Rasch Factor Analysis

Conventional Factor Analysis input xni is a test score, Likert rating or MCQ response of person n to item i. Incommensurable origins and scales are reconciled by using sample means and standard deviations to commensurate items:

This "equating" depends on complete data. When data are missing, they must be feigned or incomplete persons or items deleted. The model for Factor 1 is:

{un1} n=1,N is a vector of person "factor scores"
{vi1} i=1,L is a vector of item "factor loadings"

The Factor 1 residual is:

Whether this residual is all error or contains other factors is unknown. To find out, the residual is taken as the data for Factor 2. Matrices of decreasing residuals {{znij}} j=2,M are extracted in turn to calculate the M factor model:

Optimal values for person and item vectors {unj} and {vij} for each Factor j in turn are found by minimizing:

Decomposition to identify each Factor j is done by initializing at wnj=1 and iterating equations:

until changes in {wnj} become uninteresting.

Factor score wnj is the value predicted by Factor j's regression on "independent" variables i = 1,L with regression coefficients {vij}.

Problems with Factor Analysis of Observational Data:

1. Since, data {{xni}} are never linear measures, Equations (2) and (3) cannot work in a reproducible way.

2. The necessity for complete data is awkward.

3. The residuals {{znij}} for seeking smaller factors are awash in turbulence left by preceding larger factors.

4. There is no way to know when to stop factoring.

5. Few algorithms provide standard errors for factor loadings or factor scores.

6. When old items are refactored from new data, factor sizes, loadings and number of factors change.

Most factor analysts can tolerate problems 1-5. Problem 6, however, is impossible to swallow. As analysts notice the numerical fragility of their structures, they abandon their person factor scores and ignore everything in the factor loadings except which factor gives each item its highest loading.

Person scores for these "same factor" clusters are obtained, not from factor scores, but by summing the standardized person responses zni1 to the clustered items.

Rasch Factor Analysis:

Since data {{xni}} are labels for ordinally interpreted qualities, we address them with a probability model for ordered categories. For a dichotomy, we recognize xni=0,1 as a binomial. The error model which follows is not the ill-chosen linear error model of factor analysis, but a binomial probability Pnix for the occurrence of xni:

from which

Parameters, as with factor analysis (5), can be estimated by minimizing:

but now we get, not only least-square (maximum likelihood) estimates of linear person measures Bn and linear item calibrations Di on their common variable, but also a stochastic basis, the binomial error variances pni1pni0, for estimating useful standard errors and evaluating the probabilities of residuals:

This enables detailed misfit analysis which, in turn, allows a partition of the matrix of residuals {{zni1}} into those many zni1 which are observed to be no greater than the probability model expects, say |zni1| < 2, and the subset of more extreme residuals, say |zni1| > 2 or 3, which are sufficiently improbable to invite investigation.

It is only when improbable residuals emerge that there is an incentive to look for a second variable. The improbable residuals tell us where to look. Should there be another useful variable in these data, it will be most evident among the responses to the subset of items and persons which misfit the first analysis.

To seek a second variable, therefore, we apply the probability model again, not to the whole matrix of residuals {{zni1}}, but only to the original ordinal responses {{xni}} to this subset of items. We estimate a new set of linear item calibrations {Di2} for these items, along with a new set of additive conjoint person measures {Bn2} on this newly defined "Variable 2". To find out whether Variable 2 is useful, we plot person measures {Bn2} against person measures {Bn1}. This plot will show us the extent to which we have found a useful second variable.

Benjamin D. Wright 1994 RMT 8:1 p. 348-9

Rasch factor analysis Wright BD. … Rasch Measurement Transactions, 1994, 8:1 p.348-9

Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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