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

 From Nick Cox To statalist@hsphsun2.harvard.edu Subject Re: st: RE: Mean test in a Likert Scale Date Sat, 1 Sep 2012 02:16:09 +0100

```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.

Similarly, you could object to any summary statistic that you can get
the same summary value for different distributions. OK, but that's
pretty much the _definition_ and also the main limitation of a summary
statistic, and not all intrinsic to ordinal scales.

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.

Nick

On Fri, Aug 31, 2012 at 7:55 PM, Ulrich Kohler <kohler@wzb.eu> wrote:
> I'm sure that I don't know all the reasons why we should not take means
> over values from an ordinal scale, but _one_ reason is that conclusions
> taken from the mean are not robust against allowed transformations of
> the values.
>
> Generally, all transformations that preserve the numerical order of the
> scale are allowed for an ordinal scale. Thus we can transform the
> values
>
> 1,  2,  3,  4,  5
>
> of an ordinal scale to
>
> 0, 11, 13, 19, 42
>
> because this would not change at all the order of the scale.
>
> Now, consider two students, A and B, with grades
>
> A: 1, 1, 1, 1, 5
> B: 2, 2, 2, 2, 1
>
> where 1 means "excellent" and 5 means "very bad". If we calculate the
> students' averages we obtain 1.8 for both of them. Our substantial
> conclusion from the comparison of means would therefore be that both
> students are equal.
>
> Now, let us do the "allowed" transformation proposed above:
>
> A:  1,  1,  1,  1, 42
> B: 11, 11, 11, 11,  0
>
> In this case we obtain an average of 9.2 for student A and 8.8 for
> student B. Hence, this time our substantial conclusion would be that "B
> is better than A".
>
> Clearly we should not use a statistic that tells us different truths for
> arbitrary "allowed" transformations of scales. For the ordinal scale the
> median is better suited because conclusions from the median do not
> change for allowed transformations of values of an ordinal scale (in the
> example, student A would be better than B in either case).
>
> So far for the theory. However, in practice we might concede that we
> have measured something on an ordinal scale, but we sort of _aggree_
> that we will never do any transformations of the values. In this case
> the theoretical discussion above does not really matter. Theoretically,
> we might get different results with some transformation but as we never
> transform in practice, it will just not happen. Basically we would get
> some sort of a "conventional" absolute scale, then. It would make quite
> some sense to to use means for such scales.
>
> The question then is whether the 5-point-scale used in the original
> question can be seen as kind of a conventional absolute scale. I tend to
> say that this is the case, but I would like to leave that up to the
> questioner himself.
>
> Uli
>
>
> Am Freitag, den 31.08.2012, 18:35 +0100 schrieb Nick Cox:
>> I bow to others' expertise and experience on the minutiae here, some
>> of which seem almost theological in character. For "likert" read
>> "Likert", passim.
>>
>> My impression from the thread, however, is that some seem to think
>> that extreme views vs moderate views are the issue, and some seem to
>> think that agree vs disagree is the issue. I can't detect a consistent
>> position among posters about how intermediate points are to be handled
>> either.
>>
>> This disagreement to me adds flavour to the wording "arbitrary".
>>
>> Naturally, I am interested to learn that dichotomising a Likert scale
>> is something that researchers think is sometimes justifiable, but I
>> ever met it I would expect some discussion of quite how it was done
>> and why that made sense.
>>
>> Nick
>>
>> On Fri, Aug 31, 2012 at 5:32 PM, David Radwin <dradwin@mprinc.com> wrote:
>> > Rob,
>> >
>> > It may be the case that not labeling the middle points of a scale, as in
>> > your first example, justifies the assumption of equal spacing (deltas).
>> > But the literature suggests that verbally labeling all points on a scale,
>> > as in your second example, leads to more reliable measurement. See, for
>> > example:
>> >
>> > Alwin DF, Krosnick JA. 1991. The reliability of survey attitude
>> > measurement: The influence of question and respondent attributes. Sociol.
>> > Methods Res. 20:139-81.
>> > http://deepblue.lib.umich.edu/bitstream/2027.42/68969/2/10.1177_0049124191
>> > 020001005.pdf
>> >
>> Rob Ploutz-Snyder
>>
>> >> My 2 cents...when designing these sorts of instruments...
>> >>
>> >> I was trained that a true likert scale doesn't label each of the
>> >> points in the 5-point (or other) scale, but instead has only TWO
>> >> labels at each extreme.  For example:
>> >>
>> >> I like Statalist..............      Completely Disagree   1  2  3  4
>> >> 5    Completely Agree
>> >>
>> >> This is in CONTRAST to a scale that would label each and every point
>> >> (sometimes called "likert-type" or "modified-likert") for example:
>> >>
>> >> 1=completely disagree
>> >> 2=disagree
>> >> 3=neutral
>> >> 4=agree
>> >> 5=completely agree
>> >>
>> >> With true likert scales, while still not continuous in scale, the
>> >> distance between each category in a true likert scale is not
>> >> subjective.  The delta between "1" and "2" is the same as the delta
>> >> between "2" and "3" etc.  and it is assumed that survey respondents
>> >> can appreciate this.  The same cannot be assumed about the difference
>> >> between "completely disagree" and "disagree" being equal to the delta
>> >> between "disagree" and "neutral."
>> >>
>> >> So in that way, a  true-likert scale removes some of the subjectivity
>> >> on the deltas and seems to achieve a more proper ordinal scale as
>> >> opposed to purely categorical.
>> >>
>> >> Still doesn't justify using parametric statistical techniques...
>> >> However, most well-vetted Sociology or Psychological instruments are
>> >> designed to use multiple questions that, together, are used to measure
>> >> a particular construct.  Social scientists don't usually intend to
>> >> compare responses on single questions, but instead ask many questions
>> >> that cluster together, often verified by exploratory or confirmatory
>> >> factor analysis, where "factor scores" are then created to capture the
>> >> overall construct of interest.  These factor scores can be derived by
>> >> different methods, the simplest being a mean of the items that cluster
>> >> together, but usually by more sophisticated regression-based methods
>> >> that weigh each item according to how well it correlates with the
>> >> overall factor structure.  These factor scores are continuously
>> >> scaled, unlike the individual items that were used to derive them, and
>> >> it is these factor scores that are often analyzed by various
>> >> parametric statistical techniques.
>> >>
>> >> Whether or not the factor scores  are normally distributed in the
>> >> population (the real question) is dependent on the particulars of each
>> >> research study, but I don't categorically deny that the assumption is
>> >> invalid.
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