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I have a very high Cronbach's Alpha of $0.966$ (and composite reliability of $0.969$). My questionnaire has a 1-factor-structure, which makes it compatible with Cronbach's Alpha, but it has 22 items. So I want to know if Alpha is only that high because of the high amount of items. Is there a coefficient which is independent of the number of items? Unfortunately, I can't use omega because my n is too small ($n = 156$).

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Some thoughts

The prerequisite for using McDonald's Omega is that your one-factor model shows at least an adequate CFA model fit. To get some guidelines as to what an acceptable model fit is, check this thread. Hence, if I were you, I would fit a one-factor CFA model with 22 indicators (i.e. items). I would not necessarily call your sample too small (n=156) to try this.

Also, your Cronbach's Alpha is related to the number of items in the following way: your Alpha will increase as you keep adding extra items, only if the addition of extra items leads to an increase in the average inter-item correlation. Further, alpha is affected by a number of factors. These factors are summarized succinctly in this thread. Hopefully, you will find it enlightening.

Hence, the reason why your Alpha is very high is not, strictly speaking, related to only a high number of items per se, but probably more to the fact that these items have a high degree of inter-item correlation.

Final thought: Alpha and Omega tend to correlate highly. This means that if your CFA model fits well, my educated guess is that your Omega will almost certainly be in the region of .90, if, of course, your standardized factor loadings do not show too much discrepancy in their magnitude. This follows from my experience of running 1,000s of Omega tests.

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