MPT models

Multinomial Processing Tree (MPT) models are cognitive measurement models for categorical data. They describe observed response frequencies from a finite set of response categories (i.e., responses following a multinomial distribution) with a finite number of latent states. Each latent state is reached by particular combinations of cognitive processes; processes that are assumed to take place in an all-or-nothing fashion.

2htm

A graphical depiction of the 2HT MPT model of recognition memory.

MPT models  are task or paradigm specific measurement models and widely used (for reviews see ). Their graphical representations usually depict the item type in the root node (leftmost nodes) and  each edge depicts the occurrence or non-occurrence of a particular cognitive process leading to a specific latent state. The model parameter associated with an edge represents the probability with which a process is executed (i.e., with which a specific latent is reached). The leaf (i.e., rightmost) nodes depict the observable response categories. Each path from root to leaf depicts one possible combination of states that produces an observable response; the probability of a path is the product of the parameters associated with the edges of the path. The sum of all paths leading to the same response category is the predicted probability for that response category. The sum of all path probabilities per tree is 1.

One of the defining characteristic of MPTs is the assumption of discrete latent states precluding modeling of continuous processes. For example, the two-high threshold (2HT) model for recognition memory (depicted in the figure on the left), assumes two distinct memory states: For both old- and new-items a detection state can be reached (with either probability $$D_o$$ or $$D_n$$) in which the correct response is invariably given. In case the true status of an item is not detected (with corresponding probabilities $$1 – D_o$$ or $$1 – D_n$$) an uncertainty state is reached. In the uncertainty state a response is guessed, either “old” with probability $$g$$ or “new” with probability $$1-g$$. It is important to see that the uncertainty state is identical for old and new items: when the true status of the item is unknown the cognitive processes need to be the same. This also helps to make the model parameters identifiable (note that for a simple experiment with one pair of “old” and “new” responses for each old and new items the model parameters are nevertheless not identified).

Besides being one of the easiest way to build models that allow to dissociate latent cognitive processes, the MPT model class is statistically well developed (see , for an overview). For example, advanced model selection techniques based on the minimum description length principle are readily available (thanks to the work by ). This makes MPT models an ideal candidate for modeling categorical data.

In my work I regularly use MPT models to for measuring the contribution of cognitive processes to observed behavior (see here for a relevant list of publications) and plan to do so in the future. I have developed a package for the statistical programming language R for fitting MPT models together with David Kellen, MPTinR , and I am currently working on a successor. Overall, I am really interested in working with MPT models and in advancing the usability and further development of the MPT model class.

 

