Deviance residuals for closed capture models

[bmitchel]

I've noticed some bizarre patterning to the residuals for some closed-capture models (Huggins Full with Heterogeneity) I have been working on. These are removal models, so the histories are 10000, 01000, 00100, 00010, or 00001 and c is set to 0 (the data comes from 2-minute sections of 10-minute bird counts).

If I run the simplest model (no covariates in the input data set, identity matrix) and use the deviance residuals plot, the plot has 5 encounter histories (as expected). The first history (10000) lists an observed value of 1157 (this is the frequency of that history), an expected value of 1160, and a residual of -0.1. So far so good. But the other 4 histories list observed values equal to the frequency of the history, expected values of 0, and residuals of 0. I do not understand why the expected values would be 0, or why the residuals would be 0 with such a large discrepancy between the observed and expected values.

If I complicate things by running the same model on the same data set, but include covariates in the input file (they are not in the model), I get the same AIC and deviance (as expected). The deviance residuals plot now includes a point for each data record instead of each history. Now, each of the 10000 histories is listed with an observed value of 1, an expected value of 1160, and a residual of -48. The other 4 types of histories have observed values of 1, expected values of 0, and residuals of 0. Apparently, for the 10000 histories, the expected value is the same as for the first model, but the presence of covariates in the input file has changed the observed frequency and led to an anomalously large deviance residual (this is, after all, just 2 ways of making the same model). As before, the deviance results (or lack thereof) for the other four histories is baffling.

If I actually include covariates in the model, I get a scatter of very negative deviance residuals for data with a 10000 history, and (as before) no residuals at all for the other histories.

Does anyone have any ideas why I'm seeing these odd results?

Thanks,

Brian Mitchell
Postdoctoral Research Associate
University of Vermont

[bmitchel]

This is a follow-up to my earlier posting about strange residuals for closed capture models. It turns out that this was a bug in MARK; Gary has fixed it for data that has no individual covariates. Models with individual covariates still do not compute residuals correctly.

If you are interested in plotting your residuals, you can export the observed and expected values to Excel from MARK. The expected values then need to be divided by the number of times the history was present in the data set. The Pearson residual is then: (observed - corrected expected)/ square root (corrected expected). I could not find a formula for the deviance residual.

Gary told me that he does not think that residuals for data with individual covariates are useful for assessing GOF. My feeling is that they still are helpful for detecting unusual patterns that could prevent a good fit.

Brian Mitchell
Post-Doctoral Research Associate
University of Vermont