Hi everyone!

In this post I will talk a little bit about Cronbach's alpha in Reliability Analysis.

In reliability analysis, the internal consistency is tradicionally measured by Cronbach's alpha. That is based on average inter-item correlation, and show how closely related a set of items are as a group.

This, must be taken into account the following assumptions: The observations should be independent and errors between items shouldn't be correlated . The items should have a normal distribution, and be linearly related to the total score.

The Cronbach's alpha is obtained from the covariance between items of a scale, the total variance of the scale and the number of items that make up the scale.

It's calculated from the following formula: The number of items (questions) in the questionnaire is given by k. The variance of each question is defined by Si^2 (i = 1, ..., k). ST^2 is the variance of the sum of the responses of each subject j (j = 1, ..., n), where n is equal to the number of individuals in the sample.

The acceptable value of alpha in reliability analysis is between 0.7 and 0.8, that indicates a good reliability. (Kline, 1999).

## Where Cronbach's alpha is used?

The Cronbach's alpha is very used in social science, for instance, when you want to verify if the items really measures what you intends, or if satisfaction is really measured by these attributes. Using internal consistency measures you can evaluate this type of questions.

By the other words, with the reliability analysis, you can know how much the items in your construct are related to each other, and eventually identify problem items that should be excluded from the scale.

• How consistent is my scale of customer satisfaction?
• Is my customer satisfaction construct well represented by these items?

Let's see an example.

Next, imagine that I want to study the internal consistency of the construct "customer satisfaction" of a particular service. I want to evaluate if the customer satisfaction is really measured by a certain subset of items present in my survey?  I want to judge the consistency of results across items on the same dimension!

To illustrate, let's take into consideration is the following results:  These results show the values of Cronbach alpha, as Raw Cronbach Alpha. This is the Alpha value used to evaluate the reliability of a scale; The Standardized Cronbach Alpha, should be similar to raw Cronbach alpha. But when the k items under investigation use different measurement units, summing the variances will be problematic since they would also be expressed in different units; and Composite Reliability,  that consist in a measure of the overall reliability of a collection of heterogeneous but similar items. Values greater than 0.7 indicates a good reliability.

The table above besides the values of alpha, also presents values for Guttman's Lambda 6 (G6), and for Average inter-item correlation.

The Guttman's Lambda 6 (G6) considers the amount of variance in each item that can be accounted for the linear regression of all of the other items, and Average inter-item correlation compares correlations between all pairs of items that test the same construct by calculating the mean of all paired correlations. Values closer to 1, the better.

Analyzing these values, the table above shows a Cronbach's Alpha equal to 0.729 and Composite Reliability of the 0.770 (greater than  0.70), its indicates that the Customer Satisfaction construct in study show a good reliability. For average inter-item correlation, we have obtained 0.363, which indicates that the items on the construct, have a low association between them, on average.

However it is interesting to check if there are items that can contribute to the increase of this value, by its elimination. The graphic below shows how Cronbach's alpha can be affected. Although we initially have an alpha value greater than 0.7, this analysis shows that we can increases the Cronbach's alpha to 0.86, if item4 was dropped (look red bar). For this, we must take into account the whole literary context in which the study is inserted and validate the elimination of the respective item. In the same way that the elimination of item7 would induce alpha value decrease to 0.51.

Then, if it is possible, delete item4 and repeat the Cronbach's alpha analysis.

And that's all for now, and remember that Cronbach's Alpha gives a measure of the internal consistency of a scale or construct, that is, as all items contributes to its formation.

I hope this is useful for your analysis!