# Re: ANOVA on ordinal data

From: Jeff Sauro (jeff_sauro_at_despammed.com)
Date: 11/23/04

```Date: Tue, 23 Nov 2004 20:21:02 +0000 (UTC)

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The short answer to your question is yes you can do a t-test and ANOVA
on ordinal data. The major caveat comes from interpreting your
results. If you find a significant difference, you should only report
that one group mean is higher or lower than another group mean—an
ordinal statement. You get into trouble if you start making interval
statements such as “group one is twice as much as the other group.”
The major difference between ordinal and interval data is that the
latter has equal differences between each number, whereas with ordinal
data, you cannot say much more that the order (x is greater than y).
With interval data you can say (x is twice as likely as y).

You should know however, that there are two camps when it comes to
this issue. The more purist camp will tell you that you CANNOT use
those parametric tests with ordinal data. The other camp (most social
scientists and practitioners) will tell you it's fine.

The purist camp will cite the work of SS Stevens. He’s the guy who
came up with the whole hierarchy of data (Nominal, Ordinal, Interval
and Ratio) in his 1946 work “On the Theory of Scale Measurement”.
According to Stevens, it’s only permissible to use Interval or Ratio
data to use parametric tests (t-test, ANOVA etc). In fact, even
computing the mean and standard deviation aren’t permissible unless
you have at least interval data.

The “practitioner” camp (for lack of better word) argues that such a
rigid structure, although meaningful for classifying data should not
dictate the tests one performs. They say that “the numbers don’t know
where they come from.” One of the best works is “On the Statistical
Treatment of Football Numbers” by F. M. Lord. Lord shows how football
numbers (Nominal Data) can be averaged and manipulated to settle an
argument about whether Freshman have lower numbers than upper
classmen.

For a good discussion of both sides (with a bias toward the
practitioner camp) you should read a publicly available paper:
“Nominal, Ordinal, Interval, and Ratio Typologies are Misleading”
http://www.spss.com/research/wilkinson/Publications/Stevens.pdf

In fact, the vast majority of scales and measurements used in modern
psychology are ordinal scales! With this in mind, I’d proceed to see
what conclusions you would draw with the t-test and ANOVA and just be