Residual diagnostics for interpreting CUB models
AbstractCUB models represent a new approach for the analysis of categorical ordinal data. The relevant domain of study is the specification and estimation of the behaviour of respondents when faced to ratings by analysing the relationship among ordinal scores and observed covariates. The increasing use of such models suggests to delve into the issue of appropriate residuals to be used for diagnostic purposes. In fact, the discreteness of the response variable discourages the use of standard regression paradigms. In this context, we propose the extension and implementation of a specific graphical methodology, known as binned residual plots, in order to check the adequacy of fitted CUB models and/or infer about improvements of the maintained model. Such proposals have been exemplified through the analysis of real data.
A. AGRESTI, (2010), Analysis of ordinal categorical data, 2nd edition, J. Wiley & Sons, New Jersey.
R. ARBORETTI, S. BONNINI, M. IANNARIO, F. SOLMI, (2011), Permutation test approach for the analysis of rating data, Proceeding of SIS Meeting, Bologna.
A. C. ATKINSON, (1985), Plots, transformations, and regressions: an introduction to graphical methods of diagnostic regression analysis, Clarendon Press, Oxford.
D. J. BARTHOLOMEW, P. TZAMOURANI, (1999), The goodness of fit of latent trait models in attitude measurement, “Sociological Methods and Research”, 27:525-546.
D. A. BELSEY, E. KUH, R. E. WELSCH, (1980), Regression diagnostics: identifiying influential data and sources of collinearity, J. Wiley & Sons, New York.
R. BENDER, A. BENNER, (2000), Calculating ordinal regression models in SAS and S-Plus, “Biometrical Journal”, 42:677-699.
S. BONNINI, D. PICCOLO, L. SALMASO, F. SOLMI, (2011), Permutation inference for a class of mixture models, “Communications in Statistics. Theory and Methods”, forthcoming.
A. C. CAMERON, F. A. G. WINDMEIJER, (1997), An R-squared measure of goodness of fit for some common nonlinear regression models, “Journal of Econometrics”, 77:329-342.
W. S. CLEVELAND, (1979), Robust locally weighted regression and smoothing scatterplots, “Journal of the American Statistical Association”, 74:829-836.
R. D. COOK, S. WEISBERG, (1994), An introduction to regression graphics, J. Wiley & Sons, New York.
C. DANIEL, F. S. WOOD, (1971), Fitting equation to data, J. Wiley & Sons, New York.
A. D’ELIA, D. PICCOLO, (2005), A mixture model for preference data analysis, “Computational Statistics & Data Analysis”, 49:917-934.
F. DI IORIO, D. PICCOLO, (2009), Generalized residuals in CUB models: definition and applications, “Quaderni di Statistica”, 11:73-88.
J. FOX, (1991), Regression diagnostics: an introduction, Sage, Newbury Park, CA.
J. FOX, (1997), Applied regression analysis, linear models, and related methods, Sage, Thousand Oaks.
A. GELMAN, J. HILL, (2007), Data Analysis using Regression and Multilevel/Hierarchical Models, Cambridge University Press, New York.
T. HASTIE, (1987), A closer look at deviance, “The American Statistician”, 41:16-20.
M. IANNARIO, (2009), Fitting measures for ordinal data models, “Quaderni di Statistica”, 11:39-72.
M. IANNARIO, (2010), On the identifiability of a mixture model for ordinal data, “METRON”, LXVIII: 87-94.
M. IANNARIO, (2012), Modelling shelter choices in a class of mixture models for ordinal responses, “Statistical Methods and Applications”, 21, 1-22.
M. IANNARIO, D. PICCOLO, (2010), Statistical modelling of subjective survival probabilities, “GENUS”, LXVI:17-42.
M. IANNARIO, D. PICCOLO, (2012), CUB models: Statistical methods and empirical evidence, in: R.S.
KENETT, S. SALINI (eds.) “Modern Analysis of Customer Surveys: with application using R”, J. Wiley & Sons, New York, 231-258.
K. JÖRESKOG, I. MOUSTAKI, (2001), Factor analysis of ordinal variables: A comparison of three approaches, “Multivariate Behavioral Research”, 36:347-387.
J. M. LANWEHER, D. PREGIBON, A. C. SHOEMAKER, (1984), Graphical methods for assessing logistic regression models, “Journal of the American Statistical Association”, 79:61-83.
G. LETI, (1979), Distanze e indici statistici, La Goliardica, Roma.
B. G. LINDSAY, K. ROEDER, (1992), Residual diagnostics for mixture models, “Journal of the American Statistical Association”, 87:785-792.
I. LIU, B. MUKHERJEE, T. SUESSE, D. SPARROW, S. K. PARK, (2009), Graphical diagnostics to check model misspecification for the proportional odds regression model. “Statistics in Medicine”, 28:412-429.
P. MCCULLAGH, (1980), Regression models for ordinal data (with discussion), “Journal of the Royal Statistical Society, Series B”, 42:109-142.
P. MCCULLAGH, J. A. NELDER, (1998), Generalized linear models, 2nd edition, Chapman & Hall, London.
D. PICCOLO, (2003), On the moments of a mixture of uniform and shifted binomial random variables, “Quaderni di Statistica”, 5:85-104.
D. PICCOLO, (2006), Observed information matrix for MUB models, “Quaderni di Statistica”, 8:33-78.
D. PREGIBON, (1981), Logistic regression diagnostics, “The Annals of Statistics”, 9:705-724.
H. PURSCHA, (1994), Partial residuals in cumulative regression models for ordinal data, “Statistical Papers”, 35:273-284.
S. WEISBERG, (1980), Applied linear regression, J. Wiley & Sons, New York.
J. S. SIMONOFF, (2003), Analyzing categorical data, Springer, New York.