Mixed Models

Mixed models (aka multilevel or hierarchical models) are statistical models containing both fixed- and random-effects terms. They are useful when multiple measurements exist for each unit of observation (e.g., participant or item), or for hierarchical data. Linear mixed models (LMMs) are used for normally distributed dependent variables, generalized linear mixed models (GLMMs) are applicable to other distributions (e.g., binomial data).

lmm_exampleMixed models extend ordinary linear, ANOVA, or generalized linear models and have only relatively recently been introduced to psychology . They allow for the modeling of dependency structures in the data that are difficult to handle with ordinary linear or generalized linear models such as repeated measurements by the same individual  (e.g., allowing repeated-measures logistic regressions) or hierarchical data structures such as participants blocked in classes or sessions. Furthermore, they enable the modeling  of multiple independent random effects simultaneously, such as participant and item effects, unifying so-called F1 and F2 analysis. Together with David Kellen, I have written an introductory chapter to mixed models. Further information on how to specify mixed models for experimental designs are given by Barr and colleagues and arguments for considering both participants and items simultaneously are given by Judd et al.  .

The reference package for estimating mixed models in the statistical programming language R is lme4. A technical introduction to the models estimated by lme4 can be found here. I have developed the afex package which complements the functionality of lme4, for example by obtaining *p*-values for effects in mixed models.

 

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