# How to average ordinal items across time

We have 1 pre-stress measurement point (t1), then a severe stressor, and after many months, 3 consecutive measurement points (t2 t3 t4) that are very close together. We cannot really handle these as individual measurement points (they are very similar) - to increase reliability of these 3 timepoints t2-t4, we would like to utilize all information and merge them into one post-stress timepoint and then run longitudinal analysis with 1 pre and 1 post-stress timepoint.

The items are, unfortunately, ordinal (0,1,2,3). Averaging them leads to the problem that (except for the fact that you shouldn't average ordinal symptoms) t1 is ordinal, but the subsequent measurement point is metric (averaged t2-t4), and using these variables in statistical analyses may be very problematic.

How would you go about this? Could one justify to give a person a post-stress "2" if the person has t2="2" t3="2" and t4="3", for example? Basically, averaging metric and then rounding back to ordinal?

Thank you!

• To average, or even add - at all - you already assumed the variables were at least interval. If you average t2-t4, you assumed they were all interval. Why leave t1 out of this rather sweeping assumption? The phrase "rounding back to ordinal" makes no sense to me. A rounded interval-scale random variable is simply rounded interval scale. – Glen_b Dec 29 '13 at 16:32

## 2 Answers

As in general it is not quite proper to do interval-scaled computations on ordinal variables, it may be better to keep the ordinal nature of each measurement, and to use a repeated measures ordinal logistic model such as a proportional odds model with random effects for subjects. This can be done using the R package ordinal.

• Thanks - I've worked with the package before. The problem is that there isn't really much difference between t2 and t4, seeing that they are really close together, and very much closer than t1 and t2. So it does make sense trying to merge t2-t4 somehow and use only 2 measurement points instead of 4 (it's a reanalysis of data, so i cannot really tell the reason why the data were assessed in this fairly untypical way). That's where my question comes from. – Torvon Dec 29 '13 at 16:27
• You can cause trouble using the data to change the model in that way (multiplicity problems, falsely low standard errors). – Frank Harrell Dec 30 '13 at 3:46

Unless you are making an arbitrary decision to choose one of t2, t3 or t4 (i.e. pick one randomly for each observation of t1), I don't see how you can merge t2 through t4 in a kosher way.

If you're submitting this analysis to a journal, your version of averaging likely won't be deemed proper by the review committee. If you're doing this for a business, you may be able to get away with it (but that's not really the point, obviously).