SPSSX Discussion

Treating Ordinal Data as Continuous

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Laurie Petch

Treating Ordinal Data as Continuous

Apologies, this is more of a statistical question, though it does have an indirect bearing on SPSS. As subscribers
to this list will know, in psychology it is common practice to treat ordinal level data deriving from rating scales as if
they were continuous data and subjecting them to inferential statistical analysis. The argument I have heard in
favour of doing this is that ordinal data behave much more like continuous data when they are summed and
averaged. I have not seen this argument in writing, however, and would be grateful to anyone who can point me
in the direction of a relevant source.

Also, if anyone can suggest counter-arguments to this justification, that would be great too. It just strikes me that a
score of '120' as opposed to '118' on an anxiety measure is data of a very different kind than someone who is
120cm tall as opposed to 118cm.

To say that ordinal data behave like continuous data is surely rather like saying that, since cheese 'behaves' more
like butter when it is heated, it's okay to use cheese instead of butter to make a cake?

Laurie
--------
Laurie Petch
Chartered Educational Psychologist (British Psychological Society)

Robinson Aschoff

Re: Treating Ordinal Data as Continuous

Hi Laurie,

it is my experience too that ordinal data is commonly treated as if it
were continuous in psychology. This is not uncontroversial, however. I
found this english paper with some further references to this topic:
http://opencontent.org/docs/parametric.pdf. I would argue that one has to
decide for each particular case if using parametric statistic analysis is
justified. At least, the scale should not obviously violate equidistance.
Further discussion ranks around the question how robust the envisaged
statistical test is against the violation of the interval scale
assumption. Some argue that e.g. an ANOVA (given sufficient N) is
sufficient robust aginst this violation. Others are more sceptical.

I also have a german book chapter about this topic I could send you in
case it is of any use for you.

Regards
Robinson
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Information Management Research Group
Department of Informatics
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E-Mail: [hidden email]
Phone: +41 (0)44 635 6690
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"SPSSX(r) Discussion" <[hidden email]> wrote on 09.01.2007
11:00:37:

> Apologies, this is more of a statistical question, though it does
> have an indirect bearing on SPSS. As subscribers
> to this list will know, in psychology it is common practice to treat
> ordinal level data deriving from rating scales as if
> they were continuous data and subjecting them to inferential
> statistical analysis. The argument I have heard in
> favour of doing this is that ordinal data behave much more like
> continuous data when they are summed and
> averaged. I have not seen this argument in writing, however, and
> would be grateful to anyone who can point me
> in the direction of a relevant source.
>
> Also, if anyone can suggest counter-arguments to this justification,
> that would be great too. It just strikes me that a
> score of '120' as opposed to '118' on an anxiety measure is data of
> a very different kind than someone who is
> 120cm tall as opposed to 118cm.
>
> To say that ordinal data behave like continuous data is surely
> rather like saying that, since cheese 'behaves' more
> like butter when it is heated, it's okay to use cheese instead of
> butter to make a cake?
>
> Laurie
> --------
> Laurie Petch
> Chartered Educational Psychologist (British Psychological Society)
>

Swank, Paul R

Re: Treating Ordinal Data as Continuous

In reply to this post by Laurie Petch

The problem is compounded when one thinks of measurement scales as all
or none, it's either ordinal or interval. However, win, place, and show
in a horse race is clearly more ordinal that IQ scores. For IQ scores,
it is clear that the difference between an IQ of 50 and an IQ of 75 is
perhaps greater than th difference between 75 and 100. On the other and,
the difference between an IQ of 100 and 101 is probably pretty similar
to the difference between 99 and 100. It all relies on the relation
between the scale values and the underlying construct. I think some
scales are closer to being interval than ordianl while for others, the
opposite is true. A lot has to do with how well the scale was
constructed.

Paul R. Swank, Ph.D. Professor
Director of Reseach
Children's Learning Institute
University of Texas Health Science Center-Houston

-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
Laurie Petch
Sent: Tuesday, January 09, 2007 4:01 AM
To: [hidden email]
Subject: Treating Ordinal Data as Continuous

Apologies, this is more of a statistical question, though it does have
an indirect bearing on SPSS. As subscribers to this list will know, in
psychology it is common practice to treat ordinal level data deriving
from rating scales as if they were continuous data and subjecting them
to inferential statistical analysis. The argument I have heard in favour
of doing this is that ordinal data behave much more like continuous data
when they are summed and averaged. I have not seen this argument in
writing, however, and would be grateful to anyone who can point me in
the direction of a relevant source.

