Calculating Cronbach's alpha with systematically missing values

[Jennifer Chang]

Dear Statalist,

I have a rather obscure need that I suspect has a straightforward solution.

I am trying to calculate Cronbach's alpha for a psychometric scale. Suppose I have one scale comprising five items, v1-v5, with non-missing values denoted as "x" -- e.g.:

v1   v2   v3   v4   v5
 x    x    x
      x         x
 x         x         x
 x    x         x    x
           x    x    
 x    x              x
           x    x    x
 x              x
      x    x         x

The missingness is systematic in that the age of the respondent determines which items are administered.

Is there a way to calculate a single alpha for these items (e.g., a set of alphas for each combination of non-missing variables, which is then averaged)? At present the command won't run because there is no observation that is non-missing for all five items. The age range is 60 years, and the age bands are different for each item, so I would like to avoid doing this manually with a large number of commands along the lines of:

alpha v1 v2 if age==20
alpha v2 v3 if age==21
alpha v2 v4 if age==22
alpha v2 v5 if age==23
alpha v3 v5 v6 if age==24

etc.

Thank you!

[Dick Campbell]

You can just compute alpha on the basis of a pairwise-present covariance matrix. Thus you can get one alpha without having to compute multiple versions and averaging them. This is, in fact, the default for alpha in Stata. If you are not familiar with how alpha is computed I'd suggest you take a look at the manual entry rather than just the help file.

[daniel klein]

Is this a strictly technical question? The alpha command accepts missing values (by default), so you should be able to type

alpha v1-v5

to get one coefficient. This is some kind of weighted average, in the sense that Stata bases calculations on pairwise correlations and somehow takes into account the number of cases used in the respective calculations. Whether it is statistically sound in your case, I do not know. Conceptually it seems a bit odd to postulate one scale, yet administer different items to different groups of respondents to measure the underlying construct.

I also cannot tell whether your approach will yield meaningful results, either. I would doubt it as the scale you have in mind consists of all five items, yet you are calculating reliability for a number of pairs of items. High alpha values might mean that there is one underlying dimension, but they might also mean that you have reliable measures for a number of distinct dimensions.

Per definition your values are missing at random when conditioned on age, so another approach could be to base the reliability coefficient on the correlation matrix obtained from an EM algorithm where you impute the values with age as a covariate.

Best
Daniel

[wbuchanan]

Jennifer Chang do you have some reason to believe that the construct is invariant across age groups? If so, there are other methods that you could use to estimate the within group reliabilities and/or to test measurement invariance across the groups (e.g., multigroup SEM).