gpugesek wrote:Is there a strategy you would suggest???

First, congratulations on really doing a bit of background homework.

Second, yes, there is a robust, fairly easy way forward, that not only will answer your question, but will let you say 'MCMC' in your paper/report, letting you run with the cool kids.

1\ Take your basic model with the 2 groups. Run it using MCMC (Appendix E in the book). The MCMC.BIN file that will get created will have the posterior estimates for the two derived parameters (being, estimated abundance), for each group.

2\ suck MCMC.BIN into R. There is a script in the MARK helpfile (largely thanks to Jeff Laake) that will pull in the binary file, and massage it to the point where it creates an MCMC object called 'mcmcdata'.

3\ post-process mcmcdata, creating a variable you might call 'diff', which is the difference between derived1 and derived2 (i.e., the two abundance estimates).

4\ then, look at the HPD for this 'diff' variable, and if it doesn't bound 0, you might safely conclude there is a real difference in estimated abundance between the two groups.

The basic idea (albeit in a different context) is described at the end of the Delta method appendix (Appendix B), and is really quite straightforward.

Here is a worked example (which I should probably add to the book).

Here are some simulated closed abundance data, 2 groups - true abundance is 100, and 125, respectively. The underlying encounter probabilities were high, so there is good precision (0.75 and 0.65, respecitviely).

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`11011 6 8;`

11110 9 6;

10101 3 7;

11101 11 8;

11111 19 8;

00011 2 1;

10111 5 5;

00111 3 10;

01001 1 1;

00101 2 5;

01111 12 10;

11001 4 7;

11000 2 3;

01011 1 4;

00110 2 2;

01101 3 4;

10100 2 2;

10110 1 4;

01110 3 3;

10011 2 6;

11010 1 3;

01100 1 6;

10000 1 3;

11100 2 2;

01010 1 0;

00001 1 0;

00010 0 1;

10001 0 1;

01000 0 2;

00100 0 1;

10010 0 1;

Fit Huggins model {p=c,grp} (i.e., no time var in p or c within group, p=c within group, but separate estimates of encounter parameters between groups). Then, re-run this model using MCMC (if you've never used MCMC in MARK before, Appendix E is your starting point. But, for this example, you really don't need to use anything more than the defaults). Once run, MCMC.BIN will be created in the directory where the .inp file was located.

Then, you simply post-process MCMC.BIN using the R script from the helpfile as a starting point. For this example, the macro vars you need to put at the top of the R script would be:

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` ncovs <- 2; # Number of beta estimates`

nmeans <- 0; # Number of mean estimates

ndesigns <- 0; # Number of design matrix estimates

nsigmas <- 0; # Number of sigma estimates

nrhos <- 0; # Number of rho estimates

nlogit <- 2; # Number of real estimates

nderived <- 2; # Number of derived estimates

filename <- "C:\\USERS\\EGC\\DESKTOP\\MCMC.BIN"; # Name and path to the MCMC.BIN file

Once the script is done, you have an MCMC object called mcmcdata. From there, the following will suffice:

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`# now set things up for derived parameter calculation - difference of derived1 and derived2`

chaindata <- as.data.frame(mcmcdata);

chaindata$diff <- chaindata$derived1-chaindata$derived2;

cat("\n")

cat(" basic summary stats of diff in derived abundance estimates \n")

print(summary(chaindata$diff));

cat("variance of difference...\n")

print(var(chaindata$diff));

hold <- as.mcmc(chaindata$diff)

hpd <- HPDinterval(hold,prob=0.95)

print(hpd)

hist(chaindata$diff,xlab = expression(paste(diff((derived[1] - derived[2])))),

main="Posterior density of difference in abundance estimates")

abline(v=hpd[1],col="blue")

abline(v=hpd[2],col="blue")

If you run this additional bit of code, the summary stats for the difference between the 2 derived estimates are:

basic summary stats of diff in derived abundance estimates

Min. 1st Qu. Median Mean 3rd Qu. Max.

24.13 24.64 24.80 24.84 24.99 26.15

variance of difference...

[1] 0.07644926

with an HPD (highest posterior density) of

lower upper

var1 24.3471 25.40487

So, for this example, its very obvious that the HPD doesn't include 0, so, you can safely conclude there is a difference (and, not surprisingly, given these were 'good simulated data'), the mean of the posterior (24.84) is very close to the true difference (25).

So there you go...the MCMC approach in MARK has a lot of value added beyond estimating mu and sigma for hyperparameters. Its fairly easy to use it to generate mean/variances of all sorts of functions of derived parameters. In fact, the only difference between what MARK does, and what (say) JAGS or OpenBUGS does it that the latter calculates the derived values at each step of the chain, whereas with MARK, you post-process the chain. The former (i.e., what JAGS does, for instance) is computationally more convenient, but the latter approach (that MARK uses) yields identical results.