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# Re: st: RE: Mean test in a Likert Scale

 From Yuval Arbel To statalist@hsphsun2.harvard.edu Subject Re: st: RE: Mean test in a Likert Scale Date Sat, 1 Sep 2012 12:48:20 -0700

```David, if we return to your original question, I believe that the use
of original numerical values of ordinal variables is a very serious
error, but it has nothing to do with transformation of variables.

To examplify why this is a serious mistake, consider the following
example taken from Kmenta (1997):

Suppose our dependent variable is the level of wage and our
independent variable is the level of education with 3 categories: D= 0
for elementary-school education, D=1 for high school education and D=2

If you run regression between the wage level and D, your implicit
assumption is that wage difference between D=0 and D=1 is equal to the
wage difference between D=1 and D=2. This is a wrong and very limited
assumption.

Moreover, if for some reason you have decided that the scale of D will
be D= 0 for elementary-school education, D=10 for high school
education and D=20 for academic education, then the implicit
assumption becomes that the wage of D=20 is ten time bigger than the
wage of D=10.  In other words, the ordinal numerical value, which does
not have any quantitative interpretation, have an impact on the
regression outcomes. This is the reason we need to use binary
variables under such circumstances.

To see the correct analysis of questionnaires with ordinal questions,
you can take a look at my paper published in urban studies:

http://intl-usj.sagepub.com/content/49/11/2479.full

On Sat, Sep 1, 2012 at 9:32 AM, Nick Cox <njcoxstata@gmail.com> wrote:
> My examples -- miles per gallon, price, weight -- all qualify as ratio scales.
>
> If I understand you correctly, your view is that transformations are
> illegitimate in statistics.
>
> Is that right?
>
> How about link functions as in -glm-? Transformed scales on graphs?
>
> Nick
>
> On Sat, Sep 1, 2012 at 3:50 PM, Ulrich Kohler <kohler@wzb.eu> wrote:
>> Am Samstag, den 01.09.2012, 02:16 +0100 schrieb Nick Cox:
>>> But this objection is so strong that it rules out taking out means in
>>> most circumstances, not just for ordinal scales.
>>>
>>> It's clearly true that mean of transform is not transform of mean
>>> unless that transform is a linear function. The same argument would
>>> imply that means are invalid for measured variables (e.g. means of
>>> miles per gallon, weight, price in the auto data) because they are not
>>> equivariant under transformation. Both theory and practice tell us
>>> that means, geometric means, harmonic means, etc. can all make some
>>> sense for many measured variables. Poisson regression and generalised
>>> linear models all hinge on this.
>>
>> Sorry but I disaggree here. For an intervall scale a transformation such
>> as the one that I used in my example are not allowed because it would
>> obvioulsy distroy the equal distance characteristic of subsequent
>> values. For an intervall scale only linear transformations are allowed
>> and therfore substantive conclusions taken from the mean are robust for
>> arbitrary _allowed_ transformation of the intervall scale.
>>
>>> There's also a big difference of viewpoint here. Measurement theory
>>> loves these arguments about arbitrary order-preserving
>>> transformations, but I don't think they make much sense to scientists
>>> who actually do measurements.
>>
>>
>> But I don't think we -- that is me and you -- disaggree here. In way
>> that's what I wanted to say when I said that an ordinal scale could be
>> taken as kind of an "conventional" absolute scale in some instances.
>>
>>
>>
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--
Dr. Yuval Arbel