- ASA 2021 Statistics and Information Systems for Policy Evaluation
- Edited by Bruno Bertaccini, Luigi Fabbris, Alessandra Petrucci
Random effects regression trees for the analysis of INVALSI data
- Giulia Vannucci
- Anna Gottard
- Leonardo Grilli
- Carla Rampichini
- © 2021 Author(s) |
- CC BY 4.0
- DOI: 10.36253/978-88-5518-304-8.07
Mixed or multilevel models exploit random effects to deal with hierarchical data, where statistical units are clustered in groups and cannot be assumed as independent. Sometimes, the assumption of linear dependence of a response on a set of explanatory variables is not plausible, and model specification becomes a challenging task. Regression trees can be helpful to capture non-linear effects of the predictors. This method was extended to clustered data by modelling the fixed effects with a decision tree while accounting for the random effects with a linear mixed model in a separate step (Hajjem & Larocque, 2011; Sela & Simonoff, 2012). Random effect regression trees are shown to be less sensitive to parametric assumptions and provide improved predictive power compared to linear models with random effects and regression trees without random effects. We propose a new random effect model, called Tree embedded linear mixed model, where the regression function is piecewise-linear, consisting in the sum of a tree component and a linear component. This model can deal with both non-linear and interaction effects and cluster mean dependencies. The proposal is the mixed effect version of the semi-linear regression trees (Vannucci, 2019; Vannucci & Gottard, 2019). Model fitting is obtained by an iterative two-stage estimation procedure, where both the fixed and the random effects are jointly estimated. The proposed model allows a decomposition of the effect of a given predictor within and between clusters. We will show via a simulation study and an application to INVALSI data that these extensions improve the predictive performance of the model in the presence of quasi-linear relationships, avoiding overfitting, and facilitating interpretability.
- Keywords:
- Regression trees,
- Multilevel models,
- Random effects,
- Hierarchical data,
University of Florence, Italy - ORCID: 0000-0003-3569-6274
University of Florence, Italy - ORCID: 0000-0002-8246-4962
University of Florence, Italy - ORCID: 0000-0002-3886-7705
University of Florence, Italy - ORCID: 0000-0002-8519-083X
- Arpino, B., Bacci, S., Grilli, L., Guetto, R., Rampichini, C. (2019) Issues in prior achievement adjustment for value added analysis: an application to Invalsi tests in Italian schools. pp.17-20. In ASA Conference 2019. Statistics for Health and Well-Being. Book of Short Papers
- Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J. (1984). Classification and Regression Trees. Wadsworth, Belmont, (CA).
- Hajjem, A., Bellavance, F., Larocque, D. (2011). Mixed Effects Regression Trees for Clustered Data. Statistics and Probability Letters, 81, pp. 451–459. DOI: 10.1016/j.spl.2010.12.003
- Hajjem, A., Bellavance, F., Larocque, D. (2014). Mixed-effects Random Forest for Clustered Data. Journal of Statistical Computation and Simulation, 84, pp. 1313–1328. DOI: 10.1080/00949655.2012.741599
- Miller, P.J., McArtor, D.B., Lubke, G.H (2017). METBOOST: Exploratory Regression Analysis with Hierarchically Clustered Data. arXiv:1702.03994v1 [stat.ML]
- Sela, R.J., Simonoff, J.S. (2012). RE-EM Trees: A Data Mining Approach for Longitudinal and Clustered Data. Machine Learning, 86, pp. 169–207. DOI: 10.1007/s10994-011-5258-3
- Snijders, T.A.B., Bosker, R.J. (2012). Multilevel Analysis: An Introduction to Basic and Ad- vanced Multilevel Modeling (2nd ed.). SAGE Publications Ltd.
- Vannucci, G. (2019). Interpretable Semilinear Regression Trees. PhD Thesis, FLOrence RE- search repository.
- Wermuth, N., Cox, D.R. (1998). On Association Models defined over Independence Graphs. Bernoulli Society for Mathematical Statistics and Probability, 4, pp. 477–495. DOI: 10.2307/3318662
Chapter Information
Chapter Title
Random effects regression trees for the analysis of INVALSI data
Authors
Giulia Vannucci, Anna Gottard, Leonardo Grilli, Carla Rampichini
Language
English
DOI
10.36253/978-88-5518-304-8.07
Peer Reviewed
Publication Year
2021
Copyright Information
© 2021 Author(s)
Content License
Metadata License
Bibliographic Information
Book Title
ASA 2021 Statistics and Information Systems for Policy Evaluation
Book Subtitle
Book of short papers of the opening conference
Editors
Bruno Bertaccini, Luigi Fabbris, Alessandra Petrucci
Peer Reviewed
Publication Year
2021
Copyright Information
© 2021 Author(s)
Content License
Metadata License
Publisher Name
Firenze University Press
DOI
10.36253/978-88-5518-304-8
eISBN (pdf)
978-88-5518-304-8
eISBN (xml)
978-88-5518-305-5
Series Title
Proceedings e report
Series ISSN
2704-601X
Series E-ISSN
2704-5846