Clem has prepared well and investigated a lot so I guess Clem deserves an answer. (And not insulted anybody, so maybe I dare to suggest something.)

I think it is reasonable to think of the explanatory variables depth and location in a nested model.

But what about the response variable? Since it is a count variable I guess that, or rather I am sure that, many statisticians would suggest a Poisson model – that the response variable is Poisson distributed with “means” given by the explanatory variables depth and location. Or negative binomial distribution as a second alternative that gives a little bit more flexibility than the Poisson.

Such models can be estimated within the framework of generalized linear models (glm) and can be estimated with most standard software. Anova is a special case of glm, so it is not that strange.

Clem has taken a number of photos at within one hour and pick out the photo with the maximum number of sharks.

What does that mean? I don't know, so I throw out this question to TalkStats:

Given that the Poisson intensity is fixed within an hour, what distribution will the random variable Y = max(X1, X2,...,Xm) have? Where Xi are measurements of number of sharks (Xi~Po(mu)) taken sufficiently far away from each other in time to be considered independent random variables.

If that distribution is known then maybe it is possible to continue in the analysis.

The above was trying to answer the problem as it was given to us by Clem. But why should we accept the problem as it has been formulated? We can reformulate it.

In analysis of variance (anova) one is trying to estimate the mean, given the explanatory variables. Clem formulates the maximum above. Isn't it more natural to estimate the mean? (which is also the Poisson parameter)?

Instead of having:

5 Boo 4

I suggest to have many rows per hour like:

5 Boo 1

5 Boo 0

5 Boo 3

For, say one photo taken every minute (or every 30 second) within the hour.

This would give a three level model. Depth and in that location and in that one specific hour and the re would be the different measurements every minute.

If the model had been normally distributed if would had been easy just to average the minutes measurements to the hour level and run an anova on that. Now it is more complicated because the discrete distribution (the Poisson to start with).

Others can continue and explain this better. (Good luck!)

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And now to Clems part of this: Is is safe to swim in the North Sea? What about the Mediterranean?