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I have collected data for 30 patients for a certain test. This test was administered at 6 different but consistent intervals, and each test for each patient has lots of variables. If I want to do an analysis that compares both the differences among the intervals AND between the patients at the intervals, what sort of statistical methods are recommended?

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Sounds like a multilevel modeling approach would be useful for you. Check out this tutorial by Woltman et al. or this introduction by Snijders.

Multilevel models in a longitudinal framework allow you to describe differences within individuals (level 1) and between individuals (level 2).

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