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Garson, G. D. (2013). Factor Analysis. Asheboro, NC: Statistical Associates Publishers.

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ISBN: 978-1-62638-001-1
@c 2013 by G. David Garson and Statistical Associates Publishers. worldwide rights reserved in all languages and on all media. Permission is not granted to copy, distribute, or post e-books or passwords.



Factor analysis is used to uncover the latent structure (dimensions) of a set of variables. It reduces attribute space from a larger number of variables to a smaller number of factors and as such is a "non-dependent" procedure (that is, it does not assume a dependent variable is specified). Factor analysis could be used for any of the following purposes:

" To reduce a large number of variables to a smaller number of factors for modeling purposes, where the large number of variables precludes modeling all the measures individually. As such, factor analysis is integrated in structural equation modeling (SEM), helping confirm the latent variables modeled by SEM. However, factor analysis can be and is often used on a stand-alone basis for similar purposes.

" To establish that multiple tests measure the same factor, thereby giving justification for administering fewer tests. Factor analysis originated a century ago with Charles Spearman's attempts to show that a wide variety of mental tests could be explained by a single underlying intelligence factor (a notion now rejected, by the way).

" To validate a scale or index by demonstrating that its constituent items load on the same factor, and to drop proposed scale items which cross-load on more than one factor. " To select a subset of variables from a larger set, based on which original variables have the highest correlations with the principal component factors.

" To create a set of factors to be treated as uncorrelated variables as one approach to handling multicollinearity in such procedures as multiple regression

" To identify clusters of cases and/or outliers.

" To determine network groups by determining which sets of people cluster together (using Q-mode factor analysis, discussed below)

A non-technical analogy: A mother sees various bumps and shapes under a blanket at the bottom of a bed. When one shape moves toward the top of the bed, all the other bumps and shapes move toward the top also, so the mother concludes that what is under the blanket is a single thing - her child. Similarly, factor analysis takes as input a number of measures and tests, analogous to the bumps and shapes. Those that move together are considered a single thing, which it labels a factor. That is, in factor analysis the researcher is assuming that there is a "child" out there in the form of an underlying factor, and he or she takes simultaneous movement (correlation) as evidence of its existence. If correlation is spurious for some reason, this inference will be mistaken, of course, so it is important when conducting factor analysis that possible variables which might introduce spuriousness, such as anteceding causes, be taken into account.

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Below is the unformatted table of contents.

