Annotated Output from IML Program for Computing Coarse Factor Score Estimates
based on 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         0.069  0.186
MEQ2         0.003  0.138
MEQ3         0.088  0.018
MEQ4         0.223  0.058
MEQ5         0.265 -0.055
MEQ6         0.055 -0.013
MEQ7         0.272 -0.005
MEQ8        -0.007  0.067
MEQ9         0.107  0.065
MEQ10       -0.020  0.114
MEQ11        0.054  0.092
MEQ12       -0.029  0.096
MEQ13        0.061  0.016
MEQ14        0.006  0.071
MEQ15        0.040  0.077
MEQ16       -0.001  0.040
MEQ17        0.050  0.122
MEQ18        0.024  0.125
MEQ19        0.138  0.322
 

A two-factor solution was requested on the "proc factor" command by the user. The factor score coefficients are printed first and can be examined quickly to identify items that contribute most directly to the creation of the factor score estimates. For example, examination of the first column (i.e., the first factor) reveals that items 4, 5, and 7 contribute the most the factor score estimates; whereas item 19 reveals the largest coefficient for the second factor. A graphing procedure is shown below to aid in the selection of items with the largest factor score coefficients.
 

LABEL
Ranked Factor Score Coefficients: Factor  1

ITEMR      RANKCOEF
MEQ7          0.272
MEQ5          0.265
MEQ4          0.223
MEQ19         0.138
MEQ9          0.107
MEQ3          0.088
MEQ1          0.069
MEQ13         0.061
MEQ6          0.055
MEQ11         0.054
MEQ17         0.050
MEQ15         0.040
MEQ18         0.024
MEQ14         0.006
MEQ2          0.003
MEQ16        -0.001
MEQ8         -0.007
MEQ10        -0.020
MEQ12        -0.029
 

The factor score coefficients for the first factor are ranked in descending order. These rankings can be examined to find a sufficient "break" in the sequence. For example, the difference between MEQ4 (.223) and MEQ19 (.138) is relatively substantial. The ranked coefficients are plotted below to aid in identifying a clear break.
 
 

                                           Ranked Factor Score Coefficients: Factor  1
       
       
   0.30
       
       
            MEQ7
                  MEQ5
       
   0.25
       
       
F                       MEQ4
a      
c      
t  0.20
o      
r      
       
S      
c      
o  0.15
r                             MEQ19
e      
       
C      
o                                   MEQ9
e  0.10
f                                         MEQ3
f      
i      
c                                               MEQ1
i                                                     MEQ13 MEQ6  MEQ11
e  0.05                                                                 MEQ17
n                                                                             MEQ15
t      
                                                                                    MEQ18
       
                                                                                          MEQ14
   0.00                                                                                         MEQ2  MEQ16
                                                                                                            MEQ8
                                                                                                                  MEQ10
       
                                                                                                                        MEQ12
       
  -0.05
       
       
       
                1     2     3     4     5     6     7     8     9    10    11    12    13    14    15    16    17    18    19

                                                                    Rank

In the graph above, it is clear that items 7, 5, and 4 should be included in the computation of the coarse factor scores. However, just like reading a scree plot a degree of subjectivity is involved. One might, for instance, reasonably include item 19, and perhaps 9 and 3, as well. If all of these items were chosen, the coarse factor score estimates would be computed by summing the responses to items 7, 5, 4, 19, 9, and 3 on the MEQ. Since items on the MEQ are on different scales (for example, some items range from 1 to 4 and others range from 0 to 6), they would be converted to z-scores before being summed. The coarse factor score estimates should be assessed using the IML program provided on the main page. In the exploratory stages of research several versions of the coarse factor scores could be tried and evaluated. One might try, for example, using only items 7, 5, and 4 above and compare the resulting factor score estimates to those computed from items 7, 5, 4, 19, 9, and 3. It may be the case that including items 19, 9, and 3 does very little to improve the coarse factor score estimates.

It should also be noted that item 12 (MEQ12) has the largest negative factor score coefficient. If a substantial "break" in the plot separated item 12 from the remaining items, it would be included in the computation of the coarse factor scores as well. Since the coefficient is negative, however, it would be subtracted from rather than added to the other items in the computation process. For example, the coarse factor score estimate would appear as MEQ7 + MEQ5 + MEQ4 - MEQ12. The main point to keep in mind is that the right side of the graph must also be examined for extreme, negative coefficients and substantial breaks.
 
