Annotated Example of Output for Evaluating Factor Indeterminacy

and Factor Score Estimates Computed using Thurstone's Regression 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 0.063
0.142 0.175`
`MEQ2 0.026
0.004 0.335`
`MEQ3 0.100
0.005 0.031`
`MEQ4 0.223
0.091 0.022`
`MEQ5 0.266
0.024 -0.084`
`MEQ6 0.067 -0.030
0.017`
`MEQ7 0.309
0.016 0.001`
`MEQ8 0.003
0.002 0.112`
`MEQ9 0.099
0.087 0.034`
`MEQ10 -0.013 0.027
0.177`
`MEQ11 0.031 0.181
0.001`
`MEQ12 -0.031 0.052
0.099`
`MEQ13 0.062 0.023
0.008`
`MEQ14 0.019 -0.003
0.135`
`MEQ15 0.015 0.160
-0.010`
`MEQ16 -0.015 0.070 -0.006`
`MEQ17 0.027 0.185
0.038`
`MEQ18 0.012 0.115
0.090`
`MEQ19 0.110 0.298
0.225`

`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 regression
method (as requested) and will match the factor score coefficients printed
by the "proc factor" SAS command if they are requested with the "score"
keyword. If other types of factor score estimates are computed with the
IML programs, the above coefficients will not match those printed by "proc
factor."`

`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, 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
0.908 0.825 0.650`
` 2
0.895 0.802 0.603`
` 3
0.864 0.747 0.495`

`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
0.908 compare to MULTR -->
0.908`
` 2
0.895
0.895`
` 3
0.864
0.864`

`The validity coefficients above provide information
regarding the factor score estimates, which have been computed using Thurstone's
regression approach in the current 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. In this case, the
two sets of values are identical because the regression factor score estimates
always maximize validity. Other types of factor score estimates (for example,
the Bartlett factor scores) will not maximize validity and must hence be
compared to the MULTR values in order to assess the relative loss in validity
for the estimates for each factor.`

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

` UNIV
FACTCOR`
` . 0.647 0.438
compare to FACTCOR --> .
0.612 0.389`
` 0.638 . 0.661
0.612 . 0.618`
` 0.416 0.638 .
0.389 0.618 .`

`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 fairly similar. For example, the correlation between the first and
second factor (see the FACTCOR matrix) is .612, which compares favorably
to the correlation between the factor score estimates for the first factor
and the second factor itself (r = .647 in the UNIV matrix). The rationale
behind these values is as follows: if the first and second factor are correlated
.612, then the factor score estimates for the first factor ought to correlate
.612 with the second factor itself. If this latter correlation were found
to be .03, for instance, the factor score estimates for the first factor
would be insufficiently related to the second factor. If the same correlation
were found to be .99, then the factor score estimates for the first factor
would be overly contaminated by the second factor.`

`Correlational Accuracy`

`SCORECOR
FACTCOR`
` 1.000 . .
compare to FACTCOR --> 1.000
. .`
` 0.712 1.000 .
0.612 1.000 .`
` 0.482 0.738 1.000
0.389 0.618 1.000`

`The Correlational Accuracy information is
easier to comprehend than the Univocality criteria above. The FACTCOR matrix
again reports the correlations among the factors themselves, and the SCORECOR
matrix reports 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
do match fairly well, although the factor score estimates are generally
more highly correlated than the factors themselves.`