References

Baumann, C., Singmann, H., Gershman, S. J., & Helversen, B. von. (2020). A linear threshold model for optimal stopping behavior. Proceedings of the National Academy of Sciences, 117(23), 12750–12755. https://doi.org/10.1073/pnas.2002312117
Merkle, E. C., & Wang, T. (2018). Bayesian latent variable models for the analysis of experimental psychology data. Psychonomic Bulletin & Review, 25(1), 256–270. https://doi.org/10.3758/s13423-016-1016-7
Overstall, A. M., & Forster, J. J. (2010). Default Bayesian model determination methods for generalised linear mixed models. Computational Statistics & Data Analysis, 54(12), 3269–3288. https://doi.org/10.1016/j.csda.2010.03.008
Llorente, F., Martino, L., Delgado, D., & Lopez-Santiago, J. (2020). Marginal likelihood computation for model selection and hypothesis testing: an extensive review. ArXiv:2005.08334 [Cs, Stat]. Retrieved from http://arxiv.org/abs/2005.08334
Duersch, P., Lambrecht, M., & Oechssler, J. (2020). Measuring skill and chance in games. European Economic Review, 127, 103472. https://doi.org/10.1016/j.euroecorev.2020.103472
Lee, M. D., & Courey, K. A. (2020). Modeling Optimal Stopping in Changing Environments: a Case Study in Mate Selection. Computational Brain & Behavior. https://doi.org/10.1007/s42113-020-00085-9
Xie, W., Bainbridge, W. A., Inati, S. K., Baker, C. I., & Zaghloul, K. A. (2020). Memorability of words in arbitrary verbal associations modulates memory retrieval in the anterior temporal lobe. Nature Human Behaviour. https://doi.org/10.1038/s41562-020-0901-2
Frigg, R., & Hartmann, S. (2006). Models in Science. Retrieved from https://stanford.library.sydney.edu.au/archives/fall2012/entries/models-science/
Greenland, S., Madure, M., Schlesselman, J. J., Poole, C., & Morgenstern, H. (2020). Standardized Regression Coefficients: A Further Critique and Review of Some Alternatives, 7.
Gelman, A. (2020, June 22). Retraction of racial essentialist article that appeared in Psychological Science « Statistical Modeling, Causal Inference, and Social Science. Retrieved June 24, 2020, from https://statmodeling.stat.columbia.edu/2020/06/22/retraction-of-racial-essentialist-article-that-appeared-in-psychological-science/
Rozeboom, W. W. (1970). 2. The Art of Metascience, or, What Should a Psychological Theory Be? In J. Royce (Ed.), Toward Unification in Psychology (pp. 53–164). Toronto: University of Toronto Press. https://doi.org/10.3138/9781487577506-003
Gneiting, T., & Raftery, A. E. (2007). Strictly Proper Scoring Rules, Prediction, and Estimation. Journal of the American Statistical Association, 102(477), 359–378. https://doi.org/10.1198/016214506000001437
Rouhani, N., Norman, K. A., Niv, Y., & Bornstein, A. M. (2020). Reward prediction errors create event boundaries in memory. Cognition, 203, 104269. https://doi.org/10.1016/j.cognition.2020.104269
Robinson, M. M., Benjamin, A. S., & Irwin, D. E. (2020). Is there a K in capacity? Assessing the structure of visual short-term memory. Cognitive Psychology, 121, 101305. https://doi.org/10.1016/j.cogpsych.2020.101305
Lee, M. D., Criss, A. H., Devezer, B., Donkin, C., Etz, A., Leite, F. P., … Vandekerckhove, J. (2019). Robust Modeling in Cognitive Science. Computational Brain & Behavior, 2(3), 141–153. https://doi.org/10.1007/s42113-019-00029-y
Bailer-Jones, D. (2009). Scientific models in philosophy of science. Pittsburgh, Pa.,: University of Pittsburgh Press.
Suppes, P. (2002). Representation and invariance of scientific structures. Stanford, Calif.: CSLI Publications.
Roy, D. (2003). The Discrete Normal Distribution. Communications in Statistics - Theory and Methods, 32(10), 1871–1883. https://doi.org/10.1081/STA-120023256
Ospina, R., & Ferrari, S. L. P. (2012). A general class of zero-or-one inflated beta regression models. Computational Statistics & Data Analysis, 56(6), 1609–1623. https://doi.org/10.1016/j.csda.2011.10.005
Uygun Tunç, D., & Tunç, M. N. (2020). A Falsificationist Treatment of Auxiliary Hypotheses in Social and Behavioral Sciences: Systematic Replications Framework (preprint). PsyArXiv. https://doi.org/10.31234/osf.io/pdm7y
Murayama, K., Blake, A. B., Kerr, T., & Castel, A. D. (2016). When enough is not enough: Information overload and metacognitive decisions to stop studying information. Journal of Experimental Psychology: Learning, Memory, and Cognition, 42(6), 914–924. https://doi.org/10.1037/xlm0000213
Jefferys, W. H., & Berger, J. O. (1992). Ockham’s Razor and Bayesian Analysis. American Scientist, 80(1), 64–72. Retrieved from https://www.jstor.org/stable/29774559
Maier, S. U., Raja Beharelle, A., Polanía, R., Ruff, C. C., & Hare, T. A. (2020). Dissociable mechanisms govern when and how strongly reward attributes affect decisions. Nature Human Behaviour. https://doi.org/10.1038/s41562-020-0893-y
Nadarajah, S. (2009). An alternative inverse Gaussian distribution. Mathematics and Computers in Simulation, 79(5), 1721–1729. https://doi.org/10.1016/j.matcom.2008.08.013
Barndorff-Nielsen, O., BlÆsild, P., & Halgreen, C. (1978). First hitting time models for the generalized inverse Gaussian distribution. Stochastic Processes and Their Applications, 7(1), 49–54. https://doi.org/10.1016/0304-4149(78)90036-4
