Guttman Parameterization of Rating Scales - Revisited

"A reparameterized form of thresholds into their principal components is the method of estimation operationalized in RUMM2030. This notion of principal components is used in the sense of Guttman (1950), who rearranged ordered categories into successive principal components, beginning with the usual linear one. They are analogous to the use of orthogonal polynomials in regression where the independent variable is ordered. The term does NOT refer to the common "principal components analysis" in which a matrix of correlation coefficients is decomposed by analogy to factor analysis."

from, January, 2011.

As previously described in Guttman Parameterization of a Rating Scale, RMT 17:3, 2003, p. 944, Pender Pedler (1987, amended) constructs the Guttman decomposition of the j = 1 to m Rasch-Andrich thresholds of a rating scale with categories 0,m. He defines a series of k = 1,K orthogonal polynomials in j,

T1(j) = 1

T2(j) = 2 ( j - (m+1)/2 )

T3(j) = 3 ( j - (m+1)/2 )² - (m² - 1)/4

T4(j) = 4 ( j - (m+1)/2 )³ - ( j - (m+1)/2 )(3m² - 7)/5

In general, for polynomial k+1 of threshold j,

Tk+1(j) = [(k+1)/k] ( j - (m+1)/2 )Tk(j) - ([(m² - (k-1)²)(k² - 1)]/[4(2k - 1)(2k-3)])Tk-1(j)

So, when {Fj} are the Rasch-Andrich thresholds, and {ck} are the coefficients of the polynomials, estimated from the data by, say, Newton-Raphson iteration,

Fj = sum (ckTk(j)) for k = 1 to K

Note that there is no requirement that all the categories are observed in the data.

Andrich and Luo (2003) use cumulative thresholds, kappa(x), up to threshold x, so that

kappa(x) = - sum(Fj) for j = 1 to x,

= sum ( [ sum (ckTk(j)) for k = 1 to K ] ) for j=1 to x

= sum ((ck / Ak) Uk(x)) for k = 1 to K


Uk(x) = Ak.sum(Tk(j)) for j = 1 to x, and Ak is a constant chosen for convenience.
c1/A1 is termed the central location,
c2/A2 = θ is the dispersion,
c3/A3 = η is the skewness,
c4/A4= ζ is the kurtosis.


U1(x) = -x, with A1 = -1.

U2(x) = x(m-x), with A2 = -1

U3(x) = x(m-x)(2x-m) with A3 = -2

U4(x) = x(m-x)(5x²-5xm+m2+1) with A4 = -5

However, the utility of the orthogonal polynomials is that each higher polynomial adds to the lower ones. Accordingly, we can stop when we have estimated enough of the polynomials to give a useful definition of the threshold values. This is especially helpful when estimating long rating scales based on small datasets. The example in the Figures models the thresholds with four polynomials. It is based on ratings of Olympic Ice-Skating and is estimated by Winsteps.

John M. Linacre

Andrich, D. & Luo, G. (2003). Conditional Pairwise Estimation in the Rasch Model for Ordered Response Categories using Principal Components. Journal of Applied Measurement, 4(3), 205-221.

Guttman, L. (1950). The principal components of scale analysis. In S.A. Stouffer, L. Guttman, E.A. Suchman, P.F. Lazarsfeld, S.A. Star and J.A. Clausen (Eds.), Measurement and Prediction, pp. 312-361. New York: Wiley

Pedler, P.J. (1987) Accounting for psychometric dependence with a class of latent trait models. Ph.D. dissertation. University of Western Australia.

Guttman Parameterization of Rating Scales - Revisited, J.M. Linacre ... Rasch Measurement Transactions, 2011, 24:4, 1303

Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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