Annotated Example of Output for Evaluating Factor Indeterminacy

and Factor Score Estimates Computed using Correlation-Preserving Approach

(annotation appears in red)

The novel output generated by the IML program is printed after the results for the factor analysis (e.g., the scree plot, eigenvalues, pattern coefficients, etc.).

`**** BEGIN OUTPUT FROM PROC IML ****`

`Factor Score Coefficients for Items / Factors`

`ITEM FSCOEF`
`MEQ1 0.091
0.147 0.194`
`MEQ2 0.043 -0.051
0.427`
`MEQ3 0.131 -0.033
0.024`
`MEQ4 0.280
0.067 0.002`
`MEQ5 0.257
0.035 -0.102`
`MEQ6 0.091 -0.056
0.010`
`MEQ7 0.314
0.000 0.004`
`MEQ8 -0.003 -0.011
0.139`
`MEQ9 0.123
0.086 0.021`
`MEQ10 -0.044 0.035
0.234`
`MEQ11 0.022 0.264
-0.043`
`MEQ12 -0.062 0.071
0.124`
`MEQ13 0.092 -0.001 -0.002`
`MEQ14 0.019 -0.029
0.159`
`MEQ15 0.018 0.191
-0.054`
`MEQ16 -0.051 0.100 -0.005`
`MEQ17 0.015 0.218
0.028`
`MEQ18 -0.007 0.136
0.107`
`MEQ19 0.112 0.345
0.253`

`The factor score coefficients above are printed
first. These coefficients are the weights applied to each item in the computation
of the factor score estimates. These particular weights are from the correlation-preserving
strategy (as requested) and hence will NOT match the factor score coefficients
printed by the "proc factor" SAS command, which uses the regression strategy
for computing factor score estimates.`

`These coefficients can be examined for their
relative magnitudes per factor. For example, scanning down the first column
(i.e., the first factor), it is clear that items 3, 4, 5, 7, 9, and 19
contribute most to the computation of the factor score estimates; whereas
items 1, 11, 15, 16, 17, 18, and 19 are relatively large for the second
factor.`

`Indeterminacy / Determinacy Indices`
`(Multiple R, R-Squared, and Minimum Correlation)`

` FACTOR MULTR
RSQR MINCOR`
` 1
0.905 0.819 0.637`
` 2
0.893 0.798 0.595`
` 3
0.864 0.747 0.493`

`The indeterminacy indices are reported for
each of the three factors. As shown above, the multiple correlations (MULTR)
between the factors and items are all fairly high. These values can range
from 0 to 1.0, and values approaching 1.0 are desirable. Some authors have
suggested that the MULTR values should be substantially higher than .707
which, when squared, would equal .50. The squared multiple correlations
(RSQR) are in fact printed next to the MULTR values and, as shown, none
are close to .50. The minimum correlation (MINCOR) values are printed in
the right most column and indicate the minimum correlation that could be
obtained between two sets of equally valid factor scores for each factor.
MINCOR values approaching zero are distressing, and negative values are
disastrous. High values that approach 1.0 indicate that the factors may
be slightly indeterminate, but the infinite sets of factor scores that
could be computed will yield highly similar rankings of the individuals.
In other words, the practical impact of the indeterminacy is minimal.`

`Validity Coefficients`

` FACTOR VALID
MULTR`
` 1
0.903 compare to MULTR -->
0.905`
` 2
0.888
0.893`
` 3
0.860
0.864`

`The validity coefficients above provide information
regarding the factor score estimates, which have been computed using the
correlation-preserving approach in this example. The MULTR values represent
the maximum correlation between the factor score estimates and the factors
themselves. The VALID values represent the actual correlations between
the factor score estimates and the factors themselves. As can be seen above,
the factor score estimates lost very little in the way of validity. The
largest of the small differences is for the second factor: .888 compared
to .893. Overall then, the MULTR values (one of the determinacy indices)
are high and the factor score estimates are fairly valid representations
of the factors.`

`Univocality`
`(Rows = Factor Scores / Columns = Factors)`

` UNIV
FACTCOR`
` . 0.603 0.391
compare to FACTCOR --> .
0.616 0.393`
` 0.603 . 0.587
0.616 . 0.606`
` 0.391 0.587 .
0.393 0.606 .`

`The Univocality information indicates the
extent to which the factor score estimates are insufficiently or too highly
correlated with other factors in the analysis. The rows of
the above matrices represent the factor score estimates and the columns
represent the factors. A desirable outcome is obtained when the two matrices
above match perfectly. In this analysis the values in the two matrices
are highly similar. For example, the correlation between the second and
third factors (see the FACTCOR matrix) is .606, which compares favorably
to the correlation between the factor score estimates for the second factor
and the third factor itself (r = .587 in the UNIV matrix). The rationale
behind these values is as follows: if the second and third factors are
correlated .606, then the factor score estimates for the second factor
ought to correlate .606 with the third factor itself. If this latter correlation
were found to be .10, for instance, the factor score estimates for the
second factor would be insufficiently related to the third factor. If the
same correlation were found to be .90, then the factor score estimates
for the second factor would be overly contaminated with variance from the
third factor.`

`Correlational Accuracy`

`SCORECOR
FACTCOR`
` 1.000 . .
compare to FACTCOR --> 1.000
. .`
` 0.616 1.000 .
0.616 1.000 .`
` 0.393 0.606 1.000
0.393 0.606 1.000`

`The Correlational Accuracy information is
easier to comprehend than the Univocality criteria above. The FACTCOR matrix
again represents the correlations among the factors themselves, and the
SCORECOR matrix represents the correlations among the factor score estimates.
Clearly, if the factor score estimates are adequate representations of
the factors, the values in the two matrices should match. As shown above,
the values are matched perfectly because the correlation-preserving method
for computing the factor score estimates was chosen. In a sense, the slight
loss in validity discussed above was used to purchase the perfect match
between the two corelation matrices here.`