Dear Victoria,
the reason I've mentioned learning was your statement "In our primary analyses, we would like to combine the data", which I've interpreted as if you might be interested in differences between runs / interactions at a later point. For that purpose subjects with a single run are useless.
But this is a minor matter, which might also be irrelvant for your study. However, at some point you decided that your experiment would last x minutes, consist of y trials, and so on. For the "bad" subjects only half of the data seems to be analysable. It's not the major part, like excluding just the last few trials, it's only 50 % of your experiment. This is definitely not the way the study was conceived. So even if they have completed the study, they haven't completed it successfully. In that case I would really suggest to exclude the subjects completely.
If only the fMRI data is corrupted, then you could still present the behavioral data of all the subjects and then focus on the fMRI data of the "good" subjects. This might be misleading to some extent though, as you might detect significant effects on behavioral level due to the larger number of subjects, but fail to find anything on neural level just due to the smaller number of subjects.
Concerning "bad" runs, what do you actually mean with "low SNR"? Does this refer to the quality of the raw data (scanner artefacts, head motion, ...) and if so, how did you define the criterions? I just thought of this issue because some people interpret data as "bad" if they don't find anything in the GLM. This sort of data selection is bad science of course.
Best,
Helmut
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