# Comparing medians from two groups

Hi: I have a survey where respondents were asked to rank a series of statements in terms of how much they agreed with them. There were seven statements. So, for each of seven statements, each respondent has value that ranges from 1 to 7, depending on where they ranked that item. Two questions: first, should I conceive of this as an ordinal or an interval level variable. I'm inclined to think of it as an ordinal variable, because I'm not sure how meaningful a mean is in this context. It seems more intuitive to say the median ranking of this group of respondents' for this statement was 2, compared to 5 for a different group, rather than, say, 3.6 compared to 5.5, or whatever.

So, my first question is, is that sound logic?

The second question flows from this: If I do conceive of this as an ordinal level variable, rather than an interval, what statistical test can I use to test for difference in median values? I have read a little bit that refers to a Wilcoxon-Mann-Whitney test for difference in medians, but I'm not sure if that is proper for a cross-sectional survey of around 150 people at the same point in time.

• I would consider interpreting the ranking as the realization of a latent scoring process. Then your null hypothesis can be formulated in terms of the latent score. – Arthur B. Nov 20 '14 at 15:39
• This is not really all that helpful to me. I don't really know what a latent scoring process is. – spindoctor Nov 24 '14 at 15:31
• Sure but you can make a reasonable guess. For instance, you could assume that each group scores each item with an exponential distribution with a given mean per item and then orders the result. Null hypothesis only makes it look like you're agnostic about the hypothesis. – Arthur B. Nov 24 '14 at 15:37
• Actually, I lied. I probably won't. I'm not a statistician, in fact, I'm entirely self-taught. This analysis is for a journal that is not primarily statistical. I'm just trying to do my level best to do the best analysis possible, without going off on a wild goose chase learning some obscure statistical procedure. I did that once trying to learn how to do survival analysis properly. It took me months. Is there something wrong with doing either a difference in median or mean analysis like I presented? – spindoctor Nov 24 '14 at 21:57

I have read a little bit that refers to a Wilcoxon-Mann-Whitney test for difference in medians

The Wilcoxon-Mann-Whitney is not a test for equality of medians, it's a test for one variable being stochastically larger than another ($P(X>Y)>P(Y>X)$). If you're using it as a location test, it's actually a test for a zero (population) median of pairwise differences.

If you make the additional assumption of identical distributions apart from a location shift then it will be a test of medians (but it would also be a test of means, as long as population means exist).

I have a survey where respondents were asked to rank a series of statements in terms of how much they agreed with them. There were seven statements. So, for each of seven statements, each respondent has value that ranges from 1 to 7, depending on where they ranked that item.

So that's seven numbers per respondent. What did you do with those 7 numbers?

first, should I conceive of this as an ordinal or an interval level variable.

If you just added the scores on each item, you already assumed the 7 items were interval scale when you did that.

because I'm not sure how meaningful a mean is in this context.

If you can add a "2" and "6" and get the same result as when you got a "3" and "5", then everything needed for a mean to be meaningful was already assumed to be true.

It seems more intuitive to say the median ranking of this group of respondents' for this statement was 2, compared to 5 for a different group, rather than, say, 3.6 compared to 5.5, or whatever.

If you find a median intuitive, that's fine, there's nothing stopping you working with medians -- that doesn't require you to assume it ordinal even though you already made it interval.

You can always consider a permutation test; it assumes exchangeability under the null, which is a somewhat stronger assumption than the Wilcoxon-Mann-Whitney needs.