Also, if anyone can suggest counter-arguments to this justification,
that would be great too. It just strikes me that a score of '120' as
opposed to '118' on an anxiety measure is data of a very different kind
than someone who is 120cm tall as opposed to 118cm.

To say that ordinal data behave like continuous data is surely rather
like saying that, since cheese 'behaves' more like butter when it is
heated, it's okay to use cheese instead of butter to make a cake?

Laurie
--------
Laurie Petch
Chartered Educational Psychologist (British Psychological Society)

Bob Schacht-3

Re: Treating Ordinal Data as Continuous

In reply to this post by Laurie Petch

At 07:00 AM 1/9/2007, Swank, Paul R wrote:

>The problem is compounded when one thinks of measurement scales as all
>or none, it's either ordinal or interval. However, win, place, and show
>in a horse race is clearly more ordinal that IQ scores. For IQ scores,
>it is clear that the difference between an IQ of 50 and an IQ of 75 is
>perhaps greater than th difference between 75 and 100. On the other and,
>the difference between an IQ of 100 and 101 is probably pretty similar
>to the difference between 99 and 100. It all relies on the relation
>between the scale values and the underlying construct. I think some
>scales are closer to being interval than ordianl while for others, the
>opposite is true. A lot has to do with how well the scale was
>constructed.

bdates

Re: Treating Ordinal Data as Continuous

In reply to this post by Laurie Petch

Bob's comments summon some of the reason that I became interested in IRT as
a way of looking at ordinal measures from an interval perspective. One of
the options is to examine using IRT to develop interval data and then
analyze that.

Brian

> At 07:00 AM 1/9/2007, Swank, Paul R wrote:
> >The problem is compounded when one thinks of measurement scales as all
> >or none, it's either ordinal or interval. However, win, place, and show
> >in a horse race is clearly more ordinal that IQ scores. For IQ scores,
> >it is clear that the difference between an IQ of 50 and an IQ of 75 is
> >perhaps greater than th difference between 75 and 100. On the other and,
> >the difference between an IQ of 100 and 101 is probably pretty similar
> >to the difference between 99 and 100. It all relies on the relation
> >between the scale values and the underlying construct. I think some
> >scales are closer to being interval than ordianl while for others, the
> >opposite is true. A lot has to do with how well the scale was
> >constructed.
>
> My main concern with this issue comes from satisfaction and importance
> scales. It seems to me that we don't have enough tools regarding ordinal
> measures. The analytical tool set for interval data is far richer than
> what
> is available for ordinal data. The median can be substituted for the mean,
> but there is no analytical equivalent of variance, even though it makes
> intuitive sense that an ordinal scale with results congregating at the
> extremes ought to be more 'variable' than a scale in which the results
> congregate around the median.
>
> Bob Schacht
>
> Robert M. Schacht, Ph.D. <[hidden email]>
> Pacific Basin Rehabilitation Research & Training Center
> 1268 Young Street, Suite #204
> Research Center, University of Hawaii
> Honolulu, HI 96814
>
>

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Swank, Paul R

Re: Treating Ordinal Data as Continuous

In reply to this post by Laurie Petch

IRT is certainly one way in which we may be able to create more interval
measures. However, scales developed under classical theories may not
meet assumptions for IRT so it is not always possible to rescale an
already existing measure that was not developed from an IRT perspective.
Whenever I have compared scales developed by IRT to the same scale
scored in a more traditional way, there is usually a strong relation
between the two, especially when you are not close to the extremes. In
fact, my reason for liking IRT scales is the increased variance in the
extremes relative to classically derived scales. But I think many
clasically derived scales can be analyzed using ordinary parametric
procedures without too much loss of accuracy. I certainly would not like
to go back to the early 1950s when everyone was jumping from the
parametric ship onto the nonparametric one. Too limiting.

Paul R. Swank, Ph.D. Professor
Director of Reseach
Children's Learning Institute
University of Texas Health Science Center-Houston

-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
Dates, Brian
Sent: Tuesday, January 09, 2007 2:37 PM
To: [hidden email]
Subject: Re: Treating Ordinal Data as Continuous

Bob's comments summon some of the reason that I became interested in IRT
as a way of looking at ordinal measures from an interval perspective.
One of the options is to examine using IRT to develop interval data and
then analyze that.

