# Sample size calculation for paired ordinal data

What formula should be used for sample size calculations for paired ordinal data? Imagine we have an intervention and we want to measure the effect based on before and after measurements of an ordered outcome (say a Likert scale).

I know how to calculate for a t-test for paired continuous data and how to calculate for a Mann-Whitney test for ordinal data (Whitehead 1993), but what about paired ordinal?

Either a nonparametric solution or a parametric solution through a GLM for ordinal response would work I think.

• You may find the following paper helpful Mike Julious SA, Campbell MJ, and Altman DG (1999). Estimating sample sizes for continuous, binary, and ordinal outcomes in paired comparisons Journal of Biopharmaceutical Statistics, 9, 241-251 Jun 23, 2015 at 11:05

The standard non-parametric test for paired ordinal data is the Wilcoxon, which is sort of an augmented sign test. I don't know of a formula for power analysis for the Wilcoxon, but you can certainly get power analyses for the sign test (there are various resources listed in my question here: Free or downloadable resources for sample size calculations). Note that (as @Glen_b notes in the comment below), this would assume that there are no ties. If you expect there will be some proportion of ties, the power analysis for the sign test would give you the requisite $N$ excluding the ties, so you would inflate that estimate by multiplying it by the reciprocal of the proportion of untied data you expect to have (e.g., if you thought you might have $20\%$ tied data, and the test required $N=100$, then you'd multiply $100$ by $1/.8$ to get $125$). Unless you need the minimum $N$ to achieve a specified power, that should work for you. For example, when running power calculations for more complicated analyses, we often use a simpler calculation and then say something like 'our $N$ was calculated to achieve a minimum of 80% power on the sign test, because the Wilcoxon can be expected to be at least as powerful as the sign test, our power should meet or excede 80%'.