Displayed Output
PROC FACTOR output includes
- Mean and Std Dev (standard deviation) of each variable
and the number of observations, if you specify the SIMPLE option
- Correlations, if you specify the CORR option
- Inverse Correlation Matrix, if you specify the ALL option
- Partial Correlations Controlling all other Variables
(negative anti-image correlations), if you specify the MSA option.
If the data are appropriate for the common factor model,
the partial correlations should be small.
- Kaiser's Measure of Sampling Adequacy (Kaiser 1970;
Kaiser and Rice 1974; Cerny and Kaiser 1977)
both overall and for each variable, if you specify the
MSA option.
The MSA is a summary of how small the partial
correlations are relative to the ordinary correlations.
Values greater than 0.8 can be considered good.
Values less than 0.5 require remedial action,
either by deleting the offending variables or by
including other variables related to the offenders.
- Prior Communality Estimates, unless 1.0s are used
or unless you specify the METHOD=IMAGE, METHOD=HARRIS,
METHOD=PATTERN, or METHOD=SCORE option
- Squared Multiple Correlations of each variable with all
the other variables, if you specify the METHOD=IMAGE or
METHOD=HARRIS option
- Image Coefficients, if you specify the METHOD=IMAGE option
- Image Covariance Matrix, if you specify the METHOD=IMAGE option
- Preliminary Eigenvalues based on the prior communalities,
if you specify the METHOD=PRINIT, METHOD=ALPHA, METHOD=ML, or
METHOD=ULS option.
The table produced includes
the Total
and the Average of the eigenvalues, the Difference between
successive eigenvalues, the Proportion of variation
represented, and the Cumulative proportion of variation.
- the number of factors that are retained,
unless you specify the METHOD=PATTERN or METHOD=SCORE option
- the Scree Plot of Eigenvalues, if you specify the SCREE option.
The preliminary eigenvalues are used
if you specify the METHOD=PRINIT, METHOD=ALPHA, METHOD=ML, or
METHOD=ULS option.
- the iteration history, if you specify the
METHOD=PRINIT, METHOD=ALPHA, METHOD=ML, or
METHOD=ULS option. The table produced contains
the iteration number (Iter); the Criterion
being optimized (Joreskog 1977); the Ridge value for the
iteration if you specify the METHOD=ML or METHOD=ULS option;
the maximum Change in any
communality estimate; and the Communalities
- Significance tests, if you specify the option METHOD=ML, including
Bartlett's Chi-square, df, and Prob
for H0: No common factors and H0: factors
retained are sufficient to explain the correlations.
The variables should have an approximate multivariate
normal distribution for the probability levels to be valid.
Lawley and Maxwell (1971) suggest that the number of
observations should exceed the number of variables by
fifty or more, although Geweke and Singleton (1980)
claim that as few as ten observations are adequate
with five variables and one common factor.
Certain regularity conditions must also be
satisfied for Bartlett's test to be
valid (Geweke and Singleton 1980), but in
practice these conditions usually are satisfied.
The notation Prob>chi**2 means "the probability
under the null hypothesis of obtaining a greater
statistic than that observed."
The Chi-square value is displayed with
and without Bartlett's correction.
- Akaike's Information Criterion, if you specify the METHOD=ML option.
Akaike's information criterion (AIC) (Akaike 1973,
1974, 1987) is a general criterion for estimating
the best number of parameters to include in a
model when maximum-likelihood estimation is used.
The number of factors that yields the
smallest value of AIC is considered best.
Like the chi-square test, AIC tends to include
factors that are statistically significant but
inconsequential for practical purposes.
- Schwarz's Bayesian Criterion, if you specify the METHOD=ML option.
Schwarz's Bayesian Criterion (SBC) (Schwarz
1978) is another criterion, similar to AIC,
for determining the best number of parameters.
The number of factors that yields the
smallest value of SBC is considered best;
SBC seems to be less inclined to include trivial
factors than either AIC or the chi-square test.
- Tucker and Lewis's Reliability Coefficient, if you specify the
METHOD=ML option
(Tucker and Lewis 1973)
- Squared Canonical Correlations, if you specify the METHOD=ML option.
These are the same as the squared multiple correlations
for predicting each factor from the variables.
- Coefficient Alpha for Each Factor, if you specify the
METHOD=ALPHA option
- Eigenvectors, if you specify the EIGENVECTORS or ALL option,
unless you also specify the METHOD=PATTERN or METHOD=SCORE option
- Eigenvalues of the (Weighted) (Reduced) (Image) Correlation
or Covariance Matrix, unless you specify the METHOD=PATTERN or
METHOD=SCORE option.
