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 www.rummlab.com.au, 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,

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

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

F_j = sum (c_kT_k(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(F_j) for j = 1 to x,

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

= sum ((c_k / A_k) U_k(x)) for k = 1 to K

where

U_k(x) = A_k.sum(T_k(j)) for j = 1 to x, and A_k 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.

Specifically,

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 Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	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
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	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	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
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
Other 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

Forum

Rasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

The URL of this page is www.rasch.org/rmt/rmt244b.htm

Website: www.rasch.org/rmt/contents.htm