st: ICC vs. shared variance from common factor analysis

[Mike Lacy <Michael.Lacy@colostate.edu>]

I recently found myself puzzled by the wide disparity of  a) the 
value of the intraclass correlation calculated after doing running 
-glamm- on several 5 point Likert items vs. b) the estimate of shared 
variability from a factor analysis of the -polychoric- correlation 
matrix for those several items. While I would not expect these two 
measures be identical, I was surprised to find them differing by more 
than a factor of two.  The same phenomenon is easily reproduced for 
continuous response variables.  For example, for the following 
synthetic data, the proportion of shared variation associated with 
the first factor is 0.93, while the ICC = 0.41.

/////////////////////////////////////////////////////////////////
clear
set seed 49847
set obs 500
gen x = invnorm(uniform())
forval i = 1/5 {
 gen y`i' = 1 + `i' * x + (`i' * invnorm(uniform()))
}
factor y*, ipf
gen id = _n
reshape long y, i(id) j(j)
loneway y id
///////////////////////////////////////////////////////////////

Can anyone offer some enlightenment here?  I'd like to understand why 
these approaches to common variability would diverge so sharply, with 
the practical end of gaining a sense of when/whether to use the ICC 
vs.  the factor analysis results as an indication of the internal 
consistency of a set of measurements.

Regards,


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