Confirmatory Factor Analysis:
Evaluation
Psychology 588: Covariance structure and factor models
Feb 24, 2010

•
Evaluating a model is equivalent to testing a hypothesis of a set
of constraints (that make
whether explicit or implicit),
provided that all other assumptions fulfilled
---
similar to the
logic behind null hypothesis testing
•
All overall fits quantify
whether they are statistical or
standardized (i.e., bounded by [0,1])
•
Overall fit indicates goodness (or badness) of the whole model,
summarizing
into a scalar value
Individual fits tell goodness of fit to particular manifest DVs (as
indicated by
R
2
or SMC) and standard error of parameters
indicate how reliable parameter estimates are
---
both overall
and individual fits should be evaluated!
Evaluation
2
ˆ
,
−
S
Σ
ˆ
≠
Σ
S
ˆ
−
S
Σ

•
Residuals
indicate how the specified model predicts
individual covariances
---
note that the variances
s
ii
have no
residual in recursive models
•
Sources of residuals:
¾
Σ
≠
Σ
(
θ
),
which we want to know by the residuals
¾
sampling fluctuation
---
with large
N
, smaller residuals are
expected if the model is correct
** scale of observed variables determines the size of residuals
---
if correlations are analyzed, residuals would not have
scale dependency, varying [-2,2]
Residuals
3
ˆ
ij
ij
s
σ
−