Cronbach Alpha Assumptions

Question

I'm currently learning how to use Cronbach's alpha in R. I have a scale with 7 items and 63 respondents. The aim is just to get a practical understanding of what Cronbach's alpha is. There are some nulls in the dataset.

The method itself has a number of assumptions. I wish to test these assumptions in R. So my questions are related to a couple of these assumptions:

• Assumption of unidimensionality. I have gotten the scales from other research papers so i assume i use confirmatory analysis to test for unidimensionality?
• Error Terms are uncorrelated. How do i test this in R? Im currently using the code alpha(myscales) to generate my alpha statistics but i want to test if the scales violate this
• How to test for Tau Equivalence. This I think can be assumed based on the results of the same scale from other researchers

I realize there are better tests, for example the omega test. Its next on my list to learn.

Answer 1

All three assumptions may be studied using CFA. If a unidimensional, $\tau$-equivalent (all loadings equal) model with uncorrelated errors holds, then you have evidence that the assumptions hold. However, with only 63 people you may not have good power for a rigorous test.

The most critical of the three assumptions is unidimensionality. To investigate that issue, it's often a good starting point to use exploratory FA and to compare Alpha to measures such as $\omega_h$ or $\omega_t$. For that, you may have a look at Revelle's sources for his psych package (e.g., http://personality-project.org/r/psych/HowTo/R_for_omega.pdf and Chapter 7.2.5 of his book on psychometric theory.

Answer 2

Beyond the excellent answer given by hplieninger here, many times these issues can be surmised long before you do any statistical testing of the ideas. I have met some people that have blown past these assumptions and just plugged in their data into alpha functions with whatever software they are using, and yet speaking to them you can gather that reliability for their measures would be inaccurate.

So while hp's answer deals with the quant side, mine deals with the qual side of these assumptions:

1. Assumption of unidimensionality. This one can be quite obviously off without any stats testing involved. I've seen a number of people use some survey on something like anxiety, but then they explain that it tests sub-factors like test anxiety, foreign language anxiety, etc. It's highly likely that this sort of measure would become multi-dimensional and would require reworking. So if you see something like this, the test should be redeveloped or as hp mentioned some kind of omega coefficient can be used to assess what latent variable structure is in place that supports their measure.
2. Error terms are uncorrelated. This one is probably the least obvious of them all, but this paper does a good job of explaining the underlying causes. Some of the reasons are related to unidimensionality concerns, but there can also be some contextual effects that influence this. For example, you may have a number of items that measure the same thing but cluster together due to the way the questions are framed. As an example, if a test booklet has "item bundles", you may to a degree expect there may be some issues with errors being correlated among these items.
3. Tau equivalence. Let's say you have a test of reading ability. Its a simple composite composed of dichotomous correct and incorrect answers. The first ten items are super easy, then the next 50 items are super hard. Even though these are all reading items, its very likely that tau equivalence may not hold up in this case, as the way these 60 items will correlate will differ based on the difficulty of the items.

So these illustrate that even some heuristic pre-checks can assist in avoiding a lot of these issues in the first place.