ANOVA with non-independent observations

Question

Sorry for the verbose background to this question:

Occasionally in investigations of animal behaviour, an experimenter is interested in the amount of time that a subject spends in different, pre-defined zones in a test apparatus. I've often seen this sort of data analyzed using ANOVA; however, I have never been entirely convinced of the validity of such analyses, given that ANOVA assumes the observations are independent, and they never actually are independent in these analyses (since more time spent in one zone means that less is spent in other zones!).

For example,

D. R. Smith, C. D. Striplin, A. M. Geller, R. B. Mailman, J. Drago, C. P. Lawler, M. Gallagher, Behavioural assessment of mice lacking D1A dopamine receptors, Neuroscience, Volume 86, Issue 1, 21 May 1998, Pages 135-146

In the above article, they reduce the degrees of freedom by 1 in order to compensate for the non-independence. However, I am not sure how such a manipulation can actually ameliorate this violation of ANOVA assumptions.

Perhaps a chi-squared procedure might be more appropriate? What would you do to analyze data like this (preference for zones, based on time spent in zones)?

Thanks!

Answer 1

(Caveat Emptor: I'm not an expert in this area)

If you just want to talk about differences in time spent per location, then submitting the "time-per-location" data as counts in a multinomial mixed model (see the MCMCglmm package for R), using subject as a random effect, should do the trick.

If you want to talk about differences in location preference through time, then maybe bin time to reasonable intervals (maybe to the resolution of your timing device?), classify each interval according to the mouse's location at that time (eg. if 3 locations, each interval gets labelled either 1, 2, or 3), and again use a multinomial mixed effects model with subject as a random effect but this time add interval as a fixed effect (though possibly only after factorizing interval, which drops power but should help capture non-linearities through time).

Answer 2

Mike,

I agree that an ANOVA based on total time probably isn't the correct approach here. Further, I'm not convinced that Chi Sqaure solves your problem. Chi square will respect the idea that you can't be in two locations at the same time, but it doesn't address the problem that there are likely dependencies between time N and time N+1. In regards to this second issue, I see some analogies between your situation and what people run into with eye and mouse tracking data. A multinomial model of some sort may serve your purposes well. Unfortunately, the details of that type of model are beyond my expertise. I'm sure some statistics book somewhere has a nice little primer on that topic, but off the top of my head I'd point you towards:

• Barr D.J. (2008) Analyzing ‘visual world’ eyetracking data using multilevel logistic regression. Journal of Memory and Language, Special Issue: Emerging Data Analysis (59) pp 457-474
• https://r-forge.r-project.org/projects/gmpm/ is a non-parametric approach to the same issue being developed by Dr. Barr

If anything, both of those sources should be more than complete because they get into how to analyze the time course of the position.

Answer 3

Look into models with spatially correlated errors (and spatially correlated covariates). A brief introduction, with references to GeoDa, is available here. There are plenty of texts; good ones are by Noel Cressie, Robert Haining, and Fotheringham et al (the last link goes to a summary, not a book site). Some R code has recently been emerging but I'm unfamiliar with it.

Answer 4

I am going to suggest an answer that is very different from that of a traditional ANOVA. Let T be the total time that is available for an animal to spend in all the zones. You could define T as the total amount of waking time or some such. Suppose that you have J zones. Then by definition you have:

Sum T_j = T

You could normalize the above by dividing the lhs and the rhs by T and you get

Sum P_j = 1

where P_j is the proportion of time that an animal spends in zone j.

Now the question you have is if P_j is significantly different from 1 / J for all j.

You could assume that P_j follows a dirichlet distribution and estimate two models.

Null Model

Set the parameters of the distribution such that P_j = 1 / J. (Setting the parameters of the distribution to 1 will do.)

Alternative Model

Set the parameters of the distribution to be a function of zone specific covariates. You could then estimate the model parameters.

You would choose the alternative model if it outperforms the null model on some critera (e.g., likelihood ratio).