Sample size for t-test when the outcome is percentage point

Question

first time asking question here so I hope I will not mess up.

I am in the process of calculating the sample size for a study that utilizes an independent t-test design: comparing some outcome before and after a certain policy, subjects in the pre and post periods are not linked. Simply put, the study is about infant weaning food distributed in developing countries. The outcome is percent weight of cooking ingredients that is oil, which is fortified with nutrients.

The current recommendation is 9% weight of the baby's meal should be the fortified oil (about 1 part of oil to 10 part of powered grains). The organization is trying to promote a new amount of 16% oil, which was shown to be associated with better anthropometric outcomes.

I could use the usual t-test sample size formula, but I'm concerned that the SD of the mean 9% point may not be meaningful because the outcome is bound between 0% to 100%, the distribution of the percent point at the extremes (like 9%) is likely to be non-normal, and additionally not all caretakers follow the cooking guideline so the SD is probably huge.

So my question is: do you know of any formula is applicable to this situation of comparing mean when the individual (not group) level data are proportion?

Many thanks for reading. I appreciate any advice or reference.

Answer 1

You could use Beta regression to analyze the data, here the response variable is assumed to follow a Beta distribution rather than a normal distribution (Beta is contstrained to be between 0 and 1). Then you could use a dummy variable with before=0, after=1 as the predictor variable in the regression to look for change. You could then also include other covariates in the model if you want to adjust for their effects.

There are not simple tools for sample size/power for beta regressions, but you could compare the powers of different sample sizes by simulation. Choose a before Beta distribution, an after Beta and sample sizes, then simulate data from these and analyze it, repeat a bunch of times. The proportion of times that you reject the null of no change is the power for that combination. Repeat the whole process for different sample sizes and different after distributions and compare them to find a combination of power and sample size that you are happy with.