Factor analysis

unidimensional case

Graphic

=> In FA, indicators have an individual error term (due to measurement error and uniqueness [see next slides])

Shared / non-shared variance

graphical

Uniqueness and shared variance

Overlapping area among all variables: Variance due to the common factor

Nonoverlapping area for any variable: Variance that is unique to that particular variable: Uniqueness

Consists of (1) Specificity, a true (i.e., reliable) source of variance that is not common to any other variable (e.g., writing proficiency performance affected by different genres: argumentative vs. narrative writing); and (2) random error of measurement, i.e. unreliability (e).

Definition communality

Total amount of systematic variance of a variable, called communality (h2)

= Common factor variance + group factor variance

Variance decomposition of a unifactorial EFA at a glance

Common factor model, unifactorial case

Factor loadings

Factor Analytic Model: Multiple Factor Analysis

Choices and assumptions concerning the distributions of observed and factor variables (orthogonal factor analysis)

(6)

All of the observable random variables Zj have mean zero and variance 1, i.e., are z-standardized variables

All of the latent factors have mean zero and variance 1, i.e., are z-standardized variables

All error terms have mean zero (linearity assumption)

The factors are uncorrelated (orthogonal model)

The error terms are uncorrelated across observable variables

(i.e., assumption of random errors)

The error terms are uncorrelated with the factor variables

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