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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.

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  • $\begingroup$ Are some of the items reverse coded? When you look at alpha, do some items have negative correlation? $\endgroup$ – Peter Flom Nov 20 '13 at 19:13
  • $\begingroup$ No, none of the items are reverse coded and nor do these items have a negative correlation with each other. $\endgroup$ – user1984 Nov 20 '13 at 19:16
  • $\begingroup$ How many items do you have? Sometimes having few items (<.5) can lead to very low internal consistency. What is your average inter-item correlation? $\endgroup$ – Behacad Nov 20 '13 at 19:55
  • $\begingroup$ There are 8 items. The inter-item correlations are .15 to .30. I figured that low correlations are driving the low alpha but I am surprised to see the loadings in the CFA ranging from .45 to .69 and the good fit indices. $\endgroup$ – user1984 Nov 20 '13 at 21:04
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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?

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    $\begingroup$ Thank you for the reference and the formula. I have a sample size of 300, using imputed data and by default MPLUS is using the ML estimator. The standardized loadings are .3 to .7 approx and the inter-item correlations are .1 to .3. $\endgroup$ – user1984 Nov 22 '13 at 15:36
  • $\begingroup$ That sounds about right. Try MLM, MLR or MLMV as your estimator and see what effect that has. If it makes CFI a little worse, I'm fairly confident that your problem is that you just have low reliability. $\endgroup$ – Jeremy Miles Nov 22 '13 at 17:45
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If your instrument is evaluating two or more constructs, it is possible that your alpha could be low. I advise you to estimate one alpha for each sub-scale.

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