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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 Mon, 3 Sep 2012 00:53:42 -0700

```Regarding Nick's question, the exact reference of Kmenta (1997) is:

Jan Kmenta (1997): Elements of Econometrics, Second Edition, Chapter 11.

The particular example is taken from page 464. On the second sentence
of the first paragraph the author says that:

"It would be incorrect to use one variable with three values say, 0
for B.A., 1 for M.A. and 2 for a Ph.D."

Then he explains formally why it is wrong:

"That is, by using one variable with values 0, 1 or 2 (or any three
equidistant values) we are in fact assuming that the difference
between the salary of a Ph.D. and an M.A. is the same as that between
the salary of M.A. and B.A. Unless we know a priori this is the case,
we are not justified in making such an assumption."

On Sun, Sep 2, 2012 at 3:35 AM, Nick Cox <njcoxstata@gmail.com> wrote:
> Thanks for the clarification. I am not sure whether that leaves our
> positions as essentially similar despite apparent differences, or the
> opposite.
>
> How far researchers are willing to use transformations is strikingly
> variable between, and even within, fields of statistical science.
>
> Nick
>
> On Sun, Sep 2, 2012 at 11:11 AM, Ulrich Kohler <kohler@wzb.eu> wrote:
>> On 01.09.2012 18:32, Nick Cox 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?
>>>
>> No, that is not what I wanted to say. I wanted to say that each scale has a
>> set of transformations that are "allowed"  in the sense
>> that they don't change the defining features of the scale  (whether its
>> nominal, ordinal, etc.). You can use any
>> transformation that is allowed, but than you should use only those
>> statistics, whose conclusions are robust against
>> that transformations. You can even use transformations that change the
>> scale, but than you should use a statistic
>> that is robust against that transformation.
>>
>>
>>
>>
>>
>> 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