Example 2.1: Using Confidence Intervals to Locate Salient Factor Loadings

The FACTOR Procedure

Example 2.1: Using Confidence Intervals to Locate Salient Factor Loadings

This example illustrates how you can utilize the standard errors and confidence intervals to understand the pattern of factor loadings under the maximum likelihood estimation. There are nine tests and you want a three-factor solution for a correlation matrix based on 200 observations. You apply quartimin rotation with (default) Kaiser normalization. You define loadings with magnitudes greater than 0.45 to be salient and use 90% confidence intervals to judge the salience:

   data test(type=corr);
      title 'Quartimin-Rotated Factor Solution with Standard Errors'; 
      input _name_ $ test1-test9; 
      _type_ = 'corr';
   datalines;
   Test1      1  .561  .602  .290  .404  .328  .367  .179 -.268
   Test2   .561     1  .743  .414  .526  .442  .523  .289 -.399
   Test3   .602  .743     1  .286  .343  .361  .679  .456 -.532
   Test4   .290  .414  .286     1  .677  .446  .412  .400 -.491
   Test5   .404  .526  .343  .677     1  .584  .408  .299 -.466
   Test6   .328  .442  .361  .446  .584     1  .333  .178 -.306
   Test7   .367  .523  .679  .412  .408  .333     1  .711 -.760
   Test8   .179  .289  .456  .400  .299  .178  .711     1 -.725
   Test9  -.268 -.399 -.532 -.491 -.466 -.306 -.760 -.725     1
   ;
   proc factor data=test method=ml reorder rotate=quartimin
      nobs=200 n=3 se cover=.45 alpha=.1;
      title2 'A nine-variable-three-factor example';
   run;

Output 2.1.1: Quartimin-Rotated Factor Solution with Standard Errors

Quartimin-Rotated Factor Solution with Standard Errors

A nine-variable-three-factor example

The FACTOR Procedure

Rotation Method: Quartimin

Inter-Factor Correlations With 90% confidence limits Estimate/StdErr/LowerCL/UpperCL
	Factor1	Factor2	Factor3
Factor1	1.00000 0.00000 . .	0.41283 0.06267 0.30475 0.51041	0.38304 0.06060 0.27919 0.47804
Factor2	0.41283 0.06267 0.30475 0.51041	1.00000 0.00000 . .	0.47006 0.05116 0.38177 0.54986
Factor3	0.38304 0.06060 0.27919 0.47804	0.47006 0.05116 0.38177 0.54986	1.00000 0.00000 . .

After the quartimin rotation, the correlation matrix for factors is shown in Output 2.1.1. The factors are medium to highly correlated. The confidence intervals seem to be very wide, suggesting that the estimation of factor correlations may not be very accurate for this sample size. For example, the 90% confidence interval for the correlation between Factor1 and Factor2 is (0.30, 0.51), a range of 0.21. You may need a larger sample to get a narrower interval, or a better estimation.

Output 2.1.2: Interpretations of Factors Using Rotated Factor Pattern

A nine-variable-three-factor example

The FACTOR Procedure

Rotation Method: Quartimin

Rotated Factor Pattern (Standardized Regression Coefficients) With 90% confidence limits; Cover \|*\| = 0.45? Estimate/StdErr/LowerCL/UpperCL/Coverage Display
	Factor1	Factor2	Factor3
test8	0.86810 0.03282 0.80271 0.91286 0*[]	-0.05045 0.03185 -0.10265 0.00204 *[0]	0.00114 0.03087 -0.04959 0.05187 [0]*
test7	0.73204 0.04434 0.65040 0.79697 0*[]	0.27296 0.05292 0.18390 0.35758 0[]*	0.01098 0.03838 -0.05211 0.07399 [0]*
test9	-0.79654 0.03948 -0.85291 -0.72180 []*0	-0.01230 0.04225 -0.08163 0.05715 *[0]	-0.17307 0.04420 -0.24472 -0.09955 *[]0
test3	0.27715 0.05489 0.18464 0.36478 0[]*	0.91156 0.04877 0.78650 0.96481 0*[]	-0.19727 0.02981 -0.24577 -0.14778 *[]0
test2	0.01063 0.05060 -0.07248 0.09359 [0]*	0.71540 0.05148 0.61982 0.79007 0*[]	0.20500 0.05496 0.11310 0.29342 0[]*
test1	-0.07356 0.04245 -0.14292 -0.00348 *[]0	0.63815 0.05380 0.54114 0.71839 0*[]	0.13983 0.05597 0.04682 0.23044 0[]*
test5	0.00863 0.04394 -0.06356 0.08073 [0]*	0.03234 0.04387 -0.03986 0.10421 [0]*	0.91282 0.04509 0.80030 0.96323 0*[]
test4	0.22357 0.05956 0.12366 0.31900 0[]*	-0.07576 0.03640 -0.13528 -0.01569 *[]0	0.67925 0.05434 0.57955 0.75891 0*[]
test6	-0.04295 0.05114 -0.12656 0.04127 *[0]	0.21911 0.07481 0.09319 0.33813 0[]*	0.53183 0.06905 0.40893 0.63578 0[*]

