in depression. Arch. Gen. Psychiatry 46 89–99.
  • Kamarajan, C., Pandey, A. K., Chorlian, D. B. and Porjesz, B. (2015). The use of current source density as electrophysiological correlates in neuropsychiatric disorders: A review of human studies. Int. J. Psychophysiol. 97 310–322.
  • Khodayari-Rostamabad, A., Reilly, J. P., Hasey, G., Debruin, H. and Maccrimmon, D., (2010). Using pre-treatment EEG data to predict response to SSRI treatment for MDD. In Proceedings of the 2010 32nd Annual International Conference of the IEEE Engineering in Medicine and Biology (EBMC 2010) 6103–6106. IEEE, New York. DOI:10.1109/IEMBS.2010.5627823.
  • Kim, S., Chen, M.-H. and Dey, D. K. (2008). Flexible generalized $t$-link models for binary response data. Biometrika 95 93–106.
  • Kolda, T. G. and Bader, B. W. (2009). Tensor decompositions and applications. SIAM Rev. 51 455–500.
  • Lazarsfeld, P. F. and Henry, N. W. (1968). Latent Structure Analysis. Houghton Mifflin, Boston, MA.
  • Leuchter, A. F., Cook, I. A., Witte, E. A., Morgan, M. and Abrams, M. (2002). Changes in brain function of depressed subjects during treatment with placebo. Am. J. Psychiatr. 159 122–129.
  • Li, B., Kim, M. K. and Altman, N. (2010). On dimension folding of matrix- or array-valued statistical objects. Ann. Statist. 38 1094–1121.
  • Lu, H., Plataniotis, K. N. and Venetsanopoulos, A. N. (2008). MPCA: Multilinear principal component analysis of tensor objects. IEEE Trans. Neural Netw. 19 18–39.
  • MacCutcheon, A. L. (1987). Latent Class Analysis. Sage Publications, Thousand Oaks, CA.
  • Matthews, B. W. (1975). Comparison of the predicted and observed secondary structure of T4 phage lysozyme. Biochim. Biophys. Acta, Protein Struct. 405 442–451.
  • Mumtaz, W., Malik, A. S., Yasin, M. A. M. and Xia, L. (2015). Review on EEG and ERP predictive biomarkers for major depressive disorder. Biomed. Signal Process. Control 22 85–98.
  • Muthén, B. and Brown, H. C. (2009). Estimating drug effects in the presence of placebo response: Causal inference using growth mixture modeling. Stat. Med. 28 3363–3385.
  • Muthén, B. and Shedden, K. (1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics 55 463–469.
  • Neelon, B., O’Malley, A. J. and Normand, S.-L. T. (2011). A Bayesian two-part latent class model for longitudinal medical expenditure data: Assessing the impact of mental health and substance abuse parity. Biometrics 67 280–289.
  • Nunez, P. L. and Srinivasan, R. (2006). Electric Fields of the Brain: The Neurophysics of EEG. Oxford Univ. Press, New York.
  • Patel, M. J., Khalaf, A. and Aizenstein, H. J. (2016). Studying depression using imaging and machine learning methods. NeuroImage: Clinical 10 115–123.
  • Petkova, E., Tarpey, T. and Govindarajulu, U. (2009). Predicting potential placebo effect in drug treated subjects. Int. J. Biostat. 5 Art. 23, 27.
  • Phillips, M. L., Chase, H. W., Sheline, Y. I., Etkin, t Almeida J. R, A., Deckersbach, T. and Trivedi, M. H. (2015). Identifying predictors, moderators, and mediators of antidepressant response in major depressive disorder: Neuroimaging approaches. Am. J. Psychiatr. 172 124–138.
  • Powers, D. M. (2011). Evaluation: From precision, recall and F-measure to ROC, informedness, markedness and correlation. J. Mach. Learn. Technol. 2 37–63.
  • Quitkin, F. M., McGrath, P. J., Rabkin, J. G., Stewart, J. W., Harrison, W., Ross, D. C., Tricamo, E., Fleiss, J., Markowitz, J. and Klein, D. F. (1991). Different types of placebo response in patients receiving antidepressants. Am. J. Psychiatr. 148 197–203.
  • Shen, J. and He, X. (2015). Inference for subgroup analysis with a structured logistic-normal mixture model. J. Amer. Statist. Assoc. 110 303–312.
  • Sing, T., Sander, O. Beerenwinkel, N.and Lengauer, T. (2005). ROCR: visualizing classifier performance in R. Bioinformatics 21 3940–3941.
  • Sonawalla, S. B. and Rosenbaum, J. F. (2002). Placebo response in depression. Dialogues Clin. Neurosci. 4 105–113.
  • Stewart, J. W., Quitkin, F. M., McGrath, P. J., Amsterdam, J., Fava, M., Fawcett, J., Reimherr, F., Rosenbaum, J., Beasley, C. and Roback, P. (1998). Use of pattern analysis to predict differential relapse of remitted patients with major depression during 1 year of treatment with fluoxetine or placebo. Arch. Gen. Psychiatry 55 334–343.
  • Tarpey, T. and Petkova, E. (2010). Latent regression analysis. Stat. Model. 10 133–158.
  • Tarpey, T., Petkova, E. and Ogden, R. T. (2003). Profiling placebo responders by self-consistent partitioning of functional data. J. Amer. Statist. Assoc. 98 850–858.
  • Tarpey, T., Yun, D. and Petkova, E. (2008). Model misspecification: Finite mixture or homogeneous? Stat. Model. 8 199–218.
  • Tenke, C. E., Kayser, J., Manna, C. G., Fekri, S., Kroppmann, C. J., Schaller, J. D., Alschuler, D. M., Stewart, J. W., McGrath, P. J. and Bruder, G. E. (2011). Current source density measures of electroencephalographic alpha predict antidepressant treatment response. Biol. Psychiatry 70 388–394.
  • Vehtari, A. and Ojanen, J. (2012). A survey of Bayesian predictive methods for model assessment, selection and comparison. Stat. Surv. 6 142–228.
  • Wade, E. C. and Iosifescu, D. V. (2016). Using EEG for treatment guidance in major depressive disorder. Biol. Psychiatry: Cognitive Neurosci. Neuroimag. 1 411–422.
  • Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. J. Mach. Learn. Res. 11 3571–3594.
  • White, A. and Murphy, T. B. (2016). Mixed-membership of experts stochastic blockmodel. Netw. Sci. 4 48–80.
  • Zhou, H., Li, L. and Zhu, H. (2013). Tensor regression with applications in neuroimaging data analysis. J. Amer. Statist. Assoc. 108 540–552.

Supplemental materials

  • Supplement to “Latent class modeling using matrix covariates with application to identifying early placebo responders based on EEG signals”. Web Appendices A and B referenced in Sections 2.5 and 4 are available with this paper at the journal website.