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I have an experiment where the response to a treatment is measured over time. I now want to test for if the response profile differ between the treated group and the control group. There are measurements for 5 time points (the same for both groups), and at each time point 6 animals are killed and measured for each group.

I guess what I'd like to test for is if it's likely that the centroid profiles for the two groups come from the same distribution, given the variance in the measurements. My problem is then 1) how to define distance? and 2) how to define variance? I could do it per time point and then just do 5 t-tests and use then mean/max/number/something p-values as the representative p-value, but I have a feeling there should be a better solution.

Some assumptions: - The groups are scaled to zero mean and unit variance. - If necessary I can assume that the variance is the same for each time point, but it's quite a large simplification. - I don't want to make any assumptions or modelling regarding the shape of the profiles. - The sample sizes are too small to know anything about the underlying distributions, but I have no reason to believe that they should be very different from normal.

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  • $\begingroup$ I put this as "time-series", but maybe that requires that the measurements are done on the same sample over time? That is not the case here since there is only one sample point per animal. $\endgroup$ – Rasmus Apr 18 '17 at 9:20
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Your dependent data are not single data points, but entire curves, or functions over time. This is known as "functional data", and I'd recommend you take a look at our tag.

This is a big field, and as you already know, the first step would be to think deeply about what specifically it means that "two curves are different". Your best bet would likely be to take a look at a few textbooks to see potential ways to operationalize this "difference between two curves". For instance:

For visualization, I also found Hyndman & Shang (2010, Journal of Computational and Graphical Statistics) very useful.

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  • $\begingroup$ Thanks! Let's say that I would be fine with a rather simplistic definition of distance. I could for example use the euklidian distance between the centroids or the area between the centroids. It's where to go from there that I'm (even more) unsure about :) $\endgroup$ – Rasmus Apr 18 '17 at 9:36

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