Survival averaged across models and time

[rcurrey]

I am interested in obtaining survival estimates that include model selection uncertainty and temporal variability for inclusion in population viability analyses.

I am familiar with generating model-averaged estimates of survival but the yearly estimates vary. Is there a way to average these estimates to include the variation across years as well? I have noticed that averaging estimates over time for specific models was addressed in the White et al. paper on advanced features of Program MARK but I could not find anything that extends this across models.

[Eric Janney]

The estimated variance associated with the Phi(t) model in a CJS is a combination of sampling variance and the process variance. The process variance is an estimate of the actual temporal variability in survival. For a PVA analysis it is important to only include the process variance. It is common in PVA analyses for people to incorrectly use a variance estimate that includes sampling variance. So, how do you tease variance due to sampling error from process variance? There are several good papers that discuss methods for estimating process variance.

Gould, W. R., J. D. Nichols. 1998. Estimation of temporal variability of survival in animal populations. Ecology, 79 (7), pp. 2531-2538.

Burnham, K.P, and G.C. White. 2002. Evaluation of some random effects methodology applicable to bird ringing data. Journal of Applied Statistics, 2002, vol. 29, issue 1–4, pages 245–264

Link, W. A., and J. D. Nichols. 1994. On the importance of sampling variance to investigations of temporal variation in animal population size. Oikos 69: 539-544.

These methods are commonly referred to as variance components and the methods described by Burnham and White (2002) are incorporated in MARK. There is also a Bayesian method called MCMC that is also in MARK. There is a fair amount of literature on MCMC as well, but I don't have the references offhand. Anyway, I hope this helps.

[Jochen]

To some extent. What is available in MARK is a variance components analysis for a specific model (which seems to take a lot of time to run).

But is there a way to estimate process variance in model averaging? As I understand from the book, all the SEs reported there include sampling variance.
Did we miss anything?

[Jochen]

Yes, we actually missed something: a little more searching revealed a statement by Gary that variance components is not useful and therefore unavailable for model averaging.

For PVA we would therefore have to choose between either a model averaged estimate with its unconditional variance (including sampling variance) or the process variance of a single model (say, a time-dependent) which might not yield the best available estimate.

[cooch]

To some extent. What is available in MARK is a variance components analysis for a specific model (which seems to take a lot of time to run).

But is there a way to estimate process variance in model averaging? As I understand from the book, all the SEs reported there include sampling variance.
Did we miss anything?

I might be missing the larger purpose of this thread (probably), but some of the partitioning of sources of uncertainty in model averaging is discussed in section 4.5.1 in the current version of Chapter 4.

If what you're asking is can you estimate an estimate averaged over time for each of a series of model, some overall average over time across all those models, and a variance decomposition of same - the answer is no, I believe - at least wrt to the latter. You could generate an average estimate of survival over time from the time-specific estimates for each model (you could use a Delta method approach, for example), then derive a model averaged value of these averages (using the approaches discussed in 4.5.1), which would also give you a SE. But, I don't know of an easy way to then partition that into process and non-process variance (the operative word here being *easy*).

[Eric Janney]

I have a paper in press that uses both approaches. I used a random effects model to calculate mean annual survival over a 13 year period which I then used to calculate mean life span of the animal for that period using 1/-ln(ave. phi). I also used the variance components as a way of assessing the temporal variation in survival. For reporting annual survival rates, however, I used model ave. estimates and their unconditional variances. So, if you are interested in mean survival and the process variance around that mean, use a random effects model. But keep in mind that you need a fairly long-term data set for the random effects model to be meaningful (say >10 occasions). If you are interested in the best estimate of survival and its unconditional variance for each specific interval you should use model ave. estimates. But, Evan is correct in that the two methods do very different things and can't really be combined.

[rcurrey]

Thanks very much for all your thoughts. I have gone with the pragmatic approach of reporting the model averaged estimates of survival and incorporating the variance components of the most parsimonious CJS model into the PVA (see Slooten et al. 2000 for an example of this PVA approach). Given the underlying hypotheses for the specific models I have employed and the the fact that there is considerable support for the chosen models, I think this approach will be adequate. Thanks again for your thoughts.

Slooten, E., Fletcher, D., & Taylor, B. L. 2000. Accounting for uncertainty in risk assessment: case study of Hector's dolphin mortality due to gillnet entanglement. Conservation Biology 14(5): 1264-1270.