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
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