See
Feldt, L.S. (1965). The approximate sampling distribution of Kuder-Richardson reliability coefficient. Psychometrika, 30, 357-370.
Feldt, L.S. (1969). A test of the hypothesis that Cronbach's alpha or Kuder-Richardson coefficient is the same for two tests. Psychometrika, 34, 363-373.
Raykov and Marcoulides (2011) describe how to calculate reliability from factor loadings (McDonald's coefficient omega). If the items are tau-equivalent, omega is the same as Cronbach's alpha. Raykov and Marcoulides show how one can use Mplus to calculate the standard error of the reliability coefficient and how to calculate confidence intervals. [Introduction to psychometric theory. Routledge.]
Dear Mahmoud,
You could also install the "psychometric" package in the R framework. This package includes the function:
alpha (x)
that calculates Cronbach's alpha for a psychometric matrix (x). The package also includes the estimation of alpha confidence interval. The instruction is:
alpha.CI (alpha,k,n,level=0.90,onesided=FALSE)
where alpha is Cronbach's alpha, k is the number of items, n is the sample, level is the confidence level (.90 when default), and onesided=FALSE considers a billateral estimation. If you have obtained an alpha value of 0.85, with 50 ítems, level 0.95, and 200 people, the CI for Cronbach's alpha will be:
alpha.CI (0.85,50,200,0.95,onesided=FALSE)
Result: 0.8187288 ; 0.8782391
I hope it could help you.
All the best.
Cronbach alfa has an 1-(1-alfa) F((n-1)(k-1); n-1) distribution
So a 95% Confidence interval for it would be ]1-(1-alfa) x f(.975; n-1; (n-1)(k-1); 1-(1-alfa)xf(0.025; (n-1)(,(n-1)(k-1))[
The ref is Kistner, E. O., & Muller, K. E. (2004). Exact distributions of intraclass correlation and Cronbach’s
alpha with Gaussian data and general covariance, Psychometrika, 69(3), 459-474.
... now if you just need to calculate alfa, and know how to use R, use the Psychometric package
We should not forget that the sampling distribution of alpha depends on the properties of the data. For instance, CI's proposed by Kistner & Muller (mentioned by Joao) assume normal distribution. If the distribution is not approx. normal and the sample size is not too small, ADF estimates might be preferable, see:
Maydeu-Olivares A, Coffman DL, Hartmann WM. (2007). Asymptotically distribution-free (ADF) interval estimation of coefficient alpha. Psychol Methods. Jun;12(2):157-76.
The alpha function in the most recent release of the psych package (1.3.12) in R includes both normal theory as well as bootstrapped ci for alpha. This version was added to CRAN last week and should be available on most mirrors.
In addition, the splitHalf function will find (by brute force if the number of items 16) the highest and lowest possible split half reliabilities.
The advantage of the worst split half reliability (beta) is that it gives an estimate of the amount of general factor saturation in the test.
For finding the reliability and intercorrelations of multiple scales, i recommend the score.items function in the psych package.
For finding omega_hierarchical or omega_total, as well as alpha, you might try the omega function, also in psych.
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