# Exploratory factor analysis/Glossary

Exploratory Factor Analysis: Glossary

## Basic terms

1. Anti-image correlation matrix: Contains the negative partial covariances and correlations. Diagonals are used as a measure of sampling adequacy (MSA). Note: Be careful not to confuse this with the anti-image covariance matrix.
2. Bartlett's test of sphericity: Statistical test for the overall significance of all correlations within a correlation matrix. Used as a measure of sampling adequacy (MSA).
3. Common variance: Variance in a variable that is shared with other variables.
4. Communality: The proportion of a variable's variance explained by the extracted factor structure. Final communality estimates are the sum of squared loadings for a variable in an orthogonal factor matrix.
5. Complex variable: A variable which has notable loadings (e.g., > .4) on two or more factors.
6. Correlation: The Pearson or product-moment correlation coefficient.
7. Composite score: A variable which represents combined responses to multiple other variables. A composite score can be created as unit-weighted or regression-weighted. A composite score is created for each case for each factor.
8. Correlation matrix: A table showing the linear correlations between all pairs of variables.
9. Data reduction: Reducing the number of variables (e.g., by using factor analysis to determine a smaller number of factors to represent a larger set of factors).
10. Eigen Value: Column sum of squared loadings for a factor. Represents the variance in the variables which is accounted for by a specific factor.
11. Exploratory factor analysis: A factor analysis technique used to explore the underlying structure of a collection of observed variables.
12. Extraction: The process for determining the number of factors to retain.
13. Factor: Linear combination of the original variables. Factors represent the underlying dimensions (constructs) that summarise or account for the original set of observed variables.
14. Factor analysis: A statistical technique used to estimate factors and/or reduce the dimensionality of a large number of variables to a fewer number of factors.
15. Factor loading: Correlation between a variable and a factor, and the key to understanding the nature of a particular factor. Squared factor loadings indicate what percentage of the variance in an original variable is explained by a factor.
16. Factor matrix: Table displaying the factor loadings of all variables on each factor. Factors are presented as columns and the variables are presented as rows.
17. Factor rotation: A process of adjusting the factor axes to achieve a simpler and pragmatically more meaningful factor solution - the goal is a usually a simple factor structure.
18. Factor score: Composite score created for each observation (case) for each factor which uses factor weights in conjunction with the original variable values to calculate each observation's score. Factor scores are standardised to according to a z-score.
19. Measure of sampling adequacy (MSA): Measures which indicate the appropriateness of applying factor analysis.
20. Oblique factor rotation: Factor rotation such that the extracted factors are correlated. Rather than arbitrarily constraining the factor rotation to an orthogonal (90 degree angle), the oblique solution allows the factors to be correlated. In SPSS, this is called Oblimin rotation.
21. Orthogonal factor rotation: Factor rotation such that their axes are maintained at 90 degrees. Each factor is independent of, or orthogonal to, all other factors. In SPSS, this is called Varimax rotation.
22. Parsimony principle: When two or more theories explain the data equally well, select the simplest theory e.g., if a 2-factor and a 3-factor model explain about the same amount of variance, interpret the 2-factor model.
23. Principal axis factoring (PAF): A method of factor analysis in which the factors are based on a reduced correlation matrix using a priori communality estimates. That is, communalities are inserted in the diagonal of the correlation matrix, and the extracted factors are based only on the common variance, with unique variance excluded.
24. Principal component analysis (PC or PCA): The factors are based on the total variance of all items. [1][2]
25. Scree plot: A line graph of Eigen Values which is helpful for determining the number of factors. The Eigen Values are plotted in descending order. The number of factors is chosen where the plot levels off (or drops) from cliff to scree.
26. Simple structure: A pattern of factor loading results such that each variable loads highly onto one and only one factor.
27. Unique variance: The proportion of a variable's variance that is not shared with a factor structure. Unique variance is composed of specific and error variance.

## Other

1. Common factor: A factor on which two or more variables load.
2. Common factor analysis: A statistical technique which uses the correlations between observed variables to estimate common factors and the structural relationships linking factors to observed variables.
3. Error variance: Unreliable and inexplicable variation in a variable. Error variance is assumed to be independent of common variance, and a component of the unique variance of a variable.
4. Image of a variable: The component of a variable which is predicted from other variables. Antonym: anti-image of a variable.
5. Indeterminacy: If it is impossible to estimate population factor structures exactly because an infinite number of factor structures can produce the same correlation matrix, then there are more unknowns than equations in the common factor model, and we say that the factor structure is indeterminate.
6. Latent factor: A theoretical underlying factor hypothesised to influence a number of observed variables. Common factor analysis assumes latent variables are linearly related to observed variables.
7. Specific variance: (1) Variance of each variable unique to that variable and not explained or associated with other variables in the factor analysis. [3] (2) The component of unique variance which is reliable but not explained by common factors. [4]