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Hello I have a question regarding sample sizes:

How does Program MARK deal with unequal sample sizes of captured animals? For example, in my dataset I have city captured as a covariate. I have about 800 captured animals for one city, and then 300 for another. There is of course a higher apparent survival for the city with 800 captured animals but how does MARK deal with the potential bias of this kind of situation? I haven't found anything in the MARK book so I just assumed that it does not and my study itself is more biased.

Thank you!

Probably I'm missing something. If your capture sessions were held in two cities, say A and B, and during the course of the study you captured 300 individuals in A and 800 in B, it's not at all obvious that the apparent survival in B is higher than in A. Why should it be that way? There may be a lower abundance in A, which does not mean that there is a lower survival (maybe lower productivity, for example) or simply a lower chance of recapture.

May you please elaborate a bit further?

If you have reasonable encounter rates, your sample sizes of 300-800 should not affect your point estimates for apparent survival. With small sample sizes and low encounter rates, you can have problems with estimation where estimates of apparent survival are noisy and at the boundaries of zero or one. Note that you could easily have higher apparent survival for the city with a lower sample size if more marked individuals are recaptured on subsequent occasions. The sample size of marked individuals should have a greater effect on the precision of your parameter estimates. The confidence intervals should be tighter for the parameter estimates from your city with more marked individuals because you have more information to work with. The confidence intervals for the parameters will then determine your model rankings, and whether you have support for time-dependent models or constant models (see Fig 4.1 of the mark manual on pg 4-35). Adjustments for small sample sizes are handled by the corrected versions of the information criterion - review the difference between AIC versus AICc (pg 4-39). Unequal sample sizes are probably more common than balanced samples in ecological studies based on nonexperimental protocols. Good luck with your analysis.

Thank you very much for your quick responses!

I will elaborate now and I realized my post wasn't entirely accurate. I'm dealing with birds captured in nest boxes for a sample, and there are 800 boxes in one city and 300 boxes in another. So that was a mistake and I apologize that I said captures instead of boxes. I did have a lot more birds captured in the city with 800 boxes than captures in the city with 300, by that I mean captures to nest box abundance ratio. So MARK gave me higher apparent survival rates for the city with 800. I just wonder if MARK can specifically do anything for unequal sample sizes because I have this large difference in nest box diversity in the cities. Or is it more just ecological factors to figure out (such as lower abundance of birds or productivity like you mentioned). Thank you and I hope this is more clear now.

OK, thanks for adding information. I guess there is no exchange of marked individuals between the cities, you are using a CJS model and you are interested in comparing the apparent survival in the two cities. A situation like yours occurs very often in Ecology - the homeland of the unbalanced designs - and not just in capture-recapture analyses. I don't think there is any way to solve it, neither in MARK nor in any other environment. The sample size (see here to get an idea of what it is in capture-recapture) affects the precision of your estimates and, therefore, your ability to find "effects".

In general, I do not see why the difference you have found should be a statistical artefact. The way the unbalanced design affects your testing hypothesis is by influencing your power to detect a difference between the apparent survival in the two populations (if in one city your sample size is not enough, to put it in some way). Otherwise, the unbalanced design should not create a spurious result unless there are particular problems on other sides, such as a very low recapture probability (<10-15%) in one of the two cities which may bias the apparent survival estimate. If you did not find a difference between the apparent survival of the two cities and you suspected it was not so, you could have done a power analysis to see what effect size (what difference in apparent survival) the model would have been able to detect with (the distribution of) the given sample size. For this purpose, you would have needed to repeatedly simulate and analyze data from your model estimates to see how many times you find a biologically relevant effect (contrasting, by AIC or whatever else, the model with and without the city effect). I see many caveats to this procedure, one of them being the fact that you never know how well your data (and therefore the parameters' estimates) represent the ecological process (of which they are just one specific outcome), but this goes beyond the scope of your question…

Thank you so much for the information! This helped me understand MARK models and their sample sizes more easily, so thank you!