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Suppose I have data from a questionnaire. The questionnaire contained eight items, of which the first four comprise one scale, the second four another scale. My data now looks like this:

dat:
            | scale.1                     | scale.2
participant | item.1 item.2 item.3 item.4 | item.5 item.6 item.7 item.8
1           | 3      7      1      2      | 2      3      6      4
2           | 2      2      5      2      | 8      9      7      6
...

How do I calculate Cronbach’s alpha for this questionnaire as a whole and each individual scale?

When I calculate alpha for scale.1, do I calculate it as if the items comprising scale.2 did not exist ? Like this:

library(psy)
cronbach(data.frame(dat$item.1, dat$item.2, dat$item.3, dat$item.4))

Or is there some (better? correct?) way where I input the whole dataframe into a function, tell it which columns belong to which scale, and I get internal consistencies that considers the scales in the context of the whole questionnaire? scoreItems from the psych package seems to do something like this, but I'm too stupid to make sense of the manual.

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Cronbach's alpha is a measure of internal consistency of a scale. It makes sense to do it for scale 1 separately and for scale 2 separately but there seems no point in doing it for the whole questionnaire when there are supposed to be separate sub-scales.

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In addition to calculating two reliability or alpha coefficients (as @mdewey rightly suggested), you may also have a look at $\omega_h$ (hierarchical omega) and $\omega_h$ (omega total), which are very useful for "multidimensional" item sets. The first one is a measure of variability due to a general "g-factor", and the second one is a measure of variability due to both a general and group factors (scale.1 and scale.2 would be the group factors in your example). Definitely 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.

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