Nest survival, AICc and the effective sample size

[murray.efford]

I'm not sure to which FM Darryl refers, but Gary's approach does not seem to follow Burnham and Anderson (2002: 332) who tentatively recommend the number of animals (not sum(R_i) or a larger number) as the sample size for survival parameters in fixed effects modelling (see also their extensive chain binomial example on p207 et seq.). B&A also say "The issue of sample size can be complex... We do not pursue solutions here. We just raise the issue as a future research area". Has that research been done since? Not a big deal, but religious use of AICc is less attractive if we cannot agree on n, even to within an order of magnitude.
Murray

[gwhite]

Murray:

Ken Burnham and I just had a detailed discussion about effective sample size. For the example on page 207 that you brought up, there are 2 ways to view the models. If the set of animals alive at each interval is viewed as a sample from a multinomial distribution, then the effective sample size is n1 (which is what you are arguing for). However, if the set of animals alive at each interval is viewed as a chained binomial, where you condition on releases for each consecutive binomial, then the effective sample size is the total number of releases. The likelihood written out on page 210 makes that clear. Ken is ambivalent about which is correct – both are reasonable under the assumptions that they are derived from.

Which brings me to what you can do in MARK. Early on, the effective sample size was always taken as the number of Bernoulli trials. So, for the CJS model, the number of animals released and re-released is taken as the effective sample size, because these releases form Bernoulli trials. Similarly for dead recoveries and known fate data types. When this philosophy got questioned was with the models that had different types of parameters. First were the robust design models, and later the occupancy models. Clearly in the simple occupancy models, psi has a different sample size than p. psi is based on the number of patches, whereas p is based on the number of visits to patches. So, that is when I installed the capability to specify the effective sample size under the adjustments menu choice of the results browser.

Hence, you can use any value of effective sample size that you want, including making the value so large that AICc becomes AIC. But then you will have to justify the value, particularly if different than the conventional value.

Gary

[murray.efford]

Thanks Gary - this is all useful stuff, and I get the message that the research has not been done. I'd just add some subsidiary comments
- I think the likelihood on p 210 of B&A has a typo in the final term (final exponent should be n_r,+ - n_r)... perhaps they were saving space to get it on the line
- Although that likelihood does indeed have some bigger numbers, B&A used the number of encounter histories as their ESS in the simulations
- It's good to know MARK allows user intervention on the ESS. I had overlooked that. This is not so handy for those of us who use MARK through RMark... another job for Jeff L?
- The nice thing about nest success and known-fate is that we have only one sort of parameter - hence the value of getting that right.
- This all makes ESS sound like chasing soap in the bath. The 'conventional value' is far from clear, unless you mean the default value in MARK.
Murray

[cooch]

Thanks Gary - this is all useful stuff, and I get the message that the research has not been done. I'd just add some subsidiary comments
- I think the likelihood on p 210 of B&A has a typo in the final term (final exponent should be n_r,+ - n_r)... perhaps they were saving space to get it on the line
- Although that likelihood does indeed have some bigger numbers, B&A used the number of encounter histories as their ESS in the simulations
- It's good to know MARK allows user intervention on the ESS. I had overlooked that. This is not so handy for those of us who use MARK through RMark... another job for Jeff L?
- The nice thing about nest success and known-fate is that we have only one sort of parameter - hence the value of getting that right.
- This all makes ESS sound like chasing soap in the bath. The 'conventional value' is far from clear, unless you mean the default value in MARK.
Murray

The notion of effective sample size is a general one, not unique to multinomial problems. For example, there have been thousands of pages (literally) devoted to what does (or does not) constitute the effective population size in the context of population genetics (the math involved in the discussions is rather lovely, the histrionics of the main participants is quite entertaining). There are also similar arguments in stochastic branching process theory, which is widely used in some types of network analysis (again, which has some strong ties to some areas of genetics research, amongst other applications).

My point here is that (i) this is a never-ending story, but one worth thinking hard about, and that (ii) other than for very constrained problems, there are unlikely to be solutions which are unambiguous and universally accepted. Goodness of fit is another similar quagmire.

Most of us take some basic level of comfort in the fact that AIC and AICc are convergent for large sample sizes, and that for smaller samples, other issues are usually more problematic than ESS.