Guttman Parameterization of a Rating Scale

"A reparameterised form of thresholds into their principal components is the method of estimation operationalised in RUMM2020. 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."
Excerpted from www.rummlab.com.au

See also: Guttman Parameterization of a Rating Scale - Revisited, RMT 24:4, 2011, p. 1303.

A convenient logit-linear expression of a typical form of the Rasch polytomous model is
loge(Pnix/Pni(x-1)) = Bn - Di - Fx where Fx is the centralized (Andrich, Rasch) threshold (also called step calibration) corresponding to the point on the latent variable where categories x-1 and x are predicted to be equally likely to be observed. Categories are numbered from 0 to m.

Pedler's (1987, amended) coefficients for an orthogonal-polynomial version of the rating-scale thresholds are:
T1(x) = 1 which requires at least two categories in the rating scale
T2(x) = 2( x - (m+1)/2 ) which requires at least three categories in the rating scale
T3(x) = 3( x - (m+1)/2 )² - (m² - 1)/4 which requires at least four categories in the rating scale
T4(x) = 4( x - (m+1)/2 )³ - ( x - (m+1)/2 )(3m² - 7)/5 which requires at least five categories in the rating scale

Higher-order coefficients can be obtained from:
Tk+1(x) = [(k+1)/k] ( x - (m+1)/2 )Tk(x) - ([(m² - (k-1)²)(k² - 1)]/[4(2k - 1)(2k-3)])Tk-1(x)

In accordance with Andrich and Luo (2003), these modify the Guttman parameters, θ, η, ζ:
Di = the item difficulty
Fx =
T1(x)*0 where 0 is the rating scale central location relative to the item difficulty
+ T2(x)*θ where θ is the rating scale dispersion or unit
+ T3(x)*2*η where η is the skewness
+ T4(x)*5*ζ where ζ is the kurtosis
+ higher-order terms

Guttman Principal Component Multipliers
m x θ η ζ m x θ η ζ
2 1 -1     8 1 -7 42 -210
2 1     2 -5 6 150
3 1 -2 2   3 -3 -18 210
2 0 -4   4 -1 -30 90
3 2 2   5 1 -30 -90
4 1 -3 6 -6 6 3 -18 -210
2 -1 -6 18 7 5 6 -150
3 1 -6 -18 8 7 42 210
4 3 6 6 9 1 -8 56 -336
5 1 -4 12 -24 2 -6 14 168
2 -2 -6 48 3 -4 -16 312
3 0 -12 0 4 -2 -34 216
4 2 -6 -48 5 0 -40 0
5 4 12 24 6 2 -34 -216
6 1 -5 20 -60 7 4 -16 -312
2 -3 -4 84 8 6 14 -168
3 -1 -16 48 9 8 56 336
4 1 -16 -48 10 1 -9 72 -504
5 3 -4 -84 2 -7 24 168
6 5 20 60 3 -5 -12 420
7 1 -6 30 -120 4 -3 -36 372
2 -4 0 120 5 -1 -48 144
3 -2 -18 120 6 1 -48 -144
4 0 -24 0 7 3 -36 -372
5 2 -18 -120 8 5 -12 -420
6 4 0 -120 9 7 24 -168
7 6 30 120 10 9 72 504

This enables the Rasch threshold parameters, {Fx}, to be computed directly from the Guttman parameters, θ, η, ζ, when they are known. The numerical values of the multipliers for m = 2, 10 are shown in the Table.

Direct computation of θ, η, ζ from the {Fx} can usually be performed by means of linear regression, solving the m equations of the form above, with the {Fx} as the dependent variables, the values in the Table as the independent variables, and θ, η, ζ as the coefficients to be estimated.

Example 1: Item 14 in the RUMM2020 runAll example is a 4-category item, so m = 3. On www.rummlab.com.au, the reported estimates are θ = 2.445 and ζ = -0.160. Thus, by computation,
F1 = -2 * 2.445 + 2 * -0.160 = -5.210
F2 = 0 * 2.445 + -4 * -0.160 = 0.640
F3 = 2 * 2.445 + 2* -0.160 = 4.570

The estimates reported for the {Fx} on www.rummlab.com.au are: -5.231, .641, 4.590, indicating a close match between theoretical and empirical results.

Example 2: An m=6 rating scale has category frequencies: 96, 88, 101, 168, 210, 146, 101, The {Fx} are estimated by Winsteps at -2.30, -1.75, -1.34, 0.08, 2.08, 3.23. Excel regression analysis reports θ = 0.5794, η = 0.02786, ζ = -0.002241. According to Andrich and Luo (2003, p. 209) these values have greater stability than the {Fx}. The consequent smoothed values of {Fx} are -2.21, -2.04, -1.13, 0.24, 1.82, 3.32.

John Michael 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 a Rating Scale, Linacre J.M., Andrich D.A., Luo G. … Rasch Measurement Transactions, 2003, 17:3 p.944




Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

To be emailed about new material on www.rasch.org
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from Rasch.org

www.rasch.org welcomes your comments:

Your email address (if you want us to reply):

 

ForumRasch 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.

Coming Rasch-related Events
Aug. 11 - Sept. 8, 2023, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Aug. 29 - 30, 2023, Tue.-Wed. Pacific Rim Objective Measurement Society (PROMS), World Sports University, Macau, SAR, China https://thewsu.org/en/proms-2023
Oct. 6 - Nov. 3, 2023, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Facets), www.statistics.com
June 12 - 14, 2024, Wed.-Fri. 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden http://www.hkr.se/samc2024

 

The URL of this page is www.rasch.org/rmt/rmt173h.htm

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