 

LABEL
Ranked Factor Score Coefficients: Factor  2
 

ITEMR      RANKCOEF
MEQ19         0.322
MEQ1          0.186
MEQ2          0.138
MEQ18         0.125
MEQ17         0.122
MEQ10         0.114
MEQ12         0.096
MEQ11         0.092
MEQ15         0.077
MEQ14         0.071
MEQ8          0.067
MEQ9          0.065
MEQ4          0.058
MEQ16         0.040
MEQ3          0.018
MEQ13         0.016
MEQ7         -0.005
MEQ6         -0.013
MEQ5         -0.055
 
 

                                           Ranked Factor Score Coefficients: Factor  2
       
   0.35
       
       
            MEQ19
       
   0.30
       
       
       
       
F  0.25
a      
c      
t      
o      
r  0.20
                  MEQ1
S      
c      
o      
r  0.15
e                       MEQ2
       
C                             MEQ18 MEQ17
o                                         MEQ10
e  0.10                                         MEQ12
f                                                     MEQ11
f                                                           MEQ15
i                                                                 MEQ14 MEQ8  MEQ9
c                                                                                   MEQ4
i  0.05
e                                                                                         MEQ16
n      
t                                                                                               MEQ3  MEQ13
       
   0.00
                                                                                                            MEQ7  MEQ6
       
       
       
  -0.05
                                                                                                                        MEQ5
       
       
       
  -0.10
       
                1     2     3     4     5     6     7     8     9    10    11    12    13    14    15    16    17    18    19

                                                                    Rank
 
 

In the ranked coefficients above and accompanying graph it is clear that item 19 is key for computing coarse factor score estimates for the second factor. One might also reasonably include item 1 and possibly item 2. Item 5 reveals the most extreme negative factor score coefficient, but its absolute magnitude is less than half of item 2. Hence, although a subsantial break appears on the right side of the graph, no items will be subtracted when computing the coarse factor score estimates. As mentioned above, if one is in the early stages of analysis, several versions of scores can be computed and assessed using the evaluation program for coarse factor scores.
 

Standard Errors and t-values for Factor Score Ceofficients

ITEM        ERRORS           T_VALUES
MEQ1         0.023  0.024     3.032  7.660
MEQ2         0.021  0.023     0.148  6.124
MEQ3         0.020  0.022     4.309  0.827
MEQ4         0.024  0.025     9.400  2.288
MEQ5         0.023  0.024    11.494 -2.255
MEQ6         0.019  0.020     2.947 -0.675
MEQ7         0.024  0.025    11.449 -0.216
MEQ8         0.019  0.020    -0.346  3.325
MEQ9         0.021  0.023     4.967  2.853
MEQ10        0.020  0.021    -0.973  5.374
MEQ11        0.021  0.022     2.608  4.165
MEQ12        0.019  0.021    -1.509  4.679
MEQ13        0.019  0.021     3.122  0.792
MEQ14        0.019  0.020     0.337  3.458
MEQ15        0.020  0.022     1.959  3.562
MEQ16        0.019  0.020    -0.071  2.002
MEQ17        0.021  0.022     2.370  5.435
MEQ18        0.020  0.022     1.155  5.770
MEQ19        0.026  0.027     5.410 11.869

Standard errors and t-values for the factor score coefficients are listed above. A common criticism regarding the use of factor score coefficients for selecting items to include in coarse factor score estimates is the potential for wildly different standard errors that would obscure the interpretation of their relative magnitudes. The results for this data set, however, reveal that the standard errors within each factor are fairly homogenous, and the most extreme t-values correspond to the most extreme factor score coefficients. For example, items 7 (11.449), 5 (11.494), and 4 (9.400) have the most extreme t-values (listed in parentheses) for the first factor. Items 19 and 1 reveal the largest t-values for the second factor.

The t-values can also be tested for statistical significance with N - k - 1 degrees of freedom (N = sample size, k = number of items; df = 520 in this example). Using an unadjusted alpha of .05, the two-tailed critical value for the current example would be approximately 1.96. Using a Bonferroni adjustment (.05 / 19; considering each factor as a family) the two-tailed critical value would be approximately 3.02 The t-values above could be compared to one of these critical values and used to select the items that are included in the computation of the coarse factor score estimates. For instance, items 1, 3, 4, 5, 7, 9, 13, and 19 would be included in the computation of the coarse factor score estimates for the first factor based solely on the t-values compared to the 3.02 adjusted critical value.

Regardless of whether the t-values are compared to a critical value, the t-values and standard errors can be used in conjunction with the graphs above to select the needed items to include in the coarse factor score estimates.
 

Total Item Contribution to Squared Multiple Correlation

ITEM        CONTRIB
MEQ1         0.032  0.119
MEQ2         0.001  0.071
MEQ3         0.039  0.005
MEQ4         0.157  0.025
MEQ5         0.180 -0.011
MEQ6         0.015 -0.001
MEQ7         0.196 -0.002
MEQ8        -0.001  0.019
MEQ9         0.056  0.028
MEQ10       -0.002  0.048
MEQ11        0.021  0.043
MEQ12       -0.001  0.033
MEQ13        0.020  0.003
MEQ14        0.001  0.023
MEQ15        0.013  0.031
MEQ16       -0.000  0.008
MEQ17        0.020  0.064
MEQ18        0.007  0.064
MEQ19        0.080  0.245
.             .      .
TOTALS       0.835  0.814

The output concludes with the total contribution of each item to the squared multiple correlation for each factor. The factor score coefficients represent the direct contribution of each item to the squared multiple correlation; whereas the numbers in this final matrix include both direct and indirect effects. These values can consequently be examined to determine if additional items need to be included in the computation of the coarse factor score estimates. In this data set the largest values correspond to the largest factor score coefficients for each factor, and hence no additional items are deemed necessary.