Factor Analytic Model: Explaining Correlations

The covariance/correlation for a pair of items under the single factor model

It is commonly stated that factor analysis explains/reproduces observed item correlations

Why and how?

The task: How can we re-express the factorial model as one concerning covariances/correlations between observed variables / items?

Side note: Although we actually seek expression for correlations, we traditionally use expressions for covariances of linear combinations

However, in EFA, all observed as well as latent variables are expressed as z-standardized variables (M = 0, SD = 1)

Because 𝐶𝑜v(𝑍!,𝑍")=𝑟!"(i.e.,the covariance of z-standardized variables is equal to the correlation between X and Y) we are, in fact, dealing with correlations (see the Appendix for an explanation)

Covariance of two linear combinations

Example: Unidimensional Model

Generalizing the result to k > 1 uncorrelated factors we get

Thus, the correlations are a function of the factor loadings

! Hence, factor analysis seeks to explain correlations among p ! observed variables with k common factors (whereby k < p)

—> If the model holds, the factors explain the phenomenon (i.e., correlations among items)

Correlations as ”phenomena”?

Roots of factor analysis: Spearman (1904): Trying to discover the hidden structure of human intelligence

Observation/phenomenon: Schoolchildren’s grades in different subjects were all positively correlated with each other

Hypothesis: Reason that grades in math, English, history, etc., are all correlated: Performance in these subjects is all correlated with a general/common factor, which he named “general intelligence”

—> Reproducing / explaining the phenomenon of correlations with a single-factor model

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