# Factor Analysis versus Rasch Item Analysis of Items

Theory:
Factor AnalysisRasch Analysis
T1MotivationHopes to describe data by covarianceIntends to use data for measurement
T2ModelOrdinal scores mistaken for interval measures which have been observed without errorOrdinal responses modelled as stochastic manifestations of linear parameters, estimated with measurement error
T3Statistical basisCovariance matrix of items over examineeProbability of responses calculated from item and examinee parameters
T4Outcome of analysisFactors summarizing the covariance matrixOperational definition of variable by item calibration and person measure
T5InterpretationFactors named to represent correlated items, but with easily disputed meaningsVariable defined by item text manifests the underlying concept. Unexpected outcomes signal misconceptions
T6Principal statisticsFactor loadings (covariances of item scores with factor). Factor scores (from regressing item scores on loadings).A linear measure, error and fit statistics, for each item, examinee, and any other element modelled
T7Largest variance componentThe factor with the most variance (largest eigenvalue)The empirical manifestation of the underlying variable
T8Other variance componentsAs many factors as diagonal elements in correlation matrixModelled measurement error and unmodelled misfit
T9Other variance criterionFactor eigenvalues greater than, usually, 1.4Standardized misfit statistics greater than, usually, 2.0
T10Measurement and sampling errorFactors with small eigenvalues confound these sources of errorMeasurement error as standard errors, sampling error as standard deviations of examinee and item distributions
T11Missing dataList-wise deletion loses data. Pair-wise deletion biases factor structureMeasures, standard errors and fit statistics based on all observed data

Diagnosis:
Factor AnalysisRasch Analysis
D1Multi-modal distribution of examinees or itemsA factor for each mode, some with large eigenvalueDocumented in item and person measure distributions. No effect on misfit
D2Irregular examinee responseSlight variance increase. Not determinable from factorsLarge misfit for examinee, and somewhat increased misfit for items to which examinee responded unexpectedly
D3Identification of item bias (DIF, Differential Item Functioning) on many test itemsBy discovering factor scores correlated with group membership, and then items with loadings on those factorsExploratory: By discovering items with significant differences between their group measures
Confirmatory: By partitioning residuals of suspect items between groups to estimate bias size, significance and homogeneity
D4Identification of solitary biased itemUndetectable; bias eigenvalues insignificantAs above, plus item misfit (particularly information-weighted (INFIT) statistics)
D5Major multi-dimensionality (items: 50% math, 50% reading)After rotation, one math factor and one reading factorVariable combines math and reading items with low person separation, reliability, and patterns of poor person fit
[PCA of Rasch residuals]
D6Minor multi-dimensionality (items: 95% math, 5% reading)One math factor; insignificant reading factorVariable defined by math items, significant misfit in reading items
D7Miskeyed multiple-choice itemUndetectable; eigenvalues insignificant.Large item misfit statistic or item calibration contradicts construct

The Impact of Rasch Item Difficulty on Confirmatory Factor Analysis , S.V. Aryadoust … Rasch Measurement Transactions, 2009, 23:2 p. 1207
Confirmatory factor analysis vs. Rasch approaches: Differences and Measurement Implications, M.T. Ewing, T. Salzberger, R.R. Sinkovics … Rasch Measurement Transactions, 2009, 23:1 p. 1194-5
Conventional factor analysis vs. Rasch residual factor analysis, Wright, B.D. … 2000, 14:2 p. 753.
Rasch Analysis First or Factor Analysis First? Linacre J.M. … 1998, 11:4 p. 603.
Factor analysis and Rasch analysis, Schumacker RE, Linacre JM. … 1996, 9:4 p.470
Too many factors in Factor Analysis? Bond TG. … 1994, 8:1 p.347
Comparing factor analysis and Rasch measurement, Wright BD. … 1994, 8:1 p.350
Factor analysis vs. Rasch analysis of items, Wright BD. … 5:1 p.134

Factor Analysis versus Rasch Analysis of Items, B Wright … Rasch Measurement Transactions, 1991, 5:1 p. 134-135

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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Jan. 5 - Feb. 2, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
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May 25 - June 22, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com