zbMATH — the first resource for mathematics

A constrained regression model for an ordinal response with ordinal predictors. (English) Zbl 1430.62166
Summary: A regression model is proposed for the analysis of an ordinal response variable depending on a set of multiple covariates containing ordinal and potentially other variables. The proportional odds model [P. McCullagh, J. R. Stat. Soc., Ser. B 42, 109–142 (1980; Zbl 0483.62056)] is used for the ordinal response, and constrained maximum likelihood estimation is used to account for the ordinality of covariates. Ordinal predictors are coded by dummy variables. The parameters associated with the categories of the ordinal predictor(s) are constrained, enforcing them to be monotonic (isotonic or antitonic). A decision rule is introduced for classifying the ordinal predictors’ monotonicity directions, also providing information whether observations are compatible with both or no monotonicity direction. In addition, a monotonicity test for the parameters of any ordinal predictor is proposed. The monotonicity constrained model is proposed together with five estimation methods and compared to the unconstrained one based on simulations. The model is applied to real data explaining a 10-points Likert scale quality of life self-assessment variable by ordinal and other predictors.
62J12 Generalized linear models (logistic models)
62H12 Estimation in multivariate analysis
Full Text: DOI
[1] Agresti, A.: Analysis of Ordinal Categorical Data, vol. 656. Wiley, Hoboken (2010) · Zbl 1263.62007
[2] Bacci, S.; Bartolucci, F.; Gnaldi, M., A class of multidimensional latent class irt models for ordinal polytomous item responses, Commun. Stat. Theory Methods, 43, 787-800, (2014) · Zbl 1462.62400
[3] Barlow, R.; Brunk, H., The isotonic regression problem and its dual, J. Am. Stat. Assoc., 67, 140-147, (1972) · Zbl 0236.62050
[4] Boes, S.; Winkelmann, R., The effect of income on general life satisfaction and dissatisfaction, Soc. Indic. Res., 95, 111-128, (2010)
[5] Bonferroni, C., Teoria statistica delle classi e calcolo delle probabilita, Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commericiali di Firenze, 8, 3-62, (1936) · Zbl 0016.41103
[6] Brockett, PL, A note on the numerical assignment of scores to ranked categorical data, J. Math. Sociol., 8, 91-110, (1981) · Zbl 0469.62098
[7] Bross, ID, How to use ridit analysis, Biometrics, 14, 18-38, (1958)
[8] Casacci, S.; Pareto, A., Methods for quantifying ordinal variables: a comparative study, Qual. Quant., 49, 1859-1872, (2015)
[9] Cheung, F.; Lucas, RE, Assessing the validity of single-item life satisfaction measures: results from three large samples, Qual. Life Res., 23, 2809-2818, (2014)
[10] Leeuw, J.; Mair, P.; etal., Gifi methods for optimal scaling in R: the package homals, J. Stat. Softw., 31, 1-20, (2009)
[11] Tella, R.; MacCulloch, RJ; Oswald, AJ, The macroeconomics of happiness, Rev. Econ. Stat., 85, 809-827, (2003)
[12] Dykstra, RL; Robertson, T.; etal., An algorithm for isotonic regression for two or more independent variables, Ann. Stat., 10, 708-716, (1982) · Zbl 0485.65099
[13] Harter, HL, Expected values of normal order statistics, Biometrika, 48, 151-165, (1961) · Zbl 0134.15203
[14] Henningsen, A.; Toomet, O., maxlik: A package for maximum likelihood estimation in R, Comput. Stat., 26, 443-458, (2011) · Zbl 1304.65039
[15] Hensler, C.; Stipak, B., Estimating interval scale values for survey item response categories, Am. J. Polit. Sci., 23, 627-649, (1979)
[16] Lange, K.: Numerical Analysis for Statisticians. Springer, Berlin (2010) · Zbl 1258.62003
[17] Linting, M.; Kooij, A., Nonlinear principal components analysis with catpca: a tutorial, J. Personal. Assess., 94, 12-25, (2012)
[18] Mardia, K.V., Kent, J.T., Bibby, J.M.: Multivariate Analysis (Probability and Mathematical Statistics), 1st edn. Academic Press Inc, London (1979) · Zbl 0432.62029
[19] McCullagh, P., Regression models for ordinal data, J. R. Stat. Soc. Ser. B (Methodol.), 42, 109-142, (1980) · Zbl 0483.62056
[20] Miller, R.G.: Simultaneous Statistical Inference. Springer, Berlin (1981) · Zbl 0463.62002
[21] Mori, Y., Kuroda, M., Makino, N.: Nonlinear Principal Component Analysis and Its Applications. Springer, Berlin (2016) · Zbl 1366.62011
[22] Moustaki, I., A latent variable model for ordinal variables, Appl. Psychol. Meas., 24, 211-223, (2000)
[23] Moustaki, I., A general class of latent variable models for ordinal manifest variables with covariate effects on the manifest and latent variables, Br. J. Math. Stat. Psychol., 56, 337-357, (2003)
[24] R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna. https://www.R-project.org/ (2018). Accessed 5 Nov 2018
[25] Rufibach, K., An active set algorithm to estimate parameters in generalized linear models with ordered predictors, Comput. Stat. Data Anal., 54, 1442-1456, (2010) · Zbl 1284.62466
[26] Stevens, SS, On the theory of scales of measurement, Science, 103, 677-680, (1946) · Zbl 1226.91050
[27] Stout, QF, Isotonic regression for multiple independent variables, Algorithmica, 71, 450-470, (2015) · Zbl 1312.62084
[28] Tukey, JW, The future of data analysis, Ann. Math. Stat., 33, 1-67, (1962) · Zbl 0107.36401
[29] Tutz, G., Sequential item response models with an ordered response, Br. J. Math. Stat. Psychol., 43, 39-55, (1990) · Zbl 0718.62263
[30] Tutz, G.; Gertheiss, J., Rating scales as predictors-the old question of scale level and some answers, Psychometrika, 79, 357-376, (2014) · Zbl 1308.62151
[31] Vasdekis, VG; Cagnone, S.; Moustaki, I., A composite likelihood inference in latent variable models for ordinal longitudinal responses, Psychometrika, 77, 425-441, (2012) · Zbl 1272.62136
[32] Yee, T.W.: VGAM: Vector Generalized Linear and Additive Models. https://CRAN.R-project.org/package=VGAM, r package version 1.0-5 (2018). Accessed 5 Nov 2018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.