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.).


Factor Score Coefficients for Items / Factors

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)

       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.

(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.