## PROX with missing data, or known item or person measures

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.

Dichotomous Data

Iterative PROX: the theory

For convenience, consider the two-facet case of persons of ability {Bn} facing items of difficulty {Di}. Then, according to the Rasch model, for each dichotomous encounter there is the logistic relationship

 (1)

Sum for each item i across all Ni persons it encounters,

 (2)

where Si is the raw score of successes on item i, and Ψ is the standard logistic function.

When the {Bn} are symmetrically distributed, summing across them can be approximated by integrating across Ni normal distributions of {B} with mean μi and standard deviation σi 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 Di, the logit difficulty of item i,

 (9)

where μi is the mean and σi the standard deviation of the logit abilities of the persons encountering item i.

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

 (10)

where Rn is the raw score achieved by person n on Nn items, and μn and σn 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 μi=0, σi=0, mu;n=0, σn=0
2. Compute the item difficulties:

 (9)
If the data are complete, then μi (the mean of the person estimates) and σi (the S.D. of the person estimates) are the same for every item.
3. Subtract the mean item difficulty from all the item difficulties in order to maintain the sum of the item difficulties at 0: ΣDi=0
4. Compute the person abilities:
 (10)
If the data are complete, then μn (the mean of the item estimates) and σn (the S.D. of the item estimates) are the same for every person.
5. Repeat 2., 3. and 4. until the bigggest change in any item difficulty or person ability is less than .01 logits.
6. The item difficulties and the person abilities are the Rasch PROX estimates with standard errors:
 (11)

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
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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