Sample size calculations for paired or matched ordinal data

The problem of calculating the number of subjects in a paired or matched study in which the outcome variable is ordinal is discussed. A common approach in the case of a two category variable is to calculate the required number of discordant pairs, and then divide this by the expected proportion of discordant pairs to obtain the total sample size. An approximate solution for the number of discordant pairs is proposed for ordinal data and compared to sample sizes estimated through simulation. It is shown that the sample sizes are underestimated when the number of categories is two, but that the approximation improves as the number of categories increases. Comparison of the required discordant sample size when there are two categories with the required sample size for more than two categories would suggest that the loss of power is not great if a categorical variable is collapsed into only two categories. However, the total sample size required is likely to be greater with only two categories, since the expected proportion of discordant to concordant pairs increases. Since the expected number of discordant pairs is likely to decrease as the number of categories increases, this suggests that as a rule of thumb the required discordant sample size for the two category case be used as an approximation to the total required sample size when the number of categories is greater than two.