PROX for polytomous data. Linacre JM. … 8:4 p.400
The Normal Approximation Estimation Algorithm (PROX) was developed for dichotomous data (Cohen 1979), but can be extended to many-facet polytomous data with missing observations. The expediting specification is that each parameter (e.g., person, item, task, step) is taken to have encountered 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.
Consider the two-facet case of person abilities {B_{n}} facing item difficulties {D_{i}} on a rating scale with step calibrations {F_{k}}, k=0,m. According to Rasch, for each pair of adjacent categories, there is the logistic relationship
Label the logistic function , so that
Count the S_{ik} persons who respond to item i in category k. Then sum for each item i across all S_{ik-1}+S_{ik} persons it encounters, who respond in categories k-1 or k,
Taking the categories in pairs exhausts the data, so accumulate these sums across all odd rating scale steps, k=1,3,..,m,
For convenience, define
When the N_{i} relevant {B_{n}-F_{k}} are symmetrically distributed, summing across them can be approximated by integrating across N_{i} normal distributions of a random variable {x} with mean µ_{i} and standard deviation σ_{i} of the relevant {B_{n}-F_{k}}:
where indicates the normal cumulative distribution function.
A convenient equivalence between logistic and normal cumulative distributions (Camilli 1994) is
producing,
But, in general,
since 1.702² = 2.9,
substituting the logistic for the cumulative normal,
and rearranging, produces an estimation equation for D_{i}, the logit difficulty of item i,
with standard error
The comparable PROX estimation equation for person n with logit ability, B_{n}, is
where S_{nk} is the number of responses in category k by person n, and µ_{n} and σ_{n} summarize the distribution of relevant logit difficulties {D_{i}+F_{k}} encountered by person n.
A convenient approach is to average the results obtained by considering two sets of even numbers of categories: the upper categories, omitting category 0, with N_{iu} observations,
and the lower categories, omitting category m, with N_{il} observations,
so that
with standard error
Estimation equations for the step calibrations are
where S_{k} is the number of responses in category k. µ_{k} and _{k} summarize the logit measures {B_{n}-D_{i}} for the S_{k-i}+S_{k} responses in categories k-1 and k.
If step k is not observed, then F_{k}=∞, F_{k+1}=-∞, and F_{k}+F_{k+1} is given by
where k' indicates responses of k-1 or k+1.
These equations can be solved iteratively, with anchoring constraints like D_{i}0, Fk0, producing estimates for the measures of all elements. For more than two facets, the {µ_{i}} and {σ_{i}} summarize the distribution of the combined measures {B_{n}-..-F_{k}} of the other facets as encountered by item i, and similarly for the persons, tasks, judges, steps, etc.
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 for polytomous data. Linacre JM. … Rasch Measurement Transactions, 1995, 8:4 p.400
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 | |
---|---|
Oct. 6 - Nov. 3, 2023, Fri.-Fri. | On-line workshop: Rasch Measurement - Core Topics (E. Smith, Facets), www.statistics.com |
Oct. 12, 2023, Thursday 5 to 7 pm Colombian time | On-line workshop: Deconstruyendo el concepto de validez y Discusiones sobre estimaciones de confiabilidad SICAPSI (J. Escobar, C.Pardo) www.colpsic.org.co |
June 12 - 14, 2024, Wed.-Fri. | 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden http://www.hkr.se/samc2024 |
Aug. 9 - Sept. 6, 2024, Fri.-Fri. | On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com |
The URL of this page is www.rasch.org/rmt/rmt84k.htm
Website: www.rasch.org/rmt/contents.htm