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5 pupils are given the task of hand drawing a hundred trapezoids with a given perimeter. The vertices are marked A,B,C,D counter-clockwise starting with the longer parallel edge (A and B). The lengths for the 6 pairwise distances between the 4 vertices are recorded. These six distances are dependent variables as the lengths and angles of edges influence each other.

Assumptions: 1) None of the trapezoids drawn are truly identical. 2) The pupils do not get better or worse in the task over time, they draw the shapes with more or less the same error rate, i.e. the hundred states (i, i+1) are independently drawn.

My hypothesis is that all pupils draw trapezoids with essentially the same AB,AC,AD...CD vertex-vertex distance distributions.

The question: How can I rigorously test which vertex-vertex distances are drawn significantly longer or shorter by any two of the pupils? Please advise which statistical test to use.

First, let's assume normal distribution for the measured distance distributions, but I'm also interested how to proceed if the distances are not normally distributed.

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  • $\begingroup$ Might I suggest an alternative approach? Rather than seeking a formal test, first explore the data: do they even look like what you are hypothesizing? Simply drawing all the trapezoids in a normalized fashion (with a common center, say, and oriented with their major axes horizontal) on a single plot would likely answer most questions you might be asking, as well as suggest new ones. $\endgroup$
    – whuber
    Commented Sep 3, 2019 at 12:46

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If the distances are continuous and are not too close to 0 (truncated distribution) then a regular linear model could solve this, by comparing the means of the pupils.

You say that the pupils do not get better or worse over time, that each observation is independent. I don't know how this is possible but whatever, I would still include a random effect for each pupil to account for the fact that each pupil will draw their own version, even if it does not change over time.

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  • $\begingroup$ May I ask how you would use a regular linear model? $\endgroup$
    – Tommy
    Commented Aug 23, 2019 at 7:31
  • $\begingroup$ @Tommy Well, if you are interested in all 6 distances, then you will have to perform multivariate regression, so 6 different regression results for each distance. If you want more information you will have to supply a sample of your data. $\endgroup$ Commented Aug 23, 2019 at 7:43
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    $\begingroup$ Please correct me if I am wrong but as I understand, the use of multivariate regression is to try to understand the functional relationships between the dependent variable (Y) and independent variables (X1, X2...), to try to see what might be causing the variation in the dependent variable. But in this case the variables (X1, X2...) are the vertex-vertex distances and they are not independent. At this point it is a bit unclear how you would confirm the hypothesis or reject it by finding some outlier distances. $\endgroup$
    – Tommy
    Commented Aug 23, 2019 at 8:41
  • $\begingroup$ @Tommy I see what you are saying, your problem currently is quite complicated. You have 6 dependent variables and multiple independent variables which are not independent. The easiest option would be to get only 1 measurement out of each, to get rid of this nastiness. So for ex. 1 DV and a couple of truly IV. Otherwise you will have to perform the multivariate regression equivalent for compositional data, that is when the IV for ex. sum up to a constant. $\endgroup$ Commented Aug 23, 2019 at 12:07

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