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 {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 _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 _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_i0, Fk0, producing estimates for the measures of all elements. For more than two facets, the {µ_i} and {_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

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