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Adjusted priors for Bayes factors involving reparameterized order constraints. (English) Zbl 1396.91664

Summary: Many psychological theories that are instantiated as statistical models imply order constraints on the model parameters. To fit and test such restrictions, order constraints of the form \(\theta_i \leq \theta_j\) can be reparameterized with auxiliary parameters \(\eta \in [0, 1]\) to replace the original parameters by \(\theta_i = \eta \cdot \theta_j\). This approach is especially common in multinomial processing tree (MPT) modeling because the reparameterized, less complex model also belongs to the MPT class. Here, we discuss the importance of adjusting the prior distributions for the auxiliary parameters of a reparameterized model. This adjustment is important for computing the Bayes factor, a model selection criterion that measures the evidence in favor of an order constraint by trading off model fit and complexity. We show that uniform priors for the auxiliary parameters result in a Bayes factor that differs from the one that is obtained using a multivariate uniform prior on the order-constrained original parameters. As a remedy, we derive the adjusted priors for the auxiliary parameters of the reparameterized model. The practical relevance of the problem is underscored with a concrete example using the multi-trial pair-clustering model.

MSC:

91E45 Measurement and performance in psychology
62F15 Bayesian inference
62P15 Applications of statistics to psychology

Software:

MPTinR; JAGS; Multitree
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Full Text: DOI arXiv

References:

