Using the delta method for random effects models

[dtempel]

I've run some random effects models for a lamda analysis in MARK (reverse Pradel). I'm interested in estimating the realized population change over the course of the study to generate a graph showing how the population has changed over time relative to the initial population size. Because the realized change is a function of multiple parameters (i.e. sequential multiplication of the annual lambda estimates), I need to use the delta method to properly calculate the SE for the realized change estimates.

I have a spreadsheet that someone used calculate the SE's and plot the realized change from the same study area several years ago (we've since collected additional data each year so the data sets are not the same). From the MARK output, they used the S-hat estimates and the var-cov matrix from the random-effects means model to do this. However, it seems to me that one would rather use the S-tilde estimates because they've partitoned out the sampling variance. It actually makes a really big difference in the ending value for realized change because almost all of the population growth (lambdas > 1) occurred in the first several years of the study, and the S-hat estimates for lambda are greater than the S-tilde estimates. If I do use the S-tilde estimates, I think the appropriate var-covar matrix is an identity matrix with sigma^2 on the diagonal (referring to Burnham 2002 in the Journal of Applied Statistics).

So first, is there any reason why I shouldn't use the S-tilde estimates? Second, am I correct about constructing the var-covar matrix for the S-tilde estimates?

Thanks!

[cooch]

I've run some random effects models for a lamda analysis in MARK (reverse Pradel). I'm interested in estimating the realized population change over the course of the study to generate a graph showing how the population has changed over time relative to the initial population size. Because the realized change is a function of multiple parameters (i.e. sequential multiplication of the annual lambda estimates), I need to use the delta method to properly calculate the SE for the realized change estimates.

That works relatively well for this sort of problem. Remember to set cov for initial N and initial lambda to 0.

You might also want to consider a Link-Barker model with random effects. There are pros/cons to the various methods (see chapter 13 for the gory details...)

I have a spreadsheet that someone used calculate the SE's and plot the realized change from the same study area several years ago (we've since collected additional data each year so the data sets are not the same). From the MARK output, they used the S-hat estimates and the var-cov matrix from the random-effects means model to do this. However, it seems to me that one would rather use the S-tilde estimates because they've partitoned out the sampling variance. It actually makes a really big difference in the ending value for realized change because almost all of the population growth (lambdas > 1) occurred in the first several years of the study, and the S-hat estimates for lambda are greater than the S-tilde estimates. If I do use the S-tilde estimates, I think the appropriate var-covar matrix is an identity matrix with sigma^2 on the diagonal (referring to Burnham 2002 in the Journal of Applied Statistics).

I would agree with your initial point (being, using the S-tilde values might be more defensible).

As for the var-covar matrix, not sure why you're headed down that road, since you can output it directly from the random effects run window. Simple check the box 'Include S-tilde VC Matirx' to include it in the output.

[dtempel]

Thanks, Evan. That helps a lot. Somehow I never saw the check box for the 'S-tilde VC matrix.'