## Sunday, October 16, 2011

### The unholly alliance between $$\phi_t$$ and AIC

In the wild world of mark-recapture analysis, there is a well-worn path to publication.  Two of the key parameters are the survival and recapture probabilities in the marked population[1].  One of mile-markers is comparing two models for each parameter: either the parameter is constant, or it is different for each interval (survival) or occasion (recapture).  Sometimes an enlightened paper will skip this comparison, but there's really no guarantee.
Clearly, it's crucial to know whether the survival rate varies over time.

If you were looking at this question using hypothesis testing, you would take the following as your null hypothesis.

$H_0 : \phi_1 = \phi_2 = ... = \phi_K$
The alternative would be that each phi is independent at each time step or something in between.  If you're using hypothesis testing, this brings up, front and center, the problem that H_0 is crazy.  No ecologist would honestly accept that as a null hypothesis.  Survival varies over time, end of story.  I'm pretty sure that the reason this test comes up is that MARK has a button for it.

Besides the fact that the null hypothesis is crazy[2] (though related), small mark-recapture studies, and most are small, will lack the power to find important temporal variation in survival[3].  If biologists were willing to give their theory a little bit more weight in deciding the issue, it would be sensible to take \phi to have some distribution over time (a "random" effect) and just deal with the estimates.  Even if the estimates were nothing special, at least it would save the world from the umpteenth discussion of whether \phi varies over time.

To further muddy the waters, these days the comparison is carried out using AIC rather than hypothesis testing.  I like ranking a set of well-fitting models as well as anybody else, but in this case it just masks the fact that the study was too underpowered to even contemplate the comparison to begin with.  At least when a hypothesis-driven comparison is not significant most biologists know that you shouldn't make a big deal of it without a power analysis[4].  With a table of AIC values, an inappropriate comparison is harder to spot.  One indicator is a slew of "comparable" models which ought to remind the reader that the models are only "comparable" in light of the data on hand.

Now back to that paper...

[1] Survival in the marked and unmarked populations can be quite different, I'm sure hawks love mice with neon marker on their back.
[2] Common, there's survival variation when you look within a single manufacturing lot of CPU's, forget frogs in the wild.
[3] I just read one in a reasonable journal where the 95% confidence intervals basically stretched from 0.5-0.95, and the _title_ of the paper relied on the statistical lack of difference.
[4] Through the hard work of generations of biostatisticians, most biologists will recognize that making a big deal from a lack of significance is questionable.