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For each subject, we repeatedly measure 3 conditions for a scalar dependent variable. The research hypothesis is that there is a monotonic positive trend - that is, either 1<2<3 or 1=2<3 or 1<2=3.

Preferably, I would like both a hypothesis test for this condition relationship (i.e., can we reject the hypothesis that there is some monotonic trend?), and one for its opposite (can we reject the hypothesis that there is no or a negative or a non-monotonic trend?).

I've come up with a few naive ways of testing for this, but I wonder what the optimal way would be.

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    $\begingroup$ Optimal in what sense? Power? Under what assumptions? $\endgroup$ – Glen_b -Reinstate Monica Sep 2 '15 at 11:22
  • $\begingroup$ An UMPF would be nice, although I have a mostly pragmatic perspective - something that's the standard/most accepted method in this case would also be good to know. What kind assumptions do you mean - the distribution of the data, or the dependence between IVs and DVs? $\endgroup$ – jona Sep 2 '15 at 11:53
  • $\begingroup$ I meant in terms of distribution, yes. Or whether you sought a nonparametric test. $\endgroup$ – Glen_b -Reinstate Monica Sep 2 '15 at 11:55
  • $\begingroup$ The DVs are strongly assumed to be normally distributed. $\endgroup$ – jona Sep 2 '15 at 11:57
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    $\begingroup$ An older reference book would be the one by Barlow, Bartholomew, Bremner and Brunk, Statistical inference under order restrictions. (Not certain it has your exact case in there though. There are more recent developments of course.) $\endgroup$ – Glen_b -Reinstate Monica Sep 2 '15 at 12:00
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You need order restricted inference, there is the old book referenced by @Glen_b in comments, and newer books (and software) with links/references in this post: Hypothesis test testing for monotonic group mean change

For a more detailed answer, we would need more details about your setup.

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