Principles of factor analysis Roger Watson
Factor analysis
Factor analysis
Factor analysis
Factor analysis is a multivariate statistical method for reducing large numbers of variables to fewer underlying dimensions
Establishing validity Construct validity (unobtainable) Face validity Content validity Factorial validity Criterion concurrent predictive Convergent (divergent) and discriminant validity
Establishing validity Construct validity (unobtainable) Face validity Content validity Factorial validity Criterion concurrent predictive Convergent (divergent) and discriminant validity
EdFED Scale Supervision Physical Help Spillage Leave food on plate Refuse to eat Turn head away Refuse to open mouth Spit out food Leave mouth open Refuse to swallow food
Types of factor analysis (http://www.stat-help.com/factor.pdf) Exploratory (EFA) principal axis factoring maximum likelihood factoring principal components analysis (PCA)* Confirmatory (CFA) structural equation modelling * - not strictly EFA
Correlation or covariance Variable X Variable Y
Correlation
Correlation and covariance correlation covariance
Factor analysis describes the variance in three or more variables Sources of variance: Common Specific Error
A Unique Variance Common Variance B Unique Variance Unique Variance C
Conducting factor analysis Computer software: EFA CFA SPSS SAS STATA AMOS (SPSS) SAS STATA EQS LISREL
Procedure for EFA Ferguson & Cox (1993): Stage 1 Pre-analysis checks Stage 2 Factor extraction Stage 3 Factor rotation
Stage 1 pre-analysis checks
Data Multivariate Random Normally distributed
Tests of suitability Kaiser-Meyer-Olkin (KMO) test of common variance (>0.5) Bartlett s test of sphericity test against an identity matrix (p<0.05)
Sample size Need more respondents than variables 5 10 respondents per variable Minimum sample size 100 200 respondents
What does Principal Components Analysis do? It uses a correlation matrix to search for covariance amongst items in the questionnaire.
Stage 2 factor extraction
Unrotated matrix
How many factors are present?
How many factors are present? Eigenvalues > 1 Scree slope Parallel analysis
Eigenvalues Eigenvalues are produced by a process called principal components analysis (PCA) and represent the variance accounted for by each underlying factor.
Scree slope The scree test was suggested by Raymond B. Cattell. In this method you plot the successive eigenvalues, and look for a spot in the plot where the plot abruptly levels out. Cattell named this test after the tapering "scree" or rockpile at the bottom of a landslide.
Parallel analysis Parallel analysis is a method for determining the number of components or factors to retain from PCA or factor analysis. Essentially, the program works by creating a random dataset with the same numbers of observations and variables as the original data. A correlation matrix is computed from the randomly generated dataset and then eigenvalues of the correlation matrix are computed. When the eigenvalues from the random data are larger then the eigenvalues from the pca or factor analysis you known that the components or factors are mostly random noise.
How many factors are present?
Stage 3 factor rotation
Which rotation? Oblimin? assumes factors are correlated Varimax? assumes factors are orthogonal
Interpreting a factor matrix Examine the loadings on the factors
Loadings
Interpreting a factor matrix Examine the loadings on the factors Select loadings > X to indicate putative factors But what does X =? X = 0.3 or 0.4 usually Whatever helps you to interpret the factor matrix But you must be consistent
Factor 1
Factor 1 4
BUT! AND
Cross-loadings What should we do with them? Cross-loading items detract from a simple structure : ie one where the loadings on putative factors are high (ie > X) and those on the remainder are low (ie < X) Cross-loadings should be removed BUT the remaining items MUST be re-rotated
Confirmatory factor analysis
EFA vs CFA Is EFA adequate and/or sufficient? CFA is very demanding: Item loading of unity on putative factors Zero loading of items on non-putative factors Rarely have the opportunity: costly on data
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EFA vs CFA EFA remains valuable and even essential in establishing latent structures CFA is an excellent tool if you have the luxury of multiple and large samples
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References Bryman A, Cramer D (2005) Quantitative data analysis with SPSS 17, 18 and 19: a guide for social scientists Routledge, London Watson R, Thompson DR (2006) Use of factor analysis in Journal of Advanced Nursing: literature review Journal of Advanced Nursing 55, 330-341 Byrne BM (2001) Structural equation modelling with AMOS: basic concepts, applications, and programming, 2ndEdition Laurence Erlbaum, London Child D (1990) The essentials of factor analysis 2nd edition Cassell, London DeCoster J (1998) Overview of factor analysis http://www.stathelp.com/notes.html; retrieved 1 September 2011 Ferguson E, Cox T (1993) Exploratory factor analysis: a user s guide International Journal of Selection and Assessment 1, 84-94 Hurley AE, Scandura TA, Schriesheim CA, Brannick MT, Seers A, Vandenberg RJ, Williams LJ (1997) Exploratory and confirmatory factor analysis: guidelines, issues, and alternatives Journal of Organizational Behavior 18, 667-683 Kline P (1994) An easy guide to factor analysis Routledge, London Marsh HW, Muthén B, Asparouhov T, Lüdtke O, Robitzsch A, Morin AJS, Trautwein U (2009) Exploratory structural equation modelling, intergrating CFA and EFA: application to students evaluations of university teaching Structural Equation Modelling 16, 439-476
email: r.watson@hull.ac.uk @rwatson1955