Apparent Survival Rate for n=1

[brp]

We have been trapping foxes for the past 3 years. We have a robust design with 3 primary sessions (years: 2007, 2008, 2009) and 4 secondary occasions (days) within each year. I am running a multi-state robust design where age (pup, non-pup) are the states.

The sample sizes by age and year are given below:

Pups:
n=1 (2007)
n=77 (2008)
n=15 (2009)

Non-pups:
n=179 (2007)
n=160 (2008)
n=204 (2009)

I obtained 4 model-averaged estimates of survival: 2007-2008 and 2008-2009 apparent survival rates for pups and non-pups.

My question is: how did MARK calculate a 2007-2008 apparent survival rate for pups based on n=1?
I would be happy to hear a technical or non-technical explanation.

Thank you,
Becky

[sbonner]

Hey Becky,

What were the 4 models you fit?

If you fit a model with time and group (pup/non-pup) dependent survival rates then the estimates of the 2007--2008 pup survival rates from this models would have been nonsensical (likely 0 or 1 depending on whether that one lonely gal was seen again or not). However, constrained models with either no time effect or no group effect should have produced sensible results. The model averaged estimates may be sensible, even if you included the fully time and group dependent model in the set of plausible models, if the constrained models took most of the AIC weights.

What models did you fit, what were the estimates of the 2007--2008 survival probs for each, and what were the AIC values?

Cheers,

Simon

[brp]

Thank you, Simon. That makes sense.

The models I fit included the dot model, S(.); the year-varying model, S(year); and the age-varying model, S(age). (I excluded the year and age-varying model because the SE couldn't be estimated for the n=1 case - was that correct?) The year effect was not statistically significant, but the age effect was. I chose to report apparent survival rates by age only.

In the MARK book (Section 4.5), it states "... we calculate an average value for a parameter theta by averaging over all models in the candidate model set with common elements in the parameter structure, weighted by normalized AIC model weights."

When MARK calculates the model-averaged apparent survival rate estimates for pups and non-pups, does it use all S(.) and S(age) models in the candidate model set? Or does it only use the S(age) models?

Thanks,
Becky

[sbonner]

Hey Becky,

First off, the model S(.) models the survival probabilities for all groups and years so it is included in the model averaging. Instead of thinking of this as a model with only one survival probability, think of it as a model with 4 survival probabilities (2 for pups and 2 for non-pups) that all happen to be the same. MARK only reports one value because it would be a waste of pixels to report all 4, but you still have an estimate of the survival probability for every combination and so you can average this with the other models.
Does that make sense? Unless the structure of the models is vastly different, MARK will average over all of the fitted models.

In terms of the models for S, what you can do with the survival probabilities for the pups depends on the structure of the capture probabilities. Clearly, you aren't going to be able to estimate the survival probability for the pups in 2007--2008. In fact, I'd prefer to drop that 1 pup from the data entirely because it really isn't going to effect the results for any other parameters and just makes the models more complicated. If you do that then you only have 2 years of data for the pups, and that isn't enough to estimate survival and capture probabilities separately, so you have to make one of two assumptions:

1) That the capture probabilities are equal for the two groups, in which case you can estimate separate survival probabilities. In essence, you use the data from the non-pups to estimate their survival probabilities and the overall capture probabilities, and then use these capture probabilities to estimate the survival of the pups.

2) That survival rates are equal for the pups and the non-pups in which case you can estimate separate capture probabilities for the two groups. In this case, you use the survival probabilities from the non-pups to estimate the capture probabilities for the pups.

If you believe that one of these is biologically plausible then your OK. However, if you believe that the capture and survival probabilities vary between the years and between the age groups then you really need more data on the pups. No matter what, I think that you need to be very clear when you report your results that the year with n=1 made it impossible to fit a complete year-by-age model.

Hope that helps.

Cheers,

Simon