## OPTION 10

Ordered Categorical Data

Ref: Whitehead, J. Sample size calculations for ordered categorical data. Statistics in Medicine 1993; 12:2257-71.

Many studies yield data on an ordered categorical scale, such as, very good, good, moderate, poor. Under the assumption of proportional odds, such data can be analyzed using the techniques of logistic regression. In the comparison of two groups, this approach is equivalent to the Mann-Whitney test. Sample size and power calculations in this program use formulae consistent with an eventual logistic regression analysis.

This program allows up to ten categories of response and up to ten strata of participants. Strata frequencies may vary but must sum to unity over all strata.

Suppose that the possible categories of response are labeled C1,...,Ck, with Ci being more desirable than Cj if i<j. Let pie denote the probability that an individual receiving the "experimental" treatment gives a response in category Ci, and Qie be the probability of Ci or better:

Qie = P1e+...+Pie i=1,...,k

If Pic and Qic are similarly defined for the "control" group, the parameter:

Oi = log[Qie(1-Qic)/Qic(1-Qie)]

is the log-odds-ratio of the outcome Ci or better for an "experimental" subject relative to a "control" subject. We assume O1=...=Ok-1; this is the proportional odds model.

Strata specific category response probabilities may be individually specified or alternatively, specified as a proportional odds relative to the preceding stratum.

The formulae used are based on Normal approximations and are accurate for moderate to large sample sizes.

The program requires the user to specify the number of strata (which may be one) and the category response probabilities for at least the first stratum. The proportional odds for the "experimental" condition may be greater or less than unity.

Then given any three of the following:

1. Total number of subjects in the "experimental" group and allocation ratio (controls/experimental)
2. One-sided significance level
3. Desired power of the experiment
4. Constant proportional odds ratio associated with the alternative hypothesis,

the program can compute the fourth.