A thought about the link between Revelle's β coeff and CFA. Related?

#1
Hi stat freaks! :confused:

I've just conducted an analysis of a scale's consistency (coeffs alpha, beta, omega, theta) and CFA to inspect the scale. I was wondering about the relation between Revelle's beta coeff. and CFA results.

Beta coef. = estimates the worst possible split half of a item set.
(thus if compared to alpha /mean split half/ it can be used for diagnostics of scale's homogenity).

The scale had 25 items and I found that if I drop one item (ItX), then beta is comparable to alpha ~ 0.8. However with this item, beta ~ 0.6, alpha still ~ 0.8.
I went through the whole scale, omitting item by item and calculating Betas. The item ItX was the only one that when omitted, beta rised substantially (e.g. from 0.6 to 0.75)

Thus I thought that:
1) If the item ItX is the least homogenous withing the set of items
2) and CFA (one general factor) is estimating the scale's structure fit,

then:
1) there should be something wrong with that item ItX in CFA output. - However, it was not.
2) omitting the bad ItX item should have changed something in CFA output.
But it has not! Moreover, the mean loading (F ~ item1, ... item...24) decreased.

Please, why?