Is there any data analysis that can be performed on both paired and independent data? For example, we have scores on a questionnaire for individuals who were exposed and not exposed (control) to a particular disease, these are independent of one another. However, we also have pre/post data for individuals (they were given questionnaire before being exposed to the disease and after being exposed). I know this is a weird design but it is what we have to work with. My question is do we have to analyze the pre/post and independent samples separately (using paired and independent t tests respectively) or can we combine them together and perform some analysis? Combining them together is preferred because of the fact that we have a very low sample size. For example we would add the pre group with the non-exposed group (because the pre group wasnt exposed) and the post group with the exposed group (because the post group was exposed). What data analysis could you do here?
I found a paper (Derrick, Russ, Toher and White (2017)) that covers this case of partially overlapping data for normal data. However, what would you do if your data didn't meet the normality assumption?