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Background—617
There are a number of other measures of absolute fit. We refer you to Hu and Bentler (1995,
1999) and Browne and Cudeck (1993), McDonald and Marsh (1990), Marsh, Balla and
McDonald (1988) for details on these measures and recommendations on their use. Note
that where there are small differences in the various descriptions of the measures due to
degree-of-freedom corrections, we have used the formulae provided by Hu and Bentler
(1999).
Incremental Fit
Incremental fit indices measure the improvement in fit of the model over a more restricted
specification. Typically, the restricted specification is chosen to be the zero factor or independence model.
EViews reports up to five relative fit measures: the generalized Tucker-Lewis Nonnormed Fit
Index (NNFI), Bentler and Bonnet’s Normed Fit Index (NFI), Bollen’s Relative Fit Index
(RFI), Bollen’s Incremental Fit Index (IFI), and Bentler’s Comparative Fit Index (CFI). See
Hu and Bentler (1995)for details.
Traditionally, the rule of thumb was for acceptable models to have fit indices that exceed
0.90, but recent evidence suggests that this cutoff criterion may be inadequate. Hu and
Bentler (1999) provide some guidelines for evaluating values of the indices; for ML estimation, they recommend use of two indices, with cutoff values close to 0.95 for the NNFI, RFI,
IFI, CFI.
Rotation
The estimated loadings and factors are not unique; we may obtain others that fit the
observed covariance structure identically. This observation lies behind the notion of factor
rotation, in which we apply transformation matrices to the original factors and loadings in
the hope of obtaining a simpler factor structure.
To elaborate, we begin with the orthogonal factor model from above:
X i – m = LF i + e i
(40.8)
where E ( F i F i ¢ ) = I m . Suppose that we pre-multiply our factors by a m ¥ m rotation
matrix T¢ where T¢T = F . Then we may re-write the factor model Equation (40.1) as:
–1
X i – m = L ( T )¢T¢F i + e i = L̃F̃ i + e i
(40.9)
which is an observationally equivalent common factor model with rotated loadings
–1
L̃ = L ( T )¢ and factors F̃ i = T¢F i , where the correlation of the rotated factors is given
by:
E ( F̃ i F̃ i ¢ ) = T¢T = F
See Browne (2001) and Bernaards and Jennrich (2005) for details.
(40.10)