Re: when is data considered "continuous" for parametric testing?

espyrian wrote:
jeffrey.ellenbogen@xxxxxxxxx wrote:
Hi,
I'm new to statistics, so forgive the basic question:
What does it mean to be "continuous"? I mean, I have some data -- 2
groups of approx. 20 people -- each where any individual person's mean
score (outcome measure) is one of 4 levels: 0, .25, .5 and 1. Can I do
a t-test comparing these two groups, or is that not continous data. I
mean, I know that if it were binary (0 or 1) I would do a chi square,
but when is data considered "continuous"? 5 levels, 100, 10000? The
more I think about it, the more I think that, theoretically, no data is
truly continous. But for practical purposes, what is the lower limit
that satisfies the assumptions necessary for parametric testing (e.g.
t-test, ANOVA)?
Many many thanks.
Jeff


If your data were continuous it could take any value at all between 0 and 1 (.03, .26, .51, .98, .99, etc.). However, your data are ordinal. Only four distinct classifications are possible, and each classification ranks its position in the ordered sequence. Think of Goldilocks. The bears' porridge may have been 62.5C, 19.4C and 42.0C. But it was recorded as too hot, too cold and just right. On an ordinal scale this might be represented as +1, -1 and 0.

I would suggest the Wilcoxon Signed Rank Test for testing two groups of ordinal values.

The Wilcoxon signed ranks test is for paired samples. But I think the OP is describing 2 independent groups. So the common rank-based test would be the Mann-Whitney U, or the equivalent Wilcoxon rank-sum test. But given that there are only 4 possible scores, there would be a large number of ties. And in that case, rank-based tests are not optimal.

Going back to the original post, because there are only 4 possible values, the data are discrete, not continuous. But that doesn't preclude the possibility of measurement on an interval, or even ratio scale. (E.g., counts are discrete, but have ratio scale properties.) What is the outcome variable?

If you can make a strong case that the variable has (approximate) interval or ratio scale properties, I see nothing wrong with a t-test, despite the discrete nature of the variable. If you cannot make that case, then you could do a chi-square test on the 4x2 table. One drawback is that the ordinary chi-square test does not take advantage of the ordinal information in your outcome variable--i.e., you get the same result if you re-order the 4 rows. But there are ways to use the ordinal iformation. Dave Howell discusses one such method (i.e., test of linear-by-linear association) here:

www.uvm.edu/~dhowell/StatPages/More_Stuff/OrdinalChisq/OrdinalChiSq.html

Hope this helps.

--
Bruce Weaver
bweaver@xxxxxxxxxxxx
www.angelfire.com/wv/bwhomedir
.