difference in derived parameters when calculating by hand/MA

[Roman Luštrik]

Dear interested reader,

we are working with simulated data, where we sample "individuals" with probability p_r. Because individuals are not always within the sampling area, their capture (recapture) probability is less than expected p_r. We try to account for this using individual covariates (which is also the output of our simulation). We are running a closed captures Huggins model of {p = c * ind.cov} with Mean individual covariate values (Real parameter estimates for ind. covariates in the Run menu). MARK undershoots the Real parameter estimates (^p) by about 10% of the theoretical p_r. When we set the Real parameter estimates for ind. covariates to First encounter history covariate values, the ^p is spot on. However, the derived parameter estimate is identical to the previous setup. The help page warns us that

Note that individual covariate values specified to compute the real parameters _may_ affect the values of the derived parameters.

I'm trying to understand what's going on. I would expect (ok, hope) that when real parameter estimates change, the derived parameter estimates would also change. I tried calculating the derived estimate by hand, using the Horvitz-Thompsom estimator (page 14-5 of MARK book), and I got a different (lower) estimate compared to MARK.
Have I missed any documentation in the MARK book or the help files that elucidate this? Any thoughts?

If I was unclear in my explanation, or missed anything, please feel free to comment and I will try to rephrase my question. I am also including one file with models already waiting to be explored. The file can be snatched from my Dropbox acc (rar file, 12 kb).

[abreton]

Unless you change the model, then the derived abundance estimate will not change -- if it does, then definitely post to the forum! You fit the same model twice, model {p = c * ind.cov}, and as expected the derived abundance estimate was the same in both cases. But as you note, the real parameter estimates (capture probabilities) reported by MARK were different each time -- this makes sense.

The real parameter estimates reflect what you selected (criteria) when you ran the model each time -- first run you chose 'mean...'; second run you chose 'first...'. As expected, given these run criteria, in the first case MARK reported capture probabilities for an animal with the 'average' of your ind. cov. -- note the 'average' of this covariate may not exist in nature, such as when the covariate is binary (0,1)...the average 0.5 does not exist...something to keep in mind. In the second case, MARK reported the capture probability for an animal with the individual covariate value provided by the 'first encounter history'. The options - mean, first, user specified - are a convenience that allows users to visualize the capture probability for an individual associated with the mean, first or user specified value of an individual covariate(s). User specified is the most flexible, the capture probability for any animal can be extracted from MARK using this option.

andre

[Roman Luštrik]

Thank you very much, your helpful answer is much appreciated.