Theory:
T1 Motivation
FA: Hopes to describe data by covariance
Rasch: Intends to use data for measurement
T2 Model
FA: Ordinal scores mistaken for interval measures which have been
observed without error.
Rasch: Ordinal responses modelled as stochastic manifestations of
linear parameters, estimated with measurement error.
T3 Statistical basis
FA: Covariance matrix of items over examinees.
Rasch: Probability of responses calculated from item and examinee
parameters.
T4 Outcome of analysis
FA: Factors summarizing the covariance matrix.
Rasch: Operational definition of variable by item calibration and
person measure.
T5 Interpretation
FA: Factors named to represent correlated items, but with easily
disputed meanings.
Rasch: Variable defined by item text manifests the underlying
concept. Unexpected outcomes signal misconceptions.
T6 Principal statistics
FA: Factor loadings (covariances of item scores with factor). Factor
scores (from regressing item scores on loadings).
Rasch: A linear measure, error and fit statistics, for each item,
examinee, and any other element modelled.
T7 Largest variance component
FA: The factor with the most variance (largest eigenvalue).
Rasch: The empirical manifestation of the underlying variable.
T8 Other variance components
FA: As many factors as diagonal elements in correlation matrix.
Rasch: Modelled measurement error and unmodelled misfit.
T9 Other variance criterion
FA: Factor eigenvalues greater than, usually, 1.0.
Rasch: Standardized misfit statistics greater than, usually, 2.0.
T10 Measurement and sampling error
FA: Factors with small eigenvalues confound these sources of error.
Rasch: Measurement error as standard errors, sampling error as
standard deviations of examinee and item distributions.
T11 Missing data
FA: List-wise deletion loses data. Pair-wise deletion biases factor
structure.
Rasch: Measures, standard errors and fit statistics based on all
observed data.
Diagnosis:
D1 Multi-modal distribution of examinees or items
FA: A factor for each mode, some with large eigenvalues.
Rasch: Documented in item and person measure distributions. No
effect on misfit.
D2 Irregular examinee response
FA: Slight variance increase. Not determinable from factors.
Rasch: Large misfit for examinee, and somewhat increased misfit for
items to which examinee responded unexpectedly.
D3 Identification of item bias (Differential Item Functioning) on
many test items
FA: By discovering factor scores correlated with group membership,
and then items with loadings on those factors.
Rasch: Exploratory: By discovering items with significant differences
between their group measures.
Rasch: Confirmatory: By partitioning residuals of suspect items
between groups to estimate bias size, significance and homogeneity.
D4 Identification of solitary biased item
FA: Undetectable; bias eigenvalues insignificant.
Rasch: As above, plus item misfit (particularly information-weighted
statistics).
D5 Major multi-dimensionality (items: 50% math, 50% reading)
FA: After rotation, one math factor and one reading factor.
Rasch: Variable combines math and reading items with low person
separation, reliability, and patterns of poor person fit.
D6 Minor multi-dimensionality (items: 95% math, 5% reading)
FA: One math factor; insignificant reading factor.
Rasch: Variable defined by math items, significant misfit in reading
items.
D7 Miskeyed multiple-choice item
FA: Undetectable; eigenvalues insignificant.
Rasch: Large item misfit statistic or item calibration contradicts
construct.
For more information, see Smith & Miao (AERA 1991)
Factor Item Analysis versus Rasch Item Analysis, B Wright … Rasch Measurement Transactions, 1991, 5:1 p. 134-135
The URL of this page is www.rasch.org/rmt/rmt51i.htm
Website: www.rasch.org/rmt/contents.htm