Linacre JM. … 8:3 p.378
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.
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
Sum for each item i across all Ni persons it encounters,
where Si is the raw score of successes on item i.
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:
where Φ is the normal cumulative distribution function.
An equivalence between logistic and normal cumulative probability distributions (Camilli 1994) is
producing,
But, in general,
Then, since 1.702² = 2.9,
Substituting the logistic for the normal ogive,
and rearranging, produces an estimation equation for Di, the logit difficulty of item i,
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
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:
These equations can be solved iteratively, with an additional constraint like ΣDi=0, producing estimates for the measures of all elements. 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.
If data are complete, or responses are missing at random, 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). 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. Linacre JM. … Rasch Measurement Transactions, 1994, 8:3 p.378
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