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I have an experiment that involves testing the route-finding ability of 3 different critters. They have to travel between 5 different points (essentially a travelling salesman problem) and for each organism I measure the total length of the route. The experiment was repeated 16 times for each critter. I want to answer the question: are the crittres choosing routes that are significantly shorter than if they were selecting random routes? I was planning on running an ANOVA (or a a Welch's ANOVA) to compare the critter's route lengths to the 'random' route lengths. My question is: which is the best way to compare the 'random' paths to my organism's paths?

  1. Compare the route lengths of all critters with a column containing all possible routes (24 possibilities). (so an ANOVA with 'Critter 1 path lengths', 'critter 2 path length', 'critter 3 path lengt' and 'all possible routes' as treatments).

  2. Compare critter route lengths to a sample of possible routes, matched to sample size (so subsample 16 out of the 24 possibilities as my sample size for the other critters is 16)

  3. Choose 100 random routes from the 24 possibilities (with replacement)?

I initially thought option 1 was sufficient, then moved towards 2 and am now just confused. Any advice would be greatly appreciated.

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  • $\begingroup$ Three different types of critters, or just three individual critters? $\endgroup$ – Jay Schyler Raadt Nov 13 '18 at 11:52
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    $\begingroup$ i am not exactly sure how the question in the heading relates to the questions in the body. some months ago i researched the heading question you and found this compelling answer: researchgate.net/post/… $\endgroup$ – Winfried Nov 13 '18 at 12:33
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    $\begingroup$ Title: When you have entire population, there is no statistical analysis. $\endgroup$ – user158565 Nov 13 '18 at 14:32
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If there are 24 possible routes, then you can compute an expectation for the route lengths assuming equal probability of all routes or under some other condition. This can be a population parameter which you intend to compare the critters against.

Then pool the critters' data together and run an intercept only model. The intercept will be the average route length for all critters. And obtain a confidence interval for the intercept to see if it includes the expected average of all routes. If it doesn't, this is the first suggestion that the critters are on average behaving differently than expected on average. If it does, then you cannot distinguish their average behavior from the expected under the null. Note that your regression need not be linear regression. You have 48 data points so you might have enough data to identify an appropriate distribution for the travel times other than the normal distribution assumed during inference in linear regression.

One problem is that 16 of the 48 data points come from the same critter. So you lack statistical independence. Adjustments you can make to the general model above are:

  • a multilevel regression approach adding a random intercept for each critter. In addition to the overall intercept above, you can use this model to create a confidence interval for each critter so you can see how each specific critter differs from the expected. But some might complain that your number of critters is too small to do this.
  • a critter by critter model. This is a simple approach. You may lose statistical power though and some might complain about multiplicity issues.

I like the multilevel regression approach generally, despite the small number of critters. It is not ideal but you have the data you have. You could take a Bayesian approach, and constrain the parameters further with some priors that express skepticism of differential behavior from the average of 24 routes. This will probably be the most interesting approach to take.

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