Covariate versus group confusion

[Wijid]

Hi there

I seem to have managed to confuse myself and supervisor and am interested in any comments. I have a data set on lizards in a recaptures only set up with grouping by age: adults and juveniles. There is little info on the breeding of this species so we designated age based on snout vent length (SVL) saying less than 42mm is juv and greater is adult. We had reason to believe juveniles had a lower recapture rate and the best model suggested this was true with the juvs having a lower p value.

All fine then it was suggested to use SVL as an individual covariate and comparing this to the groups model. This I did and unsurprisingly the QAICs weren't very different although the group model tended to come out slightly better.

Here is where it gets tricky, I graphed the recapture rates against SVL (using the beta values and the equations as described in the gentle intro) and I get the opposite slope to expected from the groups models: the recapture rate actually goes down as the animals get bigger. Being that the groups are directly based on the measurement used for the individual covariate how is this possible? It just seems very unintuitive.

I have rebuilt the inp files and rerun the models twice in case I made a mistake in the set up but keep getting the same answer. I have also checked my PIMs etc with my supervisor. If I am making a mistake I would love some ideas where to look for them.

Any suggestions would be greatly appreciated!
Thanks for your time.
Wendy

[abreton]

Two questions..."in a recaptures only set up with grouping by age: adults and juveniles". This sounds like you've set up your EH with two groups representing the two ages. As long as juveniles remain juveniles throughout the study, then this strategy is okay. However, if juveniles transition into adults before the study ends then you need to drop these groups and instead use an age model (Chapter 8, 'the book'). And if the age rather than group parameterization is appropriate for your data, then you may have introduced considerable heterogeneity into your analysis. If this was the case, then your results would likely be misleading (nonsense); including, "I get the opposite slope to expected from the groups models: the recapture rate actually goes down as the animals get bigger." I'll wait for your reponse to follow up...

"...and the best model suggested this was true with the juvs having a lower p value." What p-value are you reffering to? My guess is that you may have ran a difference between proportions test using the recapture rates fom the two 'groups' as the proportions. If so, you'll want to revisit sections of Burnham and Anderson (2002) and consider using a full information-theoretic approach to model selection rather than this ad hoc test - which does not give valid results. Alternatively, you're "p value" may be in reference to the recapture parameter in the CJS model, "p"? If so, then consider replacing 'p value' with recapture rate or probability in the future.

[Wijid]

Thanks for replying so fast and I apologise for not being more clear, I was trying to be concise...

We have actually split the data set so that it works over 6 weeks to try and stop juveniles turning into adults as it were. Over the six weeks I measured SVL each time I captured an animal. In the data we averaged the SVLs and used this average SVL as the covariate and to assess what group an animal was in.

Sorry for this one: "...and the best model suggested this was true with the juvs having a lower p value." I was actually referring to the recapture rate (p) so that juveniles have a lower recapture rate than the adults, I will get this right in the future.

[abreton]

It seems clear, from what you wrote, that you did not include the individual covariate, SVL, and the group, which is based on SVL, in the same model - and this would be appropriate given the strong correlation between these predictors. Assuming this, then we're dealing with two models that differ only in whether they include SVL or group fitted to recapture probabilities. And since you have only two groups, both the SVL and group models should have the same number of parameters (k).

This means that the dichotomy that you decsribed for juvenile recapture probabilities between the two models boils down the signs of the SVL beta (which must be "-") in the SVL model and the group beta (which must be "+") in the group model. I am suspicous that, even though these models are tied for 'best' that the precision of both the SVL and group betas might be marginal or poor. Do the 95% confidence intervals for these betas (model coefficients) suggest high uncertainty (e.g., LCI -2.0, UCI 3.4; note that the true parameter might be very positive OR very negative) in the estimate? Poor precision could reasonably lead to your conflicting estimates. If so, then you may have to concede that your data are too sparse to reliably estimate the relationship between your recapture probabilities and SVL or group. I'm curious how precise your estimates are...perhaps my 'guess' is way off.

