The Normal Approximation Estimation Algorithm (PROX) is a computationally simpler Rasch estimation algorithm invented by Leslie Cohen in 1972 for responses by persons to dichotomous items with no missing data. It has been extended to many-facet polytomous data sets with missing data. The essential specification is that each element (e.g., person, item, task) encounters a symmetrically distributed sample of challenges (e.g., person facing items+tasks+judges). The distributions of challenges faced by the elements may have different means and variances.
Iterative PROX: the theory
For convenience, consider the two-facet case of persons of ability {B_{n}} facing items of difficulty {D_{i}}. Then, according to the Rasch model, for each dichotomous encounter there is the logistic relationship
(1) |
Sum for each item i across all N_{i} persons it encounters,
(2) |
where S_{i} is the raw score of successes on item i, and Ψ is the standard logistic function.
When the {B_{n}} are symmetrically distributed, summing across them can be approximated by integrating across N_{i} normal distributions of {B} with mean μ_{i} and standard deviation σ_{i} for item i:
(3) |
where Φ is the normal cumulative distribution function.
An equivalence between logistic Ψ and normal cumulative probability distributions Φ (Camilli 1994) is
(4) |
producing,
(5) |
But, in general, for the cumulative normal distribution function,
(6) |
Then, since 1.702² = 2.9,
(7) |
Substituting the logistic for the normal ogive,
(8) |
and rearranging, produces an estimation equation for D_{i}, the logit difficulty of item i,
(9) |
where μ_{i} is the mean and σ_{i} the standard deviation of the logit abilities of the persons encountering item i.
The comparable estimation equation for person n with logit ability, B_{n}, is
(10) |
where R_{n} is the raw score achieved by person n on N_{n} items, and μ_{n} and σ_{n} summarize the distribution of logit item difficulties encountered by person n.
The model standard errors of PROX measures follow:
(11) |
Iterative PROX: in practice
1. Start with all person and item difficulties set to 0, so that μ_{i}=0, σ_{i}=0, mu;_{n}=0, σ_{n}=0
2. Compute the item difficulties:
(9) |
(10) |
(11) |
For more than two facets, the {μ_{i}} and {σ_{i}} summarize the distribution of the combined measures of the other facets, as encountered by item i, and similarly for the persons, tasks, judges, etc.
If data are complete, then μ_{i} and σ_{i} can be treated as constant across items, and μ_{n} and σ_{n} 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
S_{i} is the raw score of successes on item i
S_{n} is the raw score of successes by person n
SD_{L} = sample S.D. of item raw scores
SD_{N} = sample S.D. of person raw scores
Item difficulties:
X_{L} = item difficulty expansion factor = √ [(1+SD_{N}/2.89)/(1-SD_{L}SD_{N}/8.35)]
Provisional difficulty of item i = - X_{L}*ln[(S_{i})/(N-S_{i})]
Difficulty of item i = Provisional difficulty of item i - Average provisional difficulty of all items
with S.E. of item i = X_{L}*√[N /(S_{i}*(N-S_{i}))]
Person abilities:
X_{N} = person ability expansion factor = √ [(1+SD_{L}/2.89)/(1-SD_{L}SD_{N}/8.35)]
Ability of person n = 0 + X_{N}*ln[(S_{n})/(L-S_{n})]
with S.E. of person n = X_{N}*√[L /(S_{n}*(L-S_{n}))]
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 | ||||
---|---|---|---|---|
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 |
Forum | Rasch 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 | |
---|---|
Apr. 14-17, 2020, Tue.-Fri. | International Objective Measurement Workshop (IOMW), University of California, Berkeley, https://www.iomw.org/ |
May 22 - June 19, 2020, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
June 26 - July 24, 2020, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com |
June 29 - July 1, 2020, Mon.-Wed. | Measurement at the Crossroads 2020, Milan, Italy , https://convegni.unicatt.it/mac-home |
July - November, 2020 | On-line course: An Introduction to Rasch Measurement Theory and RUMM2030Plus (Andrich & Marais), http://www.education.uwa.edu.au/ppl/courses |
July 1 - July 3, 2020, Wed.-Fri. | International Measurement Confederation (IMEKO) Joint Symposium, Warsaw, Poland, http://www.imeko-warsaw-2020.org/ |
Aug. 7 - Sept. 4, 2020, Fri.-Fri. | On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com |
Oct. 9 - Nov. 6, 2020, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
June 25 - July 23, 2021, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com |
The URL of this page is www.rasch.org/rmt/rmt83g.htm
Website: www.rasch.org/rmt/contents.htm