Here is the summary of replies from the query concerning
measure of dispersion for categorical/binary data.
Thanks to Jeremy Dawson and Jay Warner
Miguel Goncalves
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The Original Email:
Dear Allstat,
I have a set of ordered categorical variables that consist of binary
variables and variables on a four point scale. I would like to
calculate the equivalent of standard deviation for these variable.
My question is therefore to see whether there is a specific measure
of dispersion for categorical/binary data. I have been told of the
existence of the "Index of Qualitative Variation" (IQV), but I do
not know whether this is appropriate or how to calculate it.
I would be grateful for any help or references you could give me
on this matter and I will make available to Allstat a summary
of the replies I receive.
Thank you
Miguel
REPLIES
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A measure I often use for dispersion in categorical variables is
Blau's (1977) index of heterogeneity, where if you have k
categories, and the proportion of responses in category i is p(i),
Blau gives
Heterogeneity = 1 - p(1)^2 - p(2)^2 - ..... - p(k)^2
This is then a measurement between 0 and 1 with 0 being entirely
the same (i.e. all one category) and 1 being the most disperse
(though never actually achievable). The reference for this is:
Blau, P.M. (1977). "Inequality and Heterogeneity", New York: Free
Press.
Jeremy Dawson
Statistician
Organisation Studies Group
Aston Business School
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If possible, suggest you group the binary data groups into large enough
repetitions that youcan do a binary thing on them, with % (between 0 and
1, for ex.) Then go from there. I'mnot very happy with this solution
either, but...
As for the 4 level catagorical, if it is ordinal, then I urge you to
recast it as continuous/intermitant. First round: see what it gets you
as a cont. var. 2nd round: do an arcsin or 'omega' transform (a la'
Taguchi), and run it again. If it holds up the second time, you've got
something.
Jay Warner
Principal Scientist
Warner Consulting, Inc.
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