Table of Contents
Overview	8
Data	10
Key Concepts and Terms	10
Exploratory factor analysis (EFA)	10
Exploratory vs. confirmatory factor analysis (CFA)	10
Factor Analytic Data Modes	11
R-mode factor analysis	11
Q-mode factor analysis	11
Other rarer modes of factor analysis	12
Types of factor extraction	13
Principal components analysis (PCA)	13
Principal factor analysis (PFA)	14
PCA and PFA compared	14
Other Extraction Methods	16
Types of factor rotation	17
Rotation methods	17
No rotation	18
Varimax rotation	19
Quartimax rotation	20
Equamax rotation	21
Direct oblimin (oblique) rotation	22
Promax rotation	23
Other rotation methods	24
Summary	24
Factor analysis in SPSS	24
Data setup	24
The "Factor" dialog	24
Descriptives and Options	25
Extraction	26
Rotation	27
Factor Scores	28
Statistical output in SPSS	29
Factor loadings	29
Plot of factor loadings (factor space plot)	31
Factor, component, pattern, and structure matrices	33
Communality	34
Uniqueness	36
Eigenvalues	36
Extraction sums of squared loadings	37
Trace	37
Factor scores	38
Bartlett scores	39
Saving factor scores	40
Criteria for number of factors to model	40
Parallel analysis	42
Other Criteria	43
Using reproduced correlation residuals to validate the choice of number of factors	43
Summary	44
Factor analysis in SAS	44
SAS interface	44
SAS syntax	45
Rotation methods in SAS	47
Statistical output in SAS	48
Factor loadings in SAS output	48
SAS output for communalities	48
SAS output for eigenvalues	48
SAS scree plot output	49
SAS factor loadings plots	51
Factor analysis in Stata	52
Stata interface	52
Importing data into Stata	53
Stata syntax	55
Statistical output in Stata	58
Stata output for eigenvalues	58
Factor loadings in Stata output	59
Stata output for communalities	59
Stata scree plot output	59
Stata loading plots	60
Categorical principal components analysis (CATPCA)	62
Overview	62
SPSS categorical principal components analysis	63
Data considerations	63
CATPCA user interface in SPSS	63
The “Optimal Scaling” dialog	63
The main CATPCA dialog	64
The “Discretize” button dialog	66
The “Missing” button dialog	67
The “Options” button dialog	68
The “Output” button dialog	70
The “Save” button dialog	72
The “Object” button dialog	72
The “Category” button dialog	73
The “Loading” button dialog	75
SPSS CATPCA statistical output	75
The “Model Summary” table	75
The “Component Loadings” table	77
The “Component Loadings” plot	79
The “Variance Accounted For” table	80
The “Object Points Labeled by Casenumbers” plot	81
The “Object Scores” table	82
The “Biplot Component Loadings and Objects” plot	83
The “Quantifications” table	84
The “Category Points” plot	85
The “Projected Centroids” table and plot	87
SAS categorical principal components analysis	89
Overview	89
SAS syntax	89
The PROC PRINQUAL procedure	90
Principal components analysis of transformed data	94
Stata categorical principal components analysis	97
Overview	97
Example	98
The polychoric correlation matrix	98
The “Principal component analysis” table	99
The “Scoring Coefficients” table	100
Saved object scores	102
A second example using the factormat command	104
The structural equation modeling approach to factor analysis	106
Testing error in the measurement model	106
Redundancy test of one-factor vs. multi-factor models	107
Measurement invariance test comparing a model across groups	107
Orthogonality tests	107
Assumptions	107
Valid imputation of factor labels	108
Proper specification/no selection bias	108
No outliers	108
Continuous data	108
Linearity	110
Multivariate normality	110
Homoscedasticity	110
Orthogonality	111
Existence of underlying dimensions	111
Moderate to moderate-high intercorrelations without multicollinearity	111
Absence of high multicollinearity	111
No perfect multicollinearity	112
Sphericity	112
Adequate sample size	112
Frequently Asked Questions	112
How does factor analysis compare with cluster analysis and multidimensional scaling?	112
How many cases do I need to do factor analysis?	115
How do I input my data as a correlation matrix rather than raw data?	116
How many variables do I need in factor analysis? The more, the better?	117
What is KMO? What is it used for?	117
Why is normality not required for factor analysis when it is an assumption of correlation, on which factor analysis rests?	119
Is it necessary to standardize one's variables before applying factor analysis?	119
Can you pool data from two samples together in factor analysis?	120
How does factor comparison of the factor structure of two samples work?	120
Why is rotation of axes necessary?	121
Why are the factor scores I get the same when I request rotation and when I do not?	121
Why is oblique rotation less common in social science?	121
When should oblique rotation be used?	122
What is hierarchical factor analysis and how does it relate to oblique rotation?	122
How high does a factor loading have to be to consider that variable as a defining part of that factor?	123
What is simple factor structure, and is the simpler, the better?	123
How is factor analysis related to validity?	124
What is the matrix of standardized component scores, and for what might it be used in research?	124
What are the pros and cons of common factor analysis compared to PCA?	124
Why are my PCA results different in SAS compared to SPSS?	125
How do I do Q-mode factor analysis of cases rather than variables?	125
How else may I use factor analysis to identify clusters of cases and/or outliers?	125
Can factor analysis handle hierarchical/multilevel data?	125
What do I do if I want to factor categorical variables?	126
Bibliography	126
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