Rating scales with many categories often have unobserved categories or categories with very low frequencies. These make the standard estimation of Andrich thresholds (step difficulties), based on the relative frequencies of adjacent categories, subject to accidents in the data. Modeling the relationship between the thresholds with a continuous function smooths out the lumpiness in the data.
Louis Guttman (1941, etc.) proposes that responses to attitude-survey items can be summarized by the scale location and scale intensity. Guttman called these "principal components". Here named "Guttman components". The family of Guttman components was extended by Pender Pedler (1987 ), also Andrich and Luo (2003) , as a set of orthogonal polynomials summarizing the Andrich thresholds.
The Orthogonal Polynomials
Pedler's (1987) orthogonal-polynomial Guttman components of the rating-scale thresholds are:
SFUNCTION=
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Polynomial
where x is the observation
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Formula:
categories numbered 0..m
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Minimum Categories
/ Thresholds
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Interpretation
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(0)
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T0(x)
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0
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1 / 0
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rating-scale exists
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1
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T1(x)
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1
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2 / 1
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mean: central location of Andrich thresholds
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2
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T2(x)
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2( x - (m+1)/2 )
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3 / 2
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dispersion, variance: linear relationship
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3
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T3(x)
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3( x - (m+1)/2 )² - (m² - 1)/4
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4 / 3
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skewness: quadratic
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4
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T4(x)
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4( x - (m+1)/2 )³ - ( x - (m+1)/2 )(3m² - 7)/5
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5 / 4
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kurtosis: cubic
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(k+1)
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higher order:
Tk+1(x)
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[(k+1)/k] ( x - (m+1)/2 )Tk(x) - ([(m² - (k-1)²)(k² - 1)]/[4(2k - 1)(2k-3)])Tk-1(x)
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k+2 / k+1
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higher moments
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These are operationalized in Winsteps using SFUNCTION=. For instance, SFUNCTION=3:
Andrich threshold between categories x-1 and x = F(x) = p1T1(x) + p2T2(x) + p3T3(x) where p1, p2, p3 are the data-dependent coefficients.
Estimating the Coefficients of the Polynomials
The initial values of the coefficients, p, are zero. In Winsteps, the central location of the Andrich thresholds of a rating scale is set at zero logits, so p1 is stays at zero. Initially all the F(x) are also zero, so the thresholds are all at the central location.
The category probabilities are computed by applying the threshold values to the Andrich Rating-Scale Model, Partial Credit Model, etc. This produces:
1. an observed and expected frequency for each category
2. a modeled variance of the probability of each category
The likelihood of the data = Λ = Product (Probability (xni)) where xni are the observations.
Log-likelihood of the data = λ = ∑ (Ln(Pxni)) = ∑ (xni(Bn - Di) - ∑j=1,xFj) - ∑ (Ln(probability normalizer))
∂λ / ∂pk = ∑ ∑j=1,,xTk(j) - ∑Pni (∑j=1,mTk(j) )
∂2λ / ∂pk2 = ∑ Pni(∑j=1,mTk(j))2 - (∑ Pni(∑j=1,mTk(j)))2
then, by Newton-Raphson,
change in Pk = ∂λ / ∂pk / ∂2λ / ∂pk2
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.
