I am running a CFA and getting good fit indices (CFI = .99, RMSEA = .01) for a uni-dimensional scale. However, when I test for internal consistency, I get poor Cronbach's $\alpha$s ($\alpha = .6$). I have tried everything from removing outliers, to dropping items and still end up with the same problem.
I am wondering if there is anything in SEM that shows that the measure is reliable?
I know that there is some debate about whether Cronbach's $\alpha$ (or internal consistency) even measures reliability but since my field requires Cronbach's $\alpha$ to be reported as a measure of psychometric goodness, I need to find a way to show internal consistency as being adequate for this measure.
Best Answer
You can calculate the reliability of your items from the CFA.
From your standardized solution, calculate: (L1+...Lk)*2/[(L1+...Lk)*2+(Var(E1)+...+Var(Ek))]
This will give the composite reliability, which should be close to alpha.
It's harder to have good fit if you have high alpha, and it's harder to have high alpha if you have good fit. The extreme example of this is if all of the items are uncorrelated - chi-square will be zero, and RMSEA will be zero, indicating great fit. But alpha will also be zero, indicating appalling reliability. The usual flag for this is low CFI (because the null model chi-square is so low), but you don't have that. I wrote about that in this paper: http://www.sciencedirect.com/science/article/pii/S0191886906003874 (which I think is not behind a paywall).
You mention your loadings in a comment (are these standardized?). Loadings of 0.45 lead to implied correlations of 0.23, so if your loadings are that high, I don't see how your correlations can be that low, and the model still fit. (What's your sample size?)
What estimator are you using?