# How can I test for a difference between ordered groups?

Let's say I have a questionnaire about a toy and I ask:

1. The person to rate the toy on a scale of "bad, medium, good"
2. How much the person thinks it cost to make the toy

I want to test whether there's a (monotonic) relationship between the two variables. What's the best way of doing this?

1. One way is to run two different one-sided t-tests, comparing (bad, medium) and then (medium, good), but I guess this loses some information.
2. Another way is to use an ordinal logistic regression, where cost is the explanatory variable and the toy rating is the response.
3. Another way is to maybe use something like Spearman rank correlation? This seems kind of funny to me, though, so I'm not sure.
4. Something with (ordered) ANOVA? (I've never really understood ANOVA that well, so I'm not sure.)

What's a good approach to try?

Bonus 1: What are the pros/cons of the different approaches above? Bonus 2: points for pointing to an R function I could use.

• Why does Spearman rank correlation seem 'kind of funny' to you? Seems reasonable to me. – onestop Mar 11 '12 at 20:59
• onestop: I'm not sure why it seems funny, which is why it's only "kind of" :). I guess because I usually think of using Spearman rank between two numeric variables, not where one is ordered categorical -- but perhaps that's precisely because I haven't dealt with ordered categorical variables much. – raegtin Mar 11 '12 at 21:46
• Is there an issue with using ordinal logistic regression? – Michelle Mar 11 '12 at 22:52
• Michelle: not really, but wondering if there's a better way / curious about other ways. (I don't need everything that ordinal logistic regression provides me. For example, in the case of a categorical variable with just two outcomes, yeah, I guess I could use ordinary logistic regression -- but it's probably better to just use a t-test or a Wilcox test, since all I want to know is whether there's a difference, and I don't care about building a model.) – raegtin Mar 11 '12 at 22:57

Looks like Page's trend test would suit you well - it could help you to test hypothesis $H_0\colon m_{good}=m_{medium}=m_{bad},$ where $m$ is a measure of central tendency of estimated cost, against the ordered alternative $H_1\colon m_{good}>m_{medium}>m_{bad}.$ It also nonparametric, so you don't have to assume the distributions are normal. See test description and R implementation.