Chapter Contents
Chapter Contents		 Previous
Previous		 Next
Next
The FACTOR Procedure

Confidence Intervals and the Salience of Factor Loadings

The traditional approach to determining salient loadings (loadings that are considered large in absolute values) employs rules-of-thumb such as 0.3 or 0.4. However, this does not utilize the statistical evidence efficiently. The asymptotic normality of the distribution of factor loadings enables you to construct confidence intervals to gauge the salience of factor loadings. To guarantee the range-respecting properties of confidence intervals, a transformation procedure such as in CEFA (Browne, Cudeck, Tateneni, and Mels 1998) is used. For example, because the orthogonal rotated factor loading \theta must be bounded between -1 and +1, the Fisher transformation

\varphi=\frac{1}2\log(\frac{1+\theta}{1-\theta})

is employed so that \varphi is an unbounded parameter. Assuming the asymptotic normality of \hat{\varphi}, a symmetric confidence interval for \varphi is constructed. Then, a back-transformation on the confidence limits yields an asymmetric confidence interval for \theta. Applying the results of Browne (1982), a (1-\alpha)100% confidence interval for the orthogonal factor loading \theta is

( {\hat{\theta}}_{l}=\frac{a/b-1}{a/b+1}, \ {\hat{\theta}}_{u}=\frac{a x b-1}{a x b+1} )

where

a=\frac{1+\hat{\theta}}{1-\hat{\theta}}, \ b=\exp(z_{\alpha/2}x\frac{2\hat{\sigma}}{1-{\hat{\theta}}^2})

and \hat{\theta} is the estimated factor loading, \hat{\sigma} is the standard error estimate of the factor loading, and z_{\alpha/2} is the (1-\alpha/2)100 percentile point of a standard normal distribution.

Once the confidence limits are constructed, you can use the corresponding coverage displays for determining the salience of the variable-factor relationship. In a coverage display, the COVER= value is represented by an asterisk `*'. The following table summarizes the various displays and their interpretations.

Table 2.1: Interpretations of the Coverage Displays
Positive Estimate Negative Estimate COVER=0 specified Interpretation
[0]**[0] The estimate is not significantly different from zero and the CI covers a region of values that are smaller in magnitude than the COVER= value. This is strong statistical evidence for the non-salience of the variable-factor relationship.
0[ ]**[ ]0 The estimate is significantly different from zero but the CI covers a region of values that are smaller in magnitude than the COVER= value. This is strong statistical evidence for the non-salience of the variable-factor relationship.
[0*][*0][0]The estimate is not significantly different from zero or the COVER= value. The population value might have been larger or smaller in magnitude than the COVER= value. There is no statistical evidence for the salience of the variable-factor relationship.
0[*][*]0 The estimate is significantly different from zero but not from the COVER= value. This is marginal statistical evidence for the salience of the variable-factor relationship.
0*[ ][ ]*00[ ] or [ ]0The estimate is significantly different from zero and the CI covers a region of values that are larger in magnitude than the COVER= value. This is strong statistical evidence for the salience of the variable-factor relationship.

See Example 2.1 for an illustration of the use of confidence intervals for interpreting factors.

Chapter Contents
Chapter Contents		 Previous
Previous		 Next
Next		 Top
Top

Copyright © 2000 by SAS Institute Inc., Cary, NC, USA. All rights reserved.