For more than two facets, the {μi} and {σi} summarize the distribution of the combined measures of the other facets,
as encountered by item i, and similarly for the persons, tasks, judges, etc.
PROX for Complete Data
If data are complete, then μi and σi can be treated as constant across items, and μn and σn
constant across persons. This, with further simplifications, permits the non-iterative estimation equations derived by Cohen
(1979):
L = count of items
N = count of persons
Si is the raw score of successes on item i
Sn is the raw score of successes by person n
SDL = sample S.D. of item raw scores
SDN = sample S.D. of person raw scores
Item difficulties:
XL = item difficulty expansion factor = √ [(1+SDN/2.89)/(1-SDLSDN/8.35)]
Provisional difficulty of item i = - XL*ln[(Si)/(N-Si)]
Difficulty of item i = Provisional difficulty of item i - Average provisional difficulty of all items
with S.E. of item i = XL*√[N /(Si*(N-Si))]
Person abilities:
XN = person ability expansion factor = √ [(1+SDL/2.89)/(1-SDLSDN/8.35)]
Ability of person n = 0 + XN*ln[(Sn)/(L-Sn)]
with S.E. of person n = XN*√[L /(Sn*(L-Sn))]
PROX estimation equations for polytomous data will be derived in the next RMT.
John Michael Linacre
Camilli, G. 1994. Origin of the scaling constant d=1.7 in item response theory. Journal of Educational and Behavioral Statistics.
19(3) p.293-5.
Cohen, L. 1979. Approximate expressions for parameter estimates in the Rasch model. British Journal of Mathematical and
Statistical Psychology 32(1) 113-120.
PROX with missing data, or known item or person measures. Linacre JM.
Rasch Measurement Transactions, 1994, 8:3 p.378
Rasch Publications |
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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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