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Penalized Regression with Ordinal Predictors

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  • Jan Gertheiss
  • Gerhard Tutz

Abstract

Ordered categorial predictors are a common case in regression modelling. In contrast to the case of ordinal response variables, ordinal predictors have been largely neglected in the literature. In this paper, existing methods are reviewed and the use of penalized regression techniques is proposed. Based on dummy coding two types of penalization are explicitly developed; the first imposes a difference penalty, the second is a ridge type refitting procedure. Also a Bayesian motivation is provided. The concept is generalized to the case of non-normal outcomes within the framework of generalized linear models by applying penalized likelihood estimation. Simulation studies and real world data serve for illustration and to compare the approaches to methods often seen in practice, namely simple linear regression on the group labels and pure dummy coding. Especially the proposed difference penalty turns out to be highly competitive. Copyright (c) 2009 The Authors. Journal compilation (c) 2009 International Statistical Institute.

Suggested Citation

  • Jan Gertheiss & Gerhard Tutz, 2009. "Penalized Regression with Ordinal Predictors," International Statistical Review, International Statistical Institute, vol. 77(3), pages 345-365, December.
  • Handle: RePEc:bla:istatr:v:77:y:2009:i:3:p:345-365
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    References listed on IDEAS

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    1. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108.
    2. James H. Albert & Siddhartha Chib, 2001. "Sequential Ordinal Modeling with Applications to Survival Data," Biometrics, The International Biometric Society, vol. 57(3), pages 829-836, September.
    3. H. Myoken & Y. Uchida, 1977. "The generalized ridge estimator and improved adjustments for regression parameters," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 24(1), pages 113-124, December.
    4. Donald R. Jensen & Donald E. Ramirez, 2008. "Anomalies in the Foundations of Ridge Regression," International Statistical Review, International Statistical Institute, vol. 76(1), pages 89-105, April.
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    Cited by:

    1. Faisal Zahid & Gerhard Tutz, 2013. "Multinomial logit models with implicit variable selection," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 7(4), pages 393-416, December.
    2. Hess, Wolfgang & Persson, Maria & Rubenbauer, Stephanie & Gertheiss, Jan, 2013. "Using Lasso-Type Penalties to Model Time-Varying Covariate Effects in Panel Data Regressions - A Novel Approach Illustrated by the 'Death of Distance' in International Trade," Working Papers 2013:5, Lund University, Department of Economics.
    3. Gerhard Tutz & Jan Gertheiss, 2014. "Rating Scales as Predictors—The Old Question of Scale Level and Some Answers," Psychometrika, Springer;The Psychometric Society, vol. 79(3), pages 357-376, July.
    4. Stephanie Möst & Wolfgang Pößnecker & Gerhard Tutz, 2016. "Variable selection for discrete competing risks models," Quality & Quantity: International Journal of Methodology, Springer, vol. 50(4), pages 1589-1610, July.
    5. Faisal Maqbool Zahid & Gerhard Tutz, 2013. "Proportional Odds Models with High-Dimensional Data Structure," International Statistical Review, International Statistical Institute, vol. 81(3), pages 388-406, December.

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