Nest survival, AICc and the effective sample size

[murray.efford]

I'm sceptical about the effective sample size (ESS) used by MARK when computing AICc for nest survival models. As implemented, ESS depends on the time units used for intervals between nest checks: the calculated ESS adds one Bernoulli trial for each time step (e.g. day) even when the interval between checks is much longer. By using hours instead of days you can increase ESS by a factor of (nearly) 24. Surely this is a mistake? Why not use the number of encounter histories? Am I missing something?

(This is usually a small issue in practice because even with a conservative ESS, delta AIC is approximately equal to delta AICc.)

Murray

[gwhite]

Murray:
Not sure that you are missing anything, but the time interval affects sample size in almost all of the mark-recapture models. Take another example: known fate. If I release 100 animals and calculate an annual survival 1 year later, my effective sample size is 100. But, if I compute 2 6-month survival estimates, my effective sample size approaches 200 (with all the animals dying during the first 6-month interval not contributing to the second 6-month interval). The same is true of the nest survival model.

By decreasing your time interval, you increase the number of parameters estimated in the global model (i.e., hourly survival estimates for your example). Similarly, the number of binomial trials increases.

Gary

[murray.efford]

Thanks - it helps to know this is intended. I admit I'm still mystified that 'effective sample size' is determined by the model rather than the data (information) in hand. Time to do some more reading...

Murray

[murray.efford]

If I can be allowed another go at this -

Williams et al 2002:432 say "In the context of capture-recapture models, sample size [for AICc] is usually obtained by summing the releases R_i over all time periods"

That does not seem to correspond to MARK behaviour, at least for nest survival. The 'time periods' here are the sampling occasions, so the sum of the R_i is a reasonable sample size. It is largely* for computational convenience that MARK interpolates an integral number of time intervals (days, hours) between the sampling occasions - no matter how much you subdivide the intervals, the amount of data remains the same.

If I check 5 nests today and re-check them after 5 and 10 days and find no mortality, is the sample size (n for AICc)
1) 5 + 5 (R1 + R2)
2) 5 x 10
3) 5 x 10 x 24
4) something else?

MARK seems to say 'all of the above': it depends on whether you code for 5-day intervals, 1-day intervals, hourly intervals or something else. I suggest this makes the small sample correction of AIC rather dubious in this case.

OK. I'll stop now!
Murray

* although the primary reason is to allow for unequal intervals, this also allows quite neatly for covariates that vary continuously in time

[gwhite]

Murray:
The time periods in the nest survival model are the daily survival intervals, even though nests are not sampled daily. Write out the likelihood for yourself, and see the full set of parameters. The model-building process reduces this global model to a more manageable set of parameters, but all the models still produces estimates of survval for each time interval (day). Just look at the real parameter estimates.

If you change the time interval, then the set of parameters changes.

Gary

[murray.efford]

Yes, but parameters are not data. Is there any intuitive or theoretical reason why increasing the number of parameters in the global model should change the sample size i.e. the amount of data that we have? Sounds to me like pseudoreplication.
Murray

[gwhite]

Murray:

In this type of analysis, the model and associated parameters dictate how the data are used. If you use a daily interval, then each nest contributes a success or failure for each day that the nest is in the sample. Similarly, if you use an hourly interval, each nest contributes 24 observations of success or failure for each hour (excluding hours after a nest has failed). You're not thinking about the data as Bernoulli outcomes associated with a specific interval like you should be thinking about it.

Further, you can work it out for yourself, but the estimates are statistically independent, so this is not pseudo-replication in the Hurlbert sense. If you don't believe that the estimates are independent, look at the sampling covariances for the global model -- they are all zero. Of course, as soon as you impose some structure on the model (e.g., a trend model), you will introduce a sampling covariance across estimates.

Gary

[darryl]

Hi Guys,
Been watching from the sideline and am a little bit confused (a not unusual state), perhaps because I've never done a nest survival analysis.

Gary, when you talk about changing the interval to hourly, are you suggesting there are hourly observations as well, or still only daily? If you do have hourly observations then I can see your point, but if you don't then I can understand where Murray is coming from?

Darryl

[gwhite]

Darryl:
Right -- you have to do hourly observations to obtain hourly survival estimates. Of course, with nest survival data, you do not have to observe every hour, but have to be able to specify the Bernoulli outcome on an hourly basis except for intervals with a nest failure. But my point is that you can pool these observations to obtain daily survival estimates, and the effective sample size now changes.
Gary

[darryl]

Thanks for the clarification Gary (and I RTFM'd ).