"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
log_{e}(P_{nix}/P_{ni(x-1)}) = B_{n} -
D_{i} - F_{x}
where F_{x} 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:
T_{1}(x) = 1 which requires at least two categories in the rating scale
T_{2}(x) = 2( x - (m+1)/2 ) which requires at least three categories in the rating scale
T_{3}(x) = 3( x - (m+1)/2 )² - (m² - 1)/4 which requires at least four categories in the rating scale
T_{4}(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:
T_{k+1}(x) = [(k+1)/k] ( x - (m+1)/2 )T_{k}(x)
- ([(m² - (k-1)²)(k² - 1)]/[4(2k - 1)(2k-3)])T_{k-1}(x)
In accordance with Andrich and Luo (2003), these modify the Guttman parameters, θ, η, ζ:
D_{i} = the item difficulty
F_{x} =
T_{1}(x)*0 where 0 is the rating scale central location relative to the item difficulty
+ T_{2}(x)*θ where θ is the rating scale dispersion or unit
+ T_{3}(x)*2*η where η is the skewness
+ T_{4}(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, {F_{x}}, 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 {F_{x}} can usually be performed by means of linear regression, solving the m equations of the form above, with the {F_{x}} 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,
F_{1} = -2 * 2.445 + 2 * -0.160 = -5.210
F_{2} = 0 * 2.445 + -4 * -0.160 = 0.640
F_{3} = 2 * 2.445 + 2* -0.160 = 4.570
The estimates reported for the {F_{x}} 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 {F_{x}} 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 {F_{x}}. The consequent smoothed values of {F_{x}} 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 | ||||
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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 |
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