PROX for Polytomous Data

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 {Bn} facing item difficulties {Di} on a rating scale with step calibrations {Fk}, 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 Sik persons who respond to item i in category k. Then sum for each item i across all Sik-1+Sik persons it encounters, who respond in categories k-1 or k,

Rating Scales with an Even Number of Categories

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 Ni relevant {Bn-Fk} are symmetrically distributed, summing across them can be approximated by integrating across Ni normal distributions of a random variable {x} with mean µi and standard deviation σi of the relevant {Bn-Fk}:

where indicates the normal cumulative distribution function.

A convenient equivalence between logistic and normal cumulative distributions (Camilli 1994) is


But, in general,

since 1.702² = 2.9,

substituting the logistic for the cumulative normal,

and rearranging, produces an estimation equation for Di, the logit difficulty of item i,

with standard error

The comparable PROX estimation equation for person n with logit ability, Bn, is

where Snk is the number of responses in category k by person n, and µn and σn summarize the distribution of relevant logit difficulties {Di+Fk} encountered by person n.

Rating Scales with an Odd Number of Categories

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 Niu observations,

and the lower categories, omitting category m, with Nil observations,

so that

with standard error

Step Calibrations

Estimation equations for the step calibrations are

where Sk is the number of responses in category k. µk and k summarize the logit measures {Bn-Di} for the Sk-i+Sk responses in categories k-1 and k.

If step k is not observed, then Fk=∞, Fk+1=-∞, and Fk+Fk+1 is given by

where k' indicates responses of k-1 or k+1.

These equations can be solved iteratively, with anchoring constraints like Di0, Fk0, producing estimates for the measures of all elements. For more than two facets, the {µi} and {σi} summarize the distribution of the combined measures {Bn-..-Fk} 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

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