How are the expected values computed in MARK?
The mark help file provides the formulae, see also page 5-34 in the MARK gentle intro.
Are the residuals plots reliable to evaluate the fit of the models (POPAN in particular)?
For live-recapture models, the gentle intro suggests that in addition to Release and U-CARE,
Residual plots can also be quite helpful
(last line, page 5-40)
Note that the authors (Cooch and White) of the Gentle Intro tend to favor the Deviance Residuals in their examples, rather than Pearson Residuals. As well, the interrogation of residuals in the gentle intro is always (based on my quick scan) accompanied by an estimate of c-hat (overdispersion parameter). In other words, the residuals are considered but are not used exclusively in goodness-of-fit (GOF) examples provided in the text. This makes intuitive sense given the difficulty/challenges of devising GOF tests for capture-recapture models. The authors (Cooch and White) use what is available ... including deviance residuals in some cases. The collective evidence is then used to assess GOF.
On page 11-46 (sidebar) of the gentle intro,
One approach to assessing the fit of a model to a particular set of data is to consider the deviance residual plots.While this can prove useful - in particular, to assess lack of fit because the structure of the model is not appropriate given the data (e.g.,TSMmodels - seeChapter 7), if you try this approach for models with individual covariates, you’ll quickly run into a problem.
Once again, the authors report that deviance residuals "can be useful" ... meaning a valuable tool for assessing GOF when used with other approaches (especially an estimate of c-hat). Regarding individual covariates, the sidebar goes on to explain why you will "quickly get into trouble." And concludes,
the deviance residual plots for models with individual covariates are not generally interpretable
So, the res plot feature does not work properly in mark ... WHEN individual covariates (IC) are incorporated into the encounter histories and this is why (I suspect),
Gary White has told me that he does not think that the residuals are useful for examining goodness of fit for models with individual covariates
More subtly, it appears that the function does not work even when no ICs are integrated into the encounter histories but those histories are summarized for EACH animal rather than each unique capture history. I believe the sidebar text on page 11-47 is the reason why Cooch and White wrote,
Notice that a separate capture history is used for each tagged fish - unfortunately, this implies
that the residual plots and deviance plots in MARK cannot be used to assess goodness of fit.
... on page 12-14. However, I should note here that the classic 'dipper' dataset available from the MARK example files has one encounter history per dipper ... and residual plots appear to be valid. My suspicion (KEEP IN MIND I'm guessing here) is that when no ICs are identified when importing into MARK ... then when the user requests a 'residual plot' option then the encounter histories are grouped by each unique capture history and then integrated into the residual plot. However, this trick isn't possible when ICs are included in the encounter histories ... at least no one has been willing to strip the ICs from the histories, group the histories that remain and then make them available for a residual plot. And of course, if you chose a model that included one of those ICs it wouldn't make any sense to strip the IC ... so perhaps it's a good thing that when ICs are included the res plots are labeled "uninterpretable."
In conclusion, I don't have ALL of the answers you're searching for so hopefully someone else will weigh in. However, it's clear that if you're including ICs in your encounter histories then res plots WILL NOT be interpretable in MARK ... for any data type including POPAN. I would say ... not a big deal though ... you still have c-hat to assess GOF of your global model ... as well as information criteria for model selection.
andre
