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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