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.

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.

PROX for Complete Data

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_LSD_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_LSD_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

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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