Regardless of the outcome of our discussion, I suggest that you look carefully at the betas for convergence problems (all models) and to guage the reliability of the estimates (top models; perhaps those within ~6-8 AIC units).

[Morten Frederiksen]

It seems to me that the most likely cause of the unexpected difference in signs in the two models is the exact way in which the design matrix is set up for the additive group model. In such a model, the sign of the group effect depends on the exact setup. Imagine a model with two groups, juveniles (group 1) and adults (group 2), 5 occasions (4 parameters to estimate for each group) and an additive time effect. Let's further imagine that survival (or whatever parameter we're interested in) is higher for adults.

You can e.g. set up the design matrix like this, with the second column indicating the group effect:
1 1 0 0 0
1 1 1 0 0
1 1 0 1 0
1 1 0 0 1
1 0 0 0 0
1 0 1 0 0
1 0 0 1 0
1 0 0 0 1
This matrix is perfectly valid, but it will produce an estimated group effect with an opposite sign to the expected, in this case negative. This alternative matrix is totally equivalent, but will give an estimated group effect with the opposite sign (here, positive):
1 0 0 0 0
1 0 1 0 0
1 0 0 1 0
1 0 0 0 1
1 1 0 0 0
1 1 1 0 0
1 1 0 1 0
1 1 0 0 1

So, you have to be careful about how you specify your groups (which is group 1), and how you set up your design matrix. If adults have a larger SVL, you need to specify juveniles as group 1 and use the second type of design matrix above to get a comparable estimate (or conversely, adults as group 1 and the first matrix type).

If this doesn't solve your problem there's something fishy going on!

Morten

[Wijid]

Thanks for the suggestions.

I tried the design matrix as suggested and I had got it correct. However I did try swapping it around just to learn something new. I didn't calculate the group recapture rates using the beta values as Mark gives these to you in the output. The interesting thing I noticed though is although the beta values do swap as you said the recapture values are always calculated correctly.

The only reasonable explanantion I can come up with is that it is as "abreton" said that my data is too sparse. I don't have large confidence intervals in my beta values for the individual covariate but I do in the recapture probability for the groups which would lead to the same problem. My supervisor has suggested that I test this by simulating fake histories (based on the current ones) to add to the data set just to increase the numbers of individuals and see what happens. I am in the process of doing this now.

Thank you for your suggestions, it has been useful to have someone else to discuss this with.

[cooch]

I tried the design matrix as suggested and I had got it correct. However I did try swapping it around just to learn something new. I didn't calculate the group recapture rates using the beta values as Mark gives these to you in the output. The interesting thing I noticed though is although the beta values do swap as you said the recapture values are always calculated correctly.

As thy should be. Its important to understand that th value/sign of the beta variables (for, say, survival) is entirely determined by the structure of the DM you use - in particular, if you use reference cell coding (i.e., expressing estimates relative to some 'reference'; typically the intercept). Changing the reference level of (say) time, or age, or some other factor, will change the signs and magnitudes of the various beta term, but it *does not* change the value of the reconstituted estimates (provided you have well-behaved data).

And, by extension, if you *fiddle* with the DM coding for one particular parameter, this won't influence estimates for other parameters.

The basic ideas are covered in some detail in Chapter 7.

The only reasonable explanantion I can come up with is that it is as "abreton" said that my data is too sparse. I don't have large confidence intervals in my beta values for the individual covariate but I do in the recapture probability for the groups which would lead to the same problem. My supervisor has suggested that I test this by simulating fake histories (based on the current ones) to add to the data set just to increase the numbers of individuals and see what happens. I am in the process of doing this now.

Andre's suggestion is quite plausible - but increasing the number of individuals with a gven history won't help - in fact, all it will do is increase the over-dispersion. In fact, when you simulate c-hat>1 in MARK, this is essentially what MARK does - it increases the frequency of individuals with a given encounter history. So, for example, to generate an example where c-hat=2, simply double the number of individuals observed for each given encounter history.

To simulate data, try the MARK simulation capability (appendix A). You can't simulate individual covariate data, but you can handle most other design types.