The coverage displays in Output 2.1.2 show that Test8, Test7, and Test9 have salient relationships with Factor1. The coverage displays are either `0*[ ]' or `[ ]*0', indicating that the entire 90% confidence intervals for the corresponding loadings are beyond the salience value at 0.45. On the other hand, the coverage display for Test3 on Factor1 is `0[ ]*'. This indicates that even though the loading estimate is significantly larger than zero, it is not large enough to be salient. Similarly, Test3, Test2, and Test1 have salient relationships with Factor2, while Test5 and Test4 have salient relationships with Factor3. For Test6, its relationship with Factor3 is a little bit ambiguous; the 90% confidence interval covers approximately values between 0.40 and 0.64. This means that the population value might have been smaller or larger than 0.45. It is marginal evidence for a salient relationship.

Output 2.1.3: Interpretations of Factors Using Factor Structure

A nine-variable-three-factor example

The FACTOR Procedure

Rotation Method: Quartimin

Factor Structure (Correlations) With 90% confidence limits; Cover \|*\| = 0.45? Estimate/StdErr/LowerCL/UpperCL/Coverage Display
	Factor1	Factor2	Factor3
test8	0.84771 0.02871 0.79324 0.88872 0*[]	0.30847 0.06593 0.19641 0.41257 0[]*	0.30994 0.06263 0.20363 0.40904 0[]*
test7	0.84894 0.02688 0.79834 0.88764 0*[]	0.58033 0.05265 0.48721 0.66041 0*[]	0.41970 0.06060 0.31523 0.51412 0[*]
test9	-0.86791 0.02522 -0.90381 -0.81987 []*0	-0.42248 0.06187 -0.51873 -0.31567 [*]0	-0.48396 0.05504 -0.56921 -0.38841 [*]0
test3	0.57790 0.05069 0.48853 0.65528 0*[]	0.93325 0.02953 0.86340 0.96799 0*[]	0.33738 0.06779 0.22157 0.44380 0[]*
test2	0.38449 0.06143 0.27914 0.48070 0[*]	0.81615 0.03106 0.75829 0.86126 0*[]	0.54535 0.05456 0.44946 0.62883 0[*]
test1	0.24345 0.06864 0.12771 0.35264 0[]*	0.67351 0.04284 0.59680 0.73802 0*[]	0.41162 0.05995 0.30846 0.50522 0[*]
test5	0.37163 0.06092 0.26739 0.46727 0[*]	0.46498 0.04979 0.37923 0.54282 0[*]	0.93132 0.03277 0.85159 0.96894 0*[]
test4	0.45248 0.05876 0.35072 0.54367 0[*]	0.33583 0.06289 0.22867 0.43494 0[]*	0.72927 0.04061 0.65527 0.78941 0*[]
test6	0.25122 0.07140 0.13061 0.36450 0[]*	0.45137 0.05858 0.34997 0.54232 0[*]	0.61837 0.05051 0.52833 0.69465 0*[]

For oblique factor solutions, some researchers prefer to examine the factor structure loadings, which represent correlations, for determining salient relationships. In Output 2.1.3, the factor structure loadings and the associated standard error estimates and coverage displays are shown. The interpretations based on the factor structure matrix do not change much except for Test3 and Test9. Test9 now has a salient correlation with Factor3. For Test3, it has salient correlations with both Factor1 and Factor2. Fortunately, there are still tests that only have salient correlations with either Factor1 or Factor2 (but not both). This would make interpretations of factors less problematic.

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