[1] Batchelder, W. H.; Riefer, D. M., Separation of storage and retrieval factors in free recall of clusterable pairs, Psychological Review, 87, 375-397 (1980)
[2] Batchelder, W. H.; Riefer, D. M., The statistical analysis of a model for storage and retrieval processes in human memory, The British Journal of Mathematical and Statistical Psychology, 39, 129-149 (1986) · Zbl 0622.62107
[3] Batchelder, W. H.; Riefer, D. M., Theoretical and empirical review of multinomial process tree modeling, Psychonomic Bulletin & Review, 6, 57-86 (1999)
[4] Bröder, A.; Herwig, A.; Teipel, S.; Fast, K., Different storage and retrieval deficits in normal ageing and mild cognitive impairment: A multinomial modeling analysis, Psychology and Aging, 23, 353-365 (2008)
[5] Erdfelder, E.; Auer, T.-S.; Hilbig, B. E.; Assfalg, A.; Moshagen, M.; Nadarevic, L., Multinomial processing tree models: A review of the literature, Journal of Psychology, 217, 108-124 (2009)
[6] Fisher, R. A., Inverse probability, Proceedings of the Cambridge Philosophical Society, 26, 528-535 (1930) · JFM 56.1083.05
[7] Goggans, P. M.; Chan, C.-Y.; Knuth, K. H.; Caticha, A.; Center, J. L.; Giffin, A., Assigning priors for ordered and bounded parameters, (AIP conference proceedings, vol. 954 (2007), AIP Publishing), 276-282
[9] Hilbig, B. E.; Moshagen, M., Generalized outcome-based strategy classification: Comparing deterministic and probabilistic choice models, Psychonomic Bulletin & Review, 21, 1431-1443 (2014)
[10] Hoijtink, H., Informative hypotheses: theory and practice for behavioral and social scientists (2011), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL
[11] Iverson, G. J., An essay on inequalities and order-restricted inference, Journal of Mathematical Psychology, 50, 215-219 (2006) · Zbl 1186.91179
[12] Jaynes, E. T., Probability theory: the logic of science (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1045.62001
[13] Jeffreys, H., Theory of probability (1961), Oxford University Press: Oxford University Press New York · Zbl 0116.34904
[14] Kass, R. E.; Raftery, A. E., Bayes factors, Journal of the American Statistical Association, 90, 773-795 (1995) · Zbl 0846.62028
[15] Kellen, D.; Klauer, K. C., Signal detection and threshold modeling of confidence-rating ROCs: A critical test with minimal assumptions, Psychological Review, 122, 542-557 (2015)
[16] Klauer, K. C., Hierarchical multinomial processing tree models: A latent-trait approach, Psychometrika, 75, 70-98 (2010) · Zbl 1272.62126
[17] Klauer, K. C.; Singmann, H.; Kellen, D., Parametric order constraints in multinomial processing tree models: An extension of Knapp and Batchelder (2004), Journal of Mathematical Psychology, 64, 1-7 (2015) · Zbl 1318.62102
[18] Klugkist, I.; Hoijtink, H., The Bayes factor for inequality and about equality constrained models, Computational Statistics & Data Analysis, 51, 6367-6379 (2007) · Zbl 1445.62049
[19] Klugkist, I.; Laudy, O.; Hoijtink, H., Inequality constrained analysis of variance: A Bayesian approach, Psychological Methods, 10, 477 (2005)
[20] Knapp, B. R.; Batchelder, W. H., Representing parametric order constraints in multi-trial applications of multinomial processing tree models, Journal of Mathematical Psychology, 48, 215-229 (2004) · Zbl 1056.91061
[21] Lee, M. D., Bayesian outcome-based strategy classification, Behavior Research Methods, 48, 29-41 (2016)
[22] Lee, M. D.; Wagenmakers, E.-J., Bayesian cognitive modeling: a practical course (2014), Cambridge University Press
[23] Liu, C. C.; Aitkin, M., Bayes factors: Prior sensitivity and model generalizability, Journal of Mathematical Psychology, 52, 362-375 (2008) · Zbl 1152.91771
[24] Moshagen, M., multiTree: A computer program for the analysis of multinomial processing tree models, Behavior Research Methods, 42, 42-54 (2010)
[25] Myung, J. I., The importance of complexity in model selection, Journal of Mathematical Psychology, 44, 190-204 (2000) · Zbl 0946.62094
[26] Myung, J. I.; Karabatsos, G.; Iverson, G. J., A Bayesian approach to testing decision making axioms, Journal of Mathematical Psychology, 49, 205-225 (2005) · Zbl 1104.91016
[27] Myung, I. J.; Pitt, M. A., Applying Occam’s razor in modeling cognition: A Bayesian approach, Psychonomic Bulletin & Review, 4, 79-95 (1997)
[29] Riefer, D. M.; Knapp, B. R.; Batchelder, W. H.; Bamber, D.; Manifold, V., Cognitive psychometrics: Assessing storage and retrieval deficits in special populations with multinomial processing tree models, Psychological Assessment, 14, 184-201 (2002)
[30] Singmann, H.; Kellen, D., MPTinR: Analysis of multinomial processing tree models in R, Behavior Research Methods, 45, 560-575 (2013)
[31] Smith, J. B.; Batchelder, W. H., Beta-MPT: Multinomial processing tree models for addressing individual differences, Journal of Mathematical Psychology, 54, 167-183 (2010) · Zbl 1203.91264
[32] Vandekerckhove, J. S.; Matzke, D.; Wagenmakers, E., Model comparison and the principle of parsimony, (Oxford handbook of computational and mathematical psychology (2015), Oxford University Press: Oxford University Press New York, NY), 300-319
[33] Vanpaemel, W., Measuring model complexity with the prior predictive, (Bengio, Y.; Schuurmans, D.; Lafferty, J. D.; Williams, C. K.I.; Culotta, A., Advances in neural information processing systems 22 (2009), Curran Associates, Inc), 1919-1927
[34] Vanpaemel, W., Prior sensitivity in theory testing: An apologia for the Bayes factor, Journal of Mathematical Psychology, 54, 491-498 (2010) · Zbl 1203.91265
[35] Wagenmakers, E.-J., A practical solution to the pervasive problems ofp values, Psychonomic Bulletin & Review, 14, 779-804 (2007)
[36] Wagenmakers, E.-J.; Verhagen, J.; Ly, A.; Bakker, M.; Lee, M. D.; Matzke, D.; Morey, R. D., A power fallacy, Behavior Research Methods, 47, 913-917 (2015)
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