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# Using the average correlation to evaluate Cronbach's alpha and alternatives

Cronbach's alpha is used as a means of testing the reliability between a set of items and is related to the (arithmetic) average of the absolute values of the Pearson (off-diagonal) correlations. For n variables in a n by n correlation matrix there will be n(n-1)/2 distinct off-diagonal correlations ie taking either the upper or lower triangle of correlations.

As an illustration Kenny (1979, p.125) gives an example for four correlations based upon 4 variables relating to judgments of persons in a mock trial. The six distinct correlations are given in bold in the table below.

 Verdict Sentence Responsibility Innocence Verdict 1.000 Sentence 0.412 1.000 Responsibility 0.629 0.403 1.000 Innocence -0.585 -0.270 -0.500 1.000

Cronbach's alpha (Kenny, 1979 p.132-133) is equal to

$$\frac{n\bar{r}}{1 + (n-1)\bar{r}}$$ where $$\bar{r}$$ is the arithmetic absolute value of the correlations and n is the number of variables.

In the above example, n=4 and there are (4x3/2 =) 6 distinct off-diagonal Pearson correlations so the average r = (0.412+0.629+0.403+0.585+0.270+0.500)/6 = 0.4665 so Cronbach's alpha = [4 (0.4665)] / [1 + 3(0.4665)] = 0.778.

Jeremy Miles also mentions a composite reliabilty measure based upon factor loadings (on the same factor) which do not, unlike, Cronbach's alpha assume the correlations are near equal.

If you have three items with three loadings (L1, L2, L3) and three error variances (E1, E2, E3) obtained from an exploratory factor analysis then:

Composite reliability = (L1+L2+L3)**2 / [(L1+L2+L3)**2 + (E1+E2+E3)]

Just keep adding for more. Jeremy adds that unfortunately it is hard to find a reference which describes this clearly. Loehlin (1987), however, does explain the relationship between factor loadings and variance which underpin this method.

Raykov (1998) has demonstrated that Cronbach's alpha may over- or under-estimate scale reliability. Underestimation is common. He suggests obtaining item correlations using scores from a one factor model to test how well items are represented by a single factor. Due to the danger of underestimation using Cronbach's alpha, rho is now preferred and may lead to higher estimates of true reliability. Raykov's rho is not available in most standard packages but Raykov (1997) lists EQS and LISREL code for computing this measure. EQS is stand alone software and available at CBSU, as is LISREL. EQS example code is given here.

Values of Cronbach's below 0.70 are deemed unacceptable and above 0.80 good (Cicchetti, 1994).

* For other interpretations and confidence intervals for Cronbach's alpha see here.

References

Cicchetti DV (1994). Guidelines, criteria, and rules of thumb for evaluating normed and standardized assessment instruments in psychology. Psychological Assessment 6 284-290.

Kenny DA (1979). Correlation and Causality. Wiley:New York.

Loehlin JC (1987). Latent variable models: An introduction to factor, path, and structural analysis. Hillsdale, NJ: Erlbaum.

Raykov T (1997). Estimation of composite reliability for congeneric measures. Applied Psychological Measurement, 21, 173-184.

Raykov T (1998). Coefficient alpha and composite reliability with interrelated nonhomogeneous items. Applied Psychological Measurement, 22(4), 375-385.

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