Power analysis for repeated measures design with ordinal response

Question

I am designing a randomised pretest-posttest trial for which I'm trying to do a sample size calculation. My brain is about to explode from confusion, so I hope someone can help me.

Just a little summary of what the trial will look like. We have 2 interventions which will be delivered in 3 arms: intervention 1 only, intervention 1+2, intervention 2 only. The outcome of the trial is a difference in stage of change, which is a 5-category outcome reflecting readiness to change. Each participant will be in one stage at start and the same or another stage at the end of the trial. A higher number of stage is better. However, a difference between stage 1 and stage 2 can not be considered equal to a difference between stage 3 and 4.

I am not sure yet how the outcome will be presented, but I think it will look like this. I will categorise each participant as either improving (e.g. from stage 2 to 4), no effect (e.g. remaining in stage 3) or decreasing (e.g. moving from stage 4 to 1). Another option is presenting the distribution over the 5 stages (proportions).

Now this is more or less where I get lost. I need to do a sample size calculation and I just don't know how to go about this. I have found previous articles on both the distribution over the stages pre and post, as well as on numbers/proportions of participants in each of the three categories as I described above (increase, no effect, decrease) pre and post.

What type of sample size calculation should I use? And related, what statistical methods can (should) be used to analyse the data?

Answer 1

It might be easier to get a handle on this if you frame it in terms of the probability $p_2$ of being at least Stage 2, the probability $p_3$ of being at least Stage 3, etc. Then you can think about such things as the power of comparing two proportions when the true difference is $|p_{21}-p_{22}| = \delta_2$, where $p_{2j}$ is the value of $p_2$ for the $j$th treatment and $\delta$ is a difference of clinical importance. This would put everything in the framework of comparing proportions. It might be enough to do this for one of the $p_i$, the one that you think a priori will be closest to $\frac12$, where the standard errors will be the highest.

Since you have 3 arms, there are 3 comparisons, and I suggest using a simple Bonferroni correction to account for multiplicity -- e.g., use $\alpha = .05/3$ in the power calculations.