Included are the Total and the Average of the
eigenvalues, the Difference between successive
eigenvalues, the Proportion of variation represented,
and the Cumulative proportion of variation.
- the Factor Pattern, which is equal to both the matrix
of standardized regression coefficients for predicting
variables from common factors and the matrix of
correlations between variables and common factors
since the extracted factors are uncorrelated
- Variance explained by each factor, both Weighted
and Unweighted, if variable weights are used
- Final Communality Estimates, including the Total
communality; or Final Communality Estimates and Variable
Weights, including the Total communality, both Weighted
and Unweighted, if variable weights are used.
Final communality estimates are the squared multiple
correlations for predicting the variables from the
estimated factors, and they can be obtained by taking the
sum of squares of each row of the factor pattern, or a
weighted sum of squares if variable weights are used.
- Residual Correlations with Uniqueness on the Diagonal,
if you specify the RESIDUAL or ALL option
- Root Mean Square Off-diagonal Residuals, both Over-all
and for each variable, if you specify the RESIDUAL or ALL option
- Partial Correlations Controlling Factors,
if you specify the RESIDUAL or ALL option
- Root Mean Square Off-diagonal Partials, both Over-all
and for each variable, if you specify the RESIDUAL or ALL option
- Plots of Factor Pattern for unrotated factors,
if you specify the PREPLOT option. The number of plots
is determined by the NPLOT= option.
- Variable Weights for Rotation, if you specify the NORM=WEIGHT option
- Factor Weights for Rotation, if you specify the
HKPOWER= option
- Orthogonal Transformation Matrix, if you request
an orthogonal rotation
- Rotated Factor Pattern, if you request an
orthogonal rotation
- Variance explained by each factor after rotation.
If you request an orthogonal rotation and
if variable weights are used, both weighted
and unweighted values are produced.
- Target Matrix for Procrustean Transformation,
if you specify the ROTATE=PROCRUSTES or ROTATE=PROMAX option
- the Procrustean Transformation Matrix,
if you specify the ROTATE=PROCRUSTES or ROTATE=PROMAX option
- the Normalized Oblique Transformation Matrix, if you request an oblique
rotation, which, for the option ROTATE=PROMAX, is the
product of the prerotation and the Procrustean rotation
- Inter-factor Correlations, if you specify an
oblique rotation
- Rotated Factor Pattern (Std Reg Coefs), if you specify an oblique
rotation, giving standardized regression
coefficients for predicting the variables from the factors
- Reference Axis Correlations if you specify
an oblique rotation.
These are the partial correlations between the
primary factors when all factors other than
the two being correlated are partialled out.
- Reference Structure (Semipartial Correlations),
if you request an oblique rotation.
The reference structure is the matrix of semipartial
correlations (Kerlinger and Pedhazur 1973) between
variables and common factors, removing from each
common factor the effects of other common factors.
If the common factors are uncorrelated, the reference
structure is equal to the factor pattern.
- Variance explained by each factor eliminating the effects
of all other factors, if you specify an oblique rotation.
Both Weighted and Unweighted values
are produced if variable weights are used.
These variances are equal to the (weighted)
sum of the squared elements of the reference
structure corresponding to each factor.
- Factor Structure (Correlations), if you request
an oblique rotation.
The (primary) factor structure is the matrix of
correlations between variables and common factors.
If the common factors are uncorrelated, the factor
structure is equal to the factor pattern.
- Variance explained by each factor ignoring the effects
of all other factors, if you request an oblique rotation.
Both Weighted and Unweighted values
are produced if variable weights are used.
These variances are equal to the (weighted)
sum of the squared elements of the factor
structure corresponding to each factor.
- Final Communality Estimates for the rotated
factors if you specify the ROTATE= option.
The estimates should equal the unrotated communalities.
- Squared Multiple Correlations of the Variables with
Each Factor, if you specify the SCORE or ALL option, except
for unrotated principal components
- Standardized Scoring Coefficients, if you specify the
SCORE or ALL option
- Plots of the Factor Pattern for rotated factors, if you specify
the PLOT option and you request an orthogonal rotation.
The number of plots is determined by the NPLOT= option.
- Plots of the Reference Structure for rotated factors, if you
specify the PLOT option and you request an oblique rotation.
The number of plots is determined by the NPLOT= option.
Included are the Reference Axis Correlation and the Angle
between the Reference Axes for each pair of factors plotted.
If you specify the ROTATE=PROMAX option, the output includes results
for both the prerotation and the Procrustean rotation.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.