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I am not sure how the independence assumption in ANOVA applies in this scenario...

We have observational data on the foraging depth of whales across the summer separated into time intervals. We want to see if the depth is consistent over the summer, so would it be appropriate to use ANOVA with the time intervals as the grouping category? Would they be considered independent? Is there a better method to analyze this?

also, does it matter that we can't identify whether the same whales are being counted in multiple time intervals?

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Let's start with the final question.

If you believe suspect whales are homogeneous - i.e., overall behave the same way - it should not matter if they are the same whales. You can do that much more expensive research later, it here is a doubt.

You aren't clear on 'time intervals'. Should we assume you have one year, broken up into months or quarters, or many years?

Let's suppose you have one year of monthly data. The typical approach is to split it into 'summer' and 'winter'. Whether or not fall and autumn are included is a detail.

But, if you have a bunch of depth readings in each group, ANOVA should do it.

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