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 use 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 et al. 2010) is used. For example, because the orthogonal rotated factor loading theta must be bounded between –1 and +1, the Fisher transformation

phi equals one-half log left-parenthesis StartFraction 1 plus theta Over 1 minus theta EndFraction right-parenthesis

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

left-parenthesis ModifyingAbove theta With caret Subscript l Baseline equals StartFraction a slash b minus 1 Over a slash b plus 1 EndFraction comma ModifyingAbove theta With caret Subscript u Baseline equals StartFraction a times b minus 1 Over a times b plus 1 EndFraction right-parenthesis

where

a equals StartFraction 1 plus ModifyingAbove theta With caret Over 1 minus ModifyingAbove theta With caret EndFraction comma b equals exp left-parenthesis z Subscript alpha slash 2 Baseline times StartFraction 2 ModifyingAbove sigma With caret Over 1 minus ModifyingAbove theta With caret squared EndFraction right-parenthesis

and ModifyingAbove theta With caret is the estimated factor loading, ModifyingAbove sigma With caret is the standard error estimate of the factor loading, and z Subscript alpha slash 2 is the 100 left-parenthesis 1 minus alpha slash 2 right-parenthesisth 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 various displays and their interpretations.

Table 5: 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 nonsalience 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 nonsalience 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*[ ] [ ]*0 0[ ] or [ ]0 The 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 44.5 for an illustration of the use of confidence intervals for interpreting factors.

Last updated: March 08, 2022