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I have been using a between-subjects repeated measures ANOVA for data of treatment effect where different subjects provide data that vary by time, i.e., each subject has a reading for avg.sit.time1, avg.sit.time2, avg.sit.time3, etc. This has allowed me to assess the influence of time on repeated measurements across subjects.

I was wondering if this is the same approach to use to examine whether there is a difference between a treatment and control group of participants with the same data structure (i.e., repeated measurements, with readings increasing over time: time1, time2, time3, etc.), or would it be better to just take the average measurements for each group and do t-tests?

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What you mention in the second paragraph, i.e. looking for treatment effect over repeated measures on different subjects, is what repeated measures ANOVA is intended for.

However, you should still worry about whether the assumptions of repeated measures ANOVA is met; the one that comes to mind right away to me is sphericity. Given that you have more than 2 measurements over time, you should question whether you are willing to assume that within a subject, all measurements are equally correlated, or is it possible that measurements that are taken closer together in time are more highly correlated than measurements that are further away?

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