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.).


Factor Score Coefficients for Items / Factors

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)

       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.

(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.