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.
Iterative PROX: the theory
For convenience, consider the two-facet case of persons of ability {B_{n}} facing items of difficulty {D_{i}}. Then, according to the Rasch model, for each dichotomous encounter there is the logistic relationship
(1) |
Sum for each item i across all N_{i} persons it encounters,
(2) |
where S_{i} is the raw score of successes on item i, and Ψ is the standard logistic function.
When the {B_{n}} are symmetrically distributed, summing across them can be approximated by integrating across N_{i} normal distributions of {B} with mean μ_{i} and standard deviation σ_{i} 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 D_{i}, the logit difficulty of item i,
(9) |
where μ_{i} is the mean and σ_{i} the standard deviation of the logit abilities of the persons encountering item i.
The comparable estimation equation for person n with logit ability, B_{n}, is
(10) |
where R_{n} is the raw score achieved by person n on N_{n} items, and μ_{n} and σ_{n} 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 μ_{i}=0, σ_{i}=0, mu;_{n}=0, σ_{n}=0
2. Compute the item difficulties:
(9) |
(10) |
(11) |
For more than two facets, the {μ_{i}} and {σ_{i}} 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, then μ_{i} and σ_{i} can be treated as constant across items, and μ_{n} and σ_{n} 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
S_{i} is the raw score of successes on item i
S_{n} is the raw score of successes by person n
SD_{L} = sample S.D. of item raw scores
SD_{N} = sample S.D. of person raw scores
Item difficulties:
X_{L} = item difficulty expansion factor = √ [(1+SD_{N}/2.89)/(1-SD_{L}SD_{N}/8.35)]
Provisional difficulty of item i = - X_{L}*ln[(S_{i})/(N-S_{i})]
Difficulty of item i = Provisional difficulty of item i - Average provisional difficulty of all items
with S.E. of item i = X_{L}*√[N /(S_{i}*(N-S_{i}))]
Person abilities:
X_{N} = person ability expansion factor = √ [(1+SD_{L}/2.89)/(1-SD_{L}SD_{N}/8.35)]
Ability of person n = 0 + X_{N}*ln[(S_{n})/(L-S_{n})]
with S.E. of person n = X_{N}*√[L /(S_{n}*(L-S_{n}))]
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
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