modeling N directly as a function of covariates

[martin13]

I just read in White 2002 (Journal of Applied Statistics) that it is not possible to directly model N as a function of covariates using Program MARK. It seems that the best one can do is to split data into groups and compare N's, and the associated CI's, among those groups, as well as use closed modeling AIC to determine if including a grouped N increases model fit. I am conducting an analysis where it would be highly beneficial to analyze population estimates in the context of multiple habitat covariates (a la, an ANOVA). There are obvious problems with simply taking the population estimates that MARK spits out and plugging them into another statistical analysis (i.e., the population estimates are exactly that... estimates with CI's, as opposed to actual values), but I don't know what else to do. Is there another software package out there that handles this type of analysis? Am I stuck exporting the MARK estimates into another analysis and ignoring their associated error values? Is my only option to compare N's among grouped data? If this is the case, what if there are multiple variables that influence N or interaction terms? Any suggestions or advice would be greatly appreciated!

[cooch]

I just read in White 2002 (Journal of Applied Statistics) that it is not possible to directly model N as a function of covariates using Program MARK. It seems that the best one can do is to split data into groups and compare N's, and the associated CI's, among those groups, as well as use closed modeling AIC to determine if including a grouped N increases model fit. I am conducting an analysis where it would be highly beneficial to analyze population estimates in the context of multiple habitat covariates (a la, an ANOVA). There are obvious problems with simply taking the population estimates that MARK spits out and plugging them into another statistical analysis (i.e., the population estimates are exactly that... estimates with CI's, as opposed to actual values), but I don't know what else to do. Is there another software package out there that handles this type of analysis? Am I stuck exporting the MARK estimates into another analysis and ignoring their associated error values? Is my only option to compare N's among grouped data? If this is the case, what if there are multiple variables that influence N or interaction terms? Any suggestions or advice would be greatly appreciated!

See Huggins estimators - MARK Help file, Chapter 15 in the 'MARK book', and primary literature.

[martin13]

I've looked at the Huggin's estimators, however I believe according to White (2002) these don't directly model N as a function of the covariates... at least in program MARK. The covariates are entered as part of the initial dataset in MARK and are used to model f(sub 0)=N(hat) - M(t+1). This indirectly models N(hat) as well, I suppose. Does this ultimately make a difference? Or is there another software package that you could suggest to estimate N directly in the context of habitat covariates? To quote Gary White, "Incorporation of covariates to model N(hat) is not possible directly in software packages such as MARK". Thanks.[/code]

[cooch]

I've looked at the Huggin's estimators, however I believe according to White (2002) these don't directly model N as a function of the covariates... at least in program MARK. The covariates are entered as part of the initial dataset in MARK and are used to model f(sub 0)=N(hat) - M(t+1). This indirectly models N(hat) as well, I suppose. Does this ultimately make a difference? Or is there another software package that you could suggest to estimate N directly in the context of habitat covariates? To quote Gary White, "Incorporation of covariates to model N(hat) is not possible directly in software packages such as MARK". Thanks.[/code]

Then I'm missing the point of the question I suppose (what I said is correct if you're trying to model sources of heterogeneity among individuals by conditioning encounters on one or more individual covariates). If instead what you're trying to do is model variation in N over (say) time, then correct - your best approach then would be a Pradel model, or something equivalent, where you focus on rates of change (or recruitment), rather than N. You can constrain various parameters in Pradel models to be functions of covariates, but you need to be *very* careful on how you do this. You might also consider Jolly-Seber models (see the chapter on same in the MARK book).

[martin13]

I simply want to know if populations A, B, C, and D differ in size, and if so, is this attributable to habitat variables X, Y, and Z. Mark-recapture data was collected for each population simultaneously. I was trying to do a mark-recapture based analysis to improve my N estimate, rather than simply using the M(t+1) value. But if my capture protocol was the same for all populations, then perhaps I should just be using the number of individuals captured in an ANOVA?

[ganghis]

The main problem here is that MARK doesn't estimate N directly, even when N is "in the likelihood." Instead it estimates f0, where N=M_{t+1}+f0. This is to enforce the constraint that the population estimate is greater or equal to the total number of unique individuals caught.

Paul

[cschwarz@stat.sfu.ca]

I think you need to step backwards a bit.

When you say that you are interested in knowning if populations A, B, C and D differ in size, then the answer is yes. It is highly improbable that there are the exact same numbers of animals in these 4 populations.

The population sizes will vary because of different sizes of the study area, different ages of animals etc. The population sizes vary over time because of births, deaths, immigration, emigration etc. Are you interested in a fixed point of time?

Perhaps you are really interested in knowing if the DENSITY of animals varies as a function of covariates?

What is your study protocol? What species of animals are you studying?

[martin13]

Okay... I sampled cotton rats in 4 locations within a contiguous agricultural matrix. Trapping grids are identical in size, therefore I'm assuming that the grids are all sampling equal areas. Grids are far enough apart to ensure independence. Each trapping area is exposed to various quantifiable predation pressures, as well as other habitat characteristics. My intent is to use N(hat) from each of these trapping grids as a relative index to compare the cotton rat abundance indices. Rather than using the number of animals captured as the index values, I would like to estimate N(hat) values. Practically speaking, yes, I know that the index values differ among trapping areas. I now need to run an analysis to determine what factors are correlated with these differences (i.e., predation, habitat characteristics, etc.). I can estimate N(hat) and associated SE using MARK, but MARK doesn't allow for direct modeling of N(hat) using covariate information.

[cschwarz@stat.sfu.ca]

You may be able to do this with the POPAN module (I haven't tried this out yet to see if it would work or not), but in "theory" it should.

Set up the capture histories with the 4 groups as usual.

I'm assuming you are looking at closed populations in the trapping grids?

Set all the phi parameters in all groups to a single parameter index number. Before you press run, constrain this parameter to the value of 1. This will ensure that no "deaths" occur.

Similarly, set all the pent parameters in all groups to a single index number. These refer to new animals after the first capture occasion. Again, before you press run, constrain this parameter to the value of 0. This ensures no new animals enter the population. MARK has a pent_0 that is automatically set to 1-sum of the other pents.

Model the p's in the usual way.

Finally, the initial population sizes for the 4 groups (which in the case of no births/deaths is the closed population size) can be specified in the design matrix to depend upon the covariates.

This may work -- haven't had a chance to try it out with MARK yet, but give it a try.

Notice that with only 4 groups, even fitting a straight line takes up 2 parameters and so the power to detect anything but gross dependencies is likely small.