Brian

> At 07:00 AM 1/9/2007, Swank, Paul R wrote:
> >The problem is compounded when one thinks of measurement scales as
> >all or none, it's either ordinal or interval. However, win, place,
> >and show in a horse race is clearly more ordinal that IQ scores. For
> >IQ scores, it is clear that the difference between an IQ of 50 and an

> >IQ of 75 is perhaps greater than th difference between 75 and 100. On

> >the other and, the difference between an IQ of 100 and 101 is
> >probably pretty similar to the difference between 99 and 100. It all
> >relies on the relation between the scale values and the underlying
> >construct. I think some scales are closer to being interval than
> >ordianl while for others, the opposite is true. A lot has to do with
> >how well the scale was constructed.
>
> My main concern with this issue comes from satisfaction and importance

> scales. It seems to me that we don't have enough tools regarding
> ordinal measures. The analytical tool set for interval data is far
> richer than what is available for ordinal data. The median can be
> substituted for the mean, but there is no analytical equivalent of
> variance, even though it makes intuitive sense that an ordinal scale
> with results congregating at the extremes ought to be more 'variable'
> than a scale in which the results congregate around the median.
>
> Bob Schacht
>
> Robert M. Schacht, Ph.D. <[hidden email]> Pacific Basin
> Rehabilitation Research & Training Center
> 1268 Young Street, Suite #204
> Research Center, University of Hawaii
> Honolulu, HI 96814
>
>

Confidentiality Notice for Email Transmissions: The information in this
message is confidential and may be legally privileged. It is intended
solely for the addressee. Access to this message by anyone else is
unauthorised.
If you are not the intended recipient, any disclosure, copying, or
distribution of the message, or any action or omission taken by you in
reliance on it, is prohibited and may be unlawful. Please immediately
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you.

Arthur Kramer

Re: Treating Ordinal Data as Continuous

In reply to this post by Laurie Petch

Ordinal data may tend to act like interval when, (I believe) like you state,
when several items used to define a construct are summed, averaged, and
correlated to allude to an underlying factor. The resultant items, after
factor analyzing the larger set and elimination of those items with low
inter-correlations, can be said to operationally define the scale construct.
These may then be collapsed to provide a measure of the construct.

Unfortunately, oftentimes in the social sciences the methodologies neglect
the important steps of factoring out the underlying concept-as a matter of
fact, there is frequently only one (poorly constructed??) statement to used
to "operationally define" the construct in question; and this item is then
incorrectly referred to as a "Likert scale", totally disregarding the steps
Likert outlined in scale construction because it is a laborious process.

Maybe, if you took many different cheeses and extracted the primary
component(s) of the cheese (butter fat??), you could use some of that in you
recipe-but that would depend on whether it's goat milk cheese, sheep milk
cheese, or cow milk cheese, I guess.

Arthur Kramer
-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
Laurie Petch
Sent: Tuesday, January 09, 2007 5:01 AM
To: [hidden email]
Subject: Treating Ordinal Data as Continuous

Apologies, this is more of a statistical question, though it does have an
indirect bearing on SPSS. As subscribers
to this list will know, in psychology it is common practice to treat ordinal
level data deriving from rating scales as if
they were continuous data and subjecting them to inferential statistical
analysis. The argument I have heard in
favour of doing this is that ordinal data behave much more like continuous
data when they are summed and
averaged. I have not seen this argument in writing, however, and would be
grateful to anyone who can point me
in the direction of a relevant source.

Also, if anyone can suggest counter-arguments to this justification, that
would be great too. It just strikes me that a
score of '120' as opposed to '118' on an anxiety measure is data of a very
different kind than someone who is
120cm tall as opposed to 118cm.

To say that ordinal data behave like continuous data is surely rather like
saying that, since cheese 'behaves' more
like butter when it is heated, it's okay to use cheese instead of butter to
make a cake?

Laurie
--------
Laurie Petch
Chartered Educational Psychologist (British Psychological Society)

Beadle, ViAnn

Re: Treating Ordinal Data as Continuous

In reply to this post by Bob Schacht-3

The number of tools for handling ordinal data are getting richer all the time. As an example, SPSS provides the PLUM procedure which from the can be accessed under Anayze>Regression>Ordinal... in 15.0 base SPSS. In earlier versions it was available as an option.
-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of Bob Schacht
Sent: Tuesday, January 09, 2007 1:54 PM
To: [hidden email]
Subject: Re: Treating Ordinal Data as Continuous

At 07:00 AM 1/9/2007, Swank, Paul R wrote:

Kooij, A.J. van der

Re: Treating Ordinal Data as Continuous

In reply to this post by Bob Schacht-3

SPSS Categories offers several data analysis techniques for nominal, ordinal, and interval data:

CATREG: multiple regression (and one-dimensional discrimant analysis); Regression menu, Optimal Scaling.

