SFUNCTION= function to model Andrich thresholds with polynomials

By default, the structure or a rating scale (polytomy) is estimated from the data, and expressed as Rasch-Andrich thresholds. When the data are sparse, or the rating scale is long, there may not be enough observations to estimate every threshold securely. SFUNCTION= defines a smoothing function for estimating all the rating scale thresholds. If all the categories are not observed in the data, ISRANGE= defines the range of the category numbers.

 

Alternative approaches are described at Null or unobserved categories: structural and incidental zeroes, also at Unobserved and dropped categories.

 

Modeling Rasch-Andrich Thresholds for unobserved and sparse categories

Example: Olympic Ice-Skating, Exam15.txt. The rating-scale is from 0.0 to 6.0, renumbered 00 to 60 for Rasch analysis. The empirical data are from the Pairs Figure-Skating at the Salt Lake City Olympics.

 

Category frequencies:

Possible categories: 0-60 = 61 categories

Observed categories: 29, 30-59 = 30 categories

Unobserved categories: 00-28, 30, 60 = 31 categories

Analysis 1: Categories 29, 31-59 observed, renumbered 30-59. Estimated using the method of Wright, Masters (1982).

CODES=     " 29 31 32 33 34 35 36 37 38 39 40 41 42 43 44+

           + 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59"

NEWSCORE = " 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44+

           + 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59"

STKEEP=NO

Category Probability Curves

Item Characteristic Curves
Red: Model, Blue: Empirical (with confidence bands)


Analysis 2: Categories 29, 31-59 observed. Category 30 interpolated with Wilson (1991).

CODES= " 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44+

       + 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59"

STKEEP=YES


1st moment (thresholds at central location). Categories 0-60.

SFUNCTION=1 ; item thresholds are at item location. Threshold dispersion = 0.

ISRANGE=*

1 0 60 ; 1 is item in the item-group: category range 0-60

*

or
SAFILE=*
0-60 0 ; all thresholds anchored at 0
*


2nd moment (thresholds with uniform dispersion) .  Categories 0-60  with Andrich, Luo (2003), Pedler (1987)

SFUNCTION=2

ISRANGE=*

1 0-60

*


3rd moment (thresholds with dispersion and skewness). Categories 0-60  with Andrich, Luo (2003), Pedler (1987)

SFUNCTION=3

ISRANGE=*

1 0-60

*


4th moment (thresholds with dispersion, skewness and kurtosis). Categories 0-60  with Andrich, Luo (2003),  Pedler (1987)

SFUNCTION=4

ISRANGE=*

1 0-60

*


ICCs for these and higher moments

 

The 4th moment (kurtosis, red line) models the smoothest ICC for the full range of the rating scale 0-60. The higher moments (SFUNCTION=5,6,...) do not add meaningfully to the red line of the 4th moment (kurtosis, SFUNCTION=4).

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.

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

Wilson M. (1991) Unobserved or Null Categories. Rasch Measurement Transactions, 5:1, 158 - www.rasch.org/rmt/rmt51b.htm

Wright B.D., Masters G.N. (1982) Rating Scale Analysis. MESA Press.


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