Cronbach's alpha -- optimum value is 1.00?

Question

I have been brushing up on some stats and figure I'd have a look at Cronbach's alpha. As I understand it, a higher alpha coefficient means a greater interrelation between all the items.

If items are all measuring the same construct of interest, then it follows that the score on one item will predict the score on the others.

However, I came across this reference:

Multifaceted Assessment for Early Childhood Education

It states "the optimum value of an alpha coefficient is 1.00". I believe that this statement is wrong -- while a higher reliability is certainly desirable, and ideally >0.90, the only thing that could be worse than alpha = 1.0 is when alpha = 0.00.

My reasoning: suppose I had a 30 item questionnaire with an alpha coefficient of 1.00, then it means that there are 29 items that offer no predictive value, because having one item would do just as well. When alpha = 0.00, then of course, there is no point in even administering the questionnaire.

Am I missing something?

I also came across the following statement somewhere:

Cronbach's alpha will not tell you whether your scale is uni-dimensional (i.e., whether all your items measure the same construct)

I get what they're saying: a high alpha is necessary but insufficient evidence that there is one underlying construct. However, let's suppose that alpha = 0.99 -- can someone give me an example where we can meaningfully speak of two constructs? The only way I can interpret such a correlation would be to say that what we think are different constructs are actually the same construct that have been given different names.

Answer 1

What for many people was the canonical account of the issues here was published by Green and colleagues in an article entitled "Limitations of Coefficient Alpha as an Index of Test Unidimensionality" available here. They give an example of a test with five separate underlying latent variables with $\alpha = 0.811$ which is not as extreme as your request but getting close.

They state:

The fallacy of relying on Cronbach’s results as justification for the use of coefficient alpha as an index of test homogeneity lies in mistaking necessary properties of homogeneity for sufficient properties of homogeneity. Certainly high internal consistency as indicated by a high coefficient alpha will result when a general common factor runs through the items of the test. But this does not rule out obtaining high internal consistency as measured by coefficient alpha when there is no general factor running through the test items.

Their article is well worth reading in full.

Answer 2

Here’s a reply, slightly modified, I previously posted on a different forum. I agree, it is unlikely one would obtain an alpha of .99 if the items were not all highly correlated and likely generated from related processes and therefore suggest, at some level, uni-dimensionality. However…

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Cronbach's alpha is not designed to measure internal structure (think in terms of factor analysis here), but can provide a measure of internal consistency (think in terms of mean inter-item correlations here), although that appears to be questionable too according to some critics (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2792363).

Alpha is a function of covariances (and correlations), and it is also a function of number of items. Here is a formula for Cronbach's alpha in terms of mean inter-item correlations (m[r]) and the number of items (k):

$alpha = \frac{k * m[r]} { 1 + (k - 1) * m[r]}$

where

k = number of items on instrument used to calculate alpha, m[r] = mean correlation among the k items.

Given this formula, the following two scenarios are possible:

1. An instrument includes 7 items designed to measure the same construct (e.g., math interest), so there should be one factor. The mean correlation among items is m[r] = .60. Using the formula above:

$alpha = \frac{7 * .60} { 1 + (7 - 1) * .60} = .91$

2. A second instrument is designed to measure 12 unrelated or weakly related constructs. Factor analysis reveals that the internal structure demonstrates 12 distinct factors with little correlation among factors. Each construct has between 7 and 12 items to measure that construct, and the total number of items on the questionnaire is 110. The mean correlation among all 110 items is m[r] = .085. If one erroneously applies the alpha reliability formula to these 110 items, the result would be:

$alpha = \frac{110 * .085} { 1 + (110 - 1) * .085} = .91$

Note that Cronbach’s alpha is the same, within rounding error, in both situations, yet the internal structure is very different in both cases. These two examples demonstrate that Cronbach’s alpha is not designed to reveal internal structure of items, or uni-dimensionality. Better to use EFA or CFA to assess structure.

For the extreme case of alpha = .99, the above examples could be recast like this:

1. Instrument for one construct with 20 items, mean r is .84, so alpha = .99.

2. Instrument with 100 constructs and 650 items produces alpha = .99 with a mean r of .132. While this many items may seem fanciful, some national studies contain as many or more. For example, the National Longitudinal Survey of Youth 1979 started with a questionnaire with 174 pages. I estimate about 4 questions per page, so close to 700 items. It is unlikely, however, that one would mistakenly include all these items in calculation of one alpha. Link below is to a copy of the original questionnaire.

https://www.nlsinfo.org/sites/nlsinfo.org/files/attachments/121212/NLSY79_1979_Quex_size%20reduced.pdf