CATPCA: principal component analysis; Data Reduction menu, Optimal Scaling, One set.

OVERALS: canonical correlation analysis for 2 or more sets of variables (and multi-dimensional discriminant analysis); Data Reduction menu, Optimal Scaling, Multiple sets.

These procedures optimally quantify categorical variables in the context of the analysis method. That is, with CATREG the quantifications are optimal for maximizing the R-squared, with CATPCA the quantifications are optimal for maximizing the VAF, and with OVERALS the quantifications are optimal for maximizing the canonical correlation.

The quantified variables are numeric variables and can be used as input to standard analysis methods.

The researcher chooses an optimal scaling level for each variable, defining how much freedom is allowed in the in the quantifications. For example, with ordinal scaling level, the quantified variable is restriced to have a monotonic (ordinal) relation with the categorical variable (thus, a higher rank number will receive a higher quantified value than a lower rank number).

For interval variables, numeric scaling level can be choosen to analyze them at interval level (the variable is only linearly transformed to standard scores), or one of the other scaling levels to obtain optimal nonlinearly transformed variables.

If you are not sure if ordinal variables can be analyzed as if they are interval variables, you can use one of the above methods with ordinal scaling level (or monotonic spline) for all variables and compare results to the results when treating the variables as interval. For scale construction, usually there is not much difference between results of CATPCA ordinal and standard PCA.

Anita van der Kooij

Data Theory Group

Leiden University

________________________________

From: SPSSX(r) Discussion on behalf of Bob Schacht
Sent: Tue 09/01/2007 20:54
To: [hidden email]
Subject: Re: Treating Ordinal Data as Continuous

At 07:00 AM 1/9/2007, Swank, Paul R wrote:

My main concern with this issue comes from satisfaction and importance
scales. It seems to me that we don't have enough tools regarding ordinal
measures. The analytical tool set for interval data is far richer than what
is available for ordinal data. The median can be substituted for the mean,
but there is no analytical equivalent of variance, even though it makes
intuitive sense that an ordinal scale with results congregating at the
extremes ought to be more 'variable' than a scale in which the results
congregate around the median.

Bob Schacht

Robert M. Schacht, Ph.D. <[hidden email]>
Pacific Basin Rehabilitation Research & Training Center
1268 Young Street, Suite #204
Research Center, University of Hawaii
Honolulu, HI 96814

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Laurie Petch

Re: Treating Ordinal Data as Continuous

In reply to this post by Laurie Petch

Many thanks to all those who replied to this thread on and off list. This list is a mine of useful information,
populated by some very skillful miners! I think I have read all the messages in this thread, though it's a little difficult
to navigate the digest with the browser I am using.

Please can anyone tell me what IRT stands for and offer some references on how to go about it, in relation to a
previously constructed scale?

with thanks,

Laurie
---------

Laurie Petch
Chartered Educational Psychologist (British Psychological Society)

Regina, Canada

statisticsdoc

Re: Treating Ordinal Data as Continuous

Laurie,

IRT = Item Response Theory.

Here is a good website to get started (including references):

http://edres.org/irt/

Here is a tutorial from the University of Illinois IRT lab:

http://work.psych.uiuc.edu/irt/tutorial.asp

Cheers,

Stephen Brand

For personalized and professional consultation in statistics and research
design, visit
www.statisticsdoc.com

-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]]On Behalf Of
Laurie Petch
Sent: Wednesday, January 10, 2007 5:47 AM
To: [hidden email]
Subject: Re: Treating Ordinal Data as Continuous

Many thanks to all those who replied to this thread on and off list. This
list is a mine of useful information,
populated by some very skillful miners! I think I have read all the messages
in this thread, though it's a little difficult
to navigate the digest with the browser I am using.

Please can anyone tell me what IRT stands for and offer some references on
how to go about it, in relation to a
previously constructed scale?

with thanks,

Laurie
---------

Laurie Petch
Chartered Educational Psychologist (British Psychological Society)

Regina, Canada