Ghitany, M. E., Mazucheli, J., Menezes, A. F. B., & Alqallaf, F. (2019). The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval. Communications in Statistics - Theory and Methods, 48(14), 3423–3438. https://doi.org/10.1080/03610926.2018.1476717
Weichart, E. R., Turner, B. M., & Sederberg, P. B. (2020). A model of dynamic, within-trial conflict resolution for decision making. Psychological Review. https://doi.org/10.1037/rev0000191
Bates, C. J., & Jacobs, R. A. (2020). Efficient data compression in perception and perceptual memory. Psychological Review. https://doi.org/10.1037/rev0000197
Kvam, P. D., & Busemeyer, J. R. (2020). A distributional and dynamic theory of pricing and preference. Psychological Review. https://doi.org/10.1037/rev0000215
Blundell, C., Sanborn, A., & Griffiths, T. L. (2012). Look-Ahead Monte Carlo with People (p. 7). Presented at the Proceedings of the Annual Meeting of the Cognitive Science Society.
Leon-Villagra, P., Otsubo, K., Lucas, C. G., & Buchsbaum, D. (2020). Uncovering Category Representations with Linked MCMC with people. In Proceedings of the Annual Meeting of the Cognitive Science Society (p. 7).
Leon-Villagra, P., Klar, V. S., Sanborn, A. N., & Lucas, C. G. (2019). Exploring the Representation of Linear Functions. In Proceedings of the Annual Meeting of the Cognitive Science Society (p. 7).
Ramlee, F., Sanborn, A. N., & Tang, N. K. Y. (2017). What Sways People’s Judgment of Sleep Quality? A Quantitative Choice-Making Study With Good and Poor Sleepers. Sleep, 40(7). https://doi.org/10.1093/sleep/zsx091
Hsu, A. S., Martin, J. B., Sanborn, A. N., & Griffiths, T. L. (2019). Identifying category representations for complex stimuli using discrete Markov chain Monte Carlo with people. Behavior Research Methods, 51(4), 1706–1716. https://doi.org/10.3758/s13428-019-01201-9
Martin, J. B., Griffiths, T. L., & Sanborn, A. N. (2012). Testing the Efficiency of Markov Chain Monte Carlo With People Using Facial Affect Categories. Cognitive Science, 36(1), 150–162. https://doi.org/10.1111/j.1551-6709.2011.01204.x
Gronau, Q. F., Wagenmakers, E.-J., Heck, D. W., & Matzke, D. (2019). A Simple Method for Comparing Complex Models: Bayesian Model Comparison for Hierarchical Multinomial Processing Tree Models Using Warp-III Bridge Sampling. Psychometrika, 84(1), 261–284. https://doi.org/10.1007/s11336-018-9648-3
Wickelmaier, F., & Zeileis, A. (2018). Using recursive partitioning to account for parameter heterogeneity in multinomial processing tree models. Behavior Research Methods, 50(3), 1217–1233. https://doi.org/10.3758/s13428-017-0937-z
Jacobucci, R., & Grimm, K. J. (2018). Comparison of Frequentist and Bayesian Regularization in Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 25(4), 639–649. https://doi.org/10.1080/10705511.2017.1410822
Raftery, A. E. (1993). Bayesian model selection in structural equation models. In K. A. Bollen & J. S. Long (Eds.), Testing Structural Equation Models (pp. 163–180). Beverly Hills: SAGE Publications.
Lewis, S. M., & Raftery, A. E. (1997). Estimating Bayes Factors via Posterior Simulation With the Laplace-Metropolis Estimator. Journal of the American Statistical Association, 92(438), 648–655. https://doi.org/10.2307/2965712
Mair, P. (2018). Modern psychometrics with R. Cham, Switzerland: Springer.
Rosseel, Y. (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(1), 1–36. https://doi.org/10.18637/jss.v048.i02
Kaplan, D., & Lee, C. (2016). Bayesian Model Averaging Over Directed Acyclic Graphs With Implications for the Predictive Performance of Structural Equation Models. Structural Equation Modeling: A Multidisciplinary Journal, 23(3), 343–353. https://doi.org/10.1080/10705511.2015.1092088
Schoot, R. van de, Verhoeven, M., & Hoijtink, H. (2013). Bayesian evaluation of informative hypotheses in SEM using Mplus: A black bear story. European Journal of Developmental Psychology, 10(1), 81–98. https://doi.org/10.1080/17405629.2012.732719
Lin, L.-C., Huang, P.-H., & Weng, L.-J. (2017). Selecting Path Models in SEM: A Comparison of Model Selection Criteria. Structural Equation Modeling: A Multidisciplinary Journal, 24(6), 855–869. https://doi.org/10.1080/10705511.2017.1363652
Shi, D., Song, H., Liao, X., Terry, R., & Snyder, L. A. (2017). Bayesian SEM for Specification Search Problems in Testing Factorial Invariance. Multivariate Behavioral Research, 52(4), 430–444. https://doi.org/10.1080/00273171.2017.1306432
Matsueda, R. L. (2012). Key advances in the history of structural equation modeling. In Handbook of structural equation modeling (pp. 17–42). New York, NY, US: The Guilford Press.
Bollen, K. A. (2005). Structural Equation Models. In Encyclopedia of Biostatistics. American Cancer Society. https://doi.org/10.1002/0470011815.b2a13089
Tarka, P. (2018). An overview of structural equation modeling: its beginnings, historical development, usefulness and controversies in the social sciences. Quality & Quantity, 52(1), 313–354. https://doi.org/10.1007/s11135-017-0469-8
Sewell, D. K., & Stallman, A. (2020). Modeling the Effect of Speed Emphasis in Probabilistic Category Learning. Computational Brain & Behavior, 3(2), 129–152. https://doi.org/10.1007/s42113-019-00067-6