Standard Exploratory Factor Analysis Model or EFA

Specification

• every measure loads on each factor
• factors either uncorrelated (orthogonal) or correlated (oblique)
• generally factors are uncorrelated

Because the model is not unique (i.e., underidentified) it can be rotated

To test if k factors are sufficient to explain the covariation between measures estimate the following loading matrix (assuming k = 5) with orthogonal factors with unit variance:

Measure      1     2     3     4     5

If such a model fits, then k factors are sufficient.

EFA is useful when the researcher does not know how many factors there are or when it is uncertain what measures load on what factors.

• Each variable loads on one and only one factor
• Factors are correlated (conceptually useful to have correlated factors)
• Errors across indicators are independent

• Given k factors, must be k squared constraints
• Usually k of these constraints are scaling ones

Principle of nesting:  two models, one a more complicated version of another model (e.g., a one-factor model is nested within a two-factor as a one-factor model can be viewed as a two-factor model in which the correlation between factors is perfect.)

• Relative fit of a nested model: the chi square difference test, the smaller chi square and its degrees of freedom are subtracted from the larger chi square and degrees of freedom
• In principle, the more complicated model should fit for the test to be valid.

Definition of poor discriminant validity: The correlation between two factors is or is very close to one or minus one.

• Consequences
• multicollinearity
• problems of convergence
• Criteria: .85 correlation or larger indicates poor discriminant validity
• Test: estimate a model that fixes the correlation to one or collapse the two factors and see if the fit worsens

• Criteria
• Empirical
• modification index (Lagrangian multiplier in EQS)
• standardized residual
•  Theoretical
• all respecifications require some rationale
• that rationale should be extended to other cases
• More complex models
• Another factor
• Correlated errors
• definition:  Variance not explained by theoretical constructs may covary across two measures.  Such covariance is called correlated errors.
• points to remember
• Note that all of the unexplained variance is not error in the sense of random, meaningless variance.  Some of the variance is meaningful, but not due to the specified latent variables.
• Sometimes correlated errors are negative meaning that the unspecified common cause of the two indicators has effects in different directions.
• conditions for correlating errors
• Meaningfulness Rule:  only correlate errors for which there is a theoretical rationale
• Transitivity Rule: if A's errors are correlated with B's error and B's errors are correlated with C's errors then you should probably correlate A's errors with C's
• Generality Rule:  if you correlate A's and B's errors for a reason, you should correlate all other pairs of errors for which that reason applies
• Cross-variable loadings (one variable loading on two factors)
• Simpler models
• Fewer factors
• Equal loadings (should be done using the covariance matrix)

How many indicators per factor?

• 2 is the minimum
• 3 is safer, especially if factor correlations are weak
• 4 provides safety
• 5 or more is more than enough (If too many indicators then combine indicators into sets)

Perfect Indicators (measures with no measurement error)

fix loading to one

free variance

fix error variance to zero

do not correlate its "error variance" with anything 

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