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I'm trying to find the right test for to determine the trend of a set of values with some measure of statistical significance. I came across the mann-kendall test but realized it had the limiation of only one value per timepoint. I tried using the median value but I'm not sure that this is correct since I would imagine cases where the distribution for each set of values is very wide on the same timepoints should be treated differently than cases found in a very narrow distributions.

My data consists of multiple measurements per timepoint (some timepoints may have many observations others with very few or none)

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You could still test for a Kendall correlation between the observations and their time index.

However you'll have many ties, so ordinary tables for it won't work. You'll either need an asymptotic approximation with an adjusted variance or software that correctly calculates the permutation distribution, or you'll need to approximate it by simulation (sampling the permutation distributions).

However the differing spreads at different time points may raise the question of whether you'd have exchangeability under the null (or a reasonable approximation of it). The fact that the spreads look different in the data doesn't mean that you couldn't have it under the null (the point null is almost certainly to be false, so the appearance of the data doesn't tell you whether or not it's an issue) -- but you should perhaps consider its plausibility if there was truly no trend.

As an example, consider a set of observations over time whose spread is related to the mean (spread increases when the mean increases). Then if under the null you assume that all the means are the same, the spreads would presumably also be the same and the different spreads in the data would be of no consequence.

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