Factor analysis Advanced Quantitative Research Methods SongYi (Grace) Lee School of Media and Communication Temple University

Content What is Factor analysis (FA)? Taxonomy of FA Equation Factor loading / Factor score Factor extraction Factor rotation

What is factor analysis? Factor analysis (FA)? PCA (Principal component analysis)? FA and PCA both aim to reduce a set of variables into a smaller subsets of factors (PCA calls them components). FA; Common variables; Smaller number of constructs; Parsimony PCA; maximum of total variance; linear components “FA derives a mathematical model from which factors are estimated, whereas PCA decomposes original data into a set of linear variates” (Field, 2013, p. 796) Variables? Interval/Ratio

Latent variables Latent variables refers to variables that cannot be directly measured. When variables consist of a cluster of explanatory construct, it is called latent variable, alternatively, latent variable. (e.g., What latent variables can you assume from the scores of mathematics, physics, and chemistry?)

Terms of Factor analysis Factor: Explanatory constructs that consist of variables. Factor loading: Coefficient of variables and factor. Factor matrix: A matrix that shows correlation coefficient which allows us to match variables with higher coefficients. Figure 17.2 from Field(2013) An R-Matrix

Terms continue.. Communality: Percentage of variance of a variable which is accounted for by a factor. (e.g., Variable with no unique/random variance = 1; variable that shares none of its variance (Unique variance)= 0) It is calculated by sum of squares of factor loading Eigenvalue: The sum of variance of variables that are accounted for by a factor.

Factor analysis equation Linear model 𝑦=𝑎+ 𝑏 1 𝑥 1𝑖 + 𝑏 2 𝑥 2𝑖 +…+ 𝑏 𝑛 𝑥 𝑛𝑖 Factor analysis 𝑌 𝑖 = 𝑏 1 𝑋 𝑖 + 𝑏 2 𝑋 2𝑖 +…+ 𝑏 𝑛 𝑋 𝑛𝑖 Variable with factors

Factor Loading Factor loading refers to the coefficient of variables and factor. There are two factors; consideration, and sociability. The equations for the factors would look like these. Figure 17.3 from the textbook (Field, 2013)

Factor loading vs. Factor score Factor score is calculated by each score that individuals gain from the measurement and factor loadings for each variable. This method is known as “weighted average”. Rarely used, because it is simplistic. However it is the easiest way to explain the principles of factor analysis. Factor score is using factor loading as coefficient, and scores from the measurement are multiplied by the factor loadings. Regression (general linear model) has issues of ### so in order to alleviate those issues, there are two adjustments; Bartlett’s test & Anderson-Rubin method

Correlation between variables Bartlett’s test Unbiased factor score Anderson-Rubin method Best method when uncorrelated scores are required. Modification of Bartlett’s test. No correlation between predictors – avoiding multi colinearity

Factor Extraction Factor extraction means deciding the number of factors you would keep. Eigenvalues/ Scree plot Eigenvalues associated with a variate indicating the substantive importance of that factor ∴ Retain only factors with higher eigenvalues. Scree plot is used to determine whether the eigenvalue is large enough or not.

Eigenvalue with a scree plot The closer the communalities are to 1, the better our factors are at explaining the original data. The more factors retained, the greater the communalities will be (Field, 2013, p.799).

Factor rotation Factor rotation is performed in order to improve the interpretation of data. Orthogonal (When factors are independent) Varimax (Widely used) Quartimax Equamax Oblique (When factors are correlated) Direct Oblimin (Widely used) Promax

Example Let’s do some SPSS.!