TSM model with 1st time period as the reference

[GDistiller]

Hi
I am new to Mark and capture-recapture models and have worked my way through the manual. I follow why when one constrains a TSM model there is no Age*T1 interaction term but essentially my query concerns what one does if the reference category is not the last time period, but rather the 1st time period. So essentially I am referring to the manual at the bottom of page 278 and top of page 279 where it says "Now, for a real test - how would the design matrix change if you had made age 1 the reference age class in your dummy variable coding scheme..." and then goes on to say "We’ll leave that to you as an ‘exercise’".

My confusion arises because if T1 is the reference then there will not be an interaction term and so it seems like there then ends up being some sort of redundancy. By writing out the linear model it looks like Beta 0 might be redundant since it would never appear by itself and so I have thought about dropping the intercept from the model which would then give me the right number of columns in the DM...but this doesn't seem right as then it seems I would be forcing the model through the origin.

I know I can just choose to parameterise it differently but I am intrigued as to how one would implement this. Any help would be much appreciated.

Thanks

Greg

[cooch]

Hi
I am new to Mark and capture-recapture models and have worked my way through the manual. I follow why when one constrains a TSM model there is no Age*T1 interaction term but essentially my query concerns what one does if the reference category is not the last time period, but rather the 1st time period. So essentially I am referring to the manual at the bottom of page 278 and top of page 279

For future reference, please cite the page within chapter (rather than the page of the full PDF file, as you must be doing). Each chapter has page numbering within chapter, to make referring to a particular page independent of changes in length in any of the preceding chapters (which happens with some frequency). So, for this thread, you are referring to pp. 20-21 in chapter 7.

where it says "Now, for a real test - how would the design matrix change if you had made age 1 the reference age class in your dummy variable coding scheme..." and then goes on to say "We’ll leave that to you as an ‘exercise’".

My confusion arises because if T1 is the reference then there will not be an interaction term and so it seems like there then ends up being some sort of redundancy. By writing out the linear model it looks like Beta 0 might be redundant since it would never appear by itself and so I have thought about dropping the intercept from the model which would then give me the right number of columns in the DM...but this doesn't seem right as then it seems I would be forcing the model through the origin.

I know I can just choose to parameterise it differently but I am intrigued as to how one would implement this. Any help would be much appreciated.

Thanks

Greg

Quick answer (I'm late for a meeting). I believe you're missing the point (in the referenced text), which has nothing to do so much with specifying the time interval used as the reference, but rather, which age class is the reference. At the bottom of p. 19, you'll see that the dummy coding discussed uses age 3 (of the 3 age classes) as the reference, such that beta's for each other age class (age 1 and 2) reflect change ('effect size') relative to age 3+ individuals. The 'take-home problem' (which I think I used on an exam once) was 'what happens if you make age 1 the reference age class'. This is indicated in how the beta dummy coding is specified at the bottom of p. 20.

[GDistiller]

Thanks very much for the quick reply - I will reference the page within chapter from now on but yes I was referring to pgs 20-21 witin chapter 7. OK I see that the take home exercise is referring to making age 1 the reference so perhaps I can rather reference page 15 earlier in the chapter but in the context of a TSM marking (to account for transients):

In that model there are 5 time dummy variables (T1-T5) and 4 interaction variables since Age*T1 falls away ie a total of 10 parameters. My question then refers to how one would approach this if one wanted to use the first time period as a ref - since there would not be an Age*T1 interaction term in the first place, does one not get 5 time dummy variables (T2-T6) and 5 interaction terms for a total of 11 parameters? This is the essence of my confusion about there being a redundant parameter somewhere.

Hope I'm making sense!

Thanks again

Greg

[cooch]

Thanks very much for the quick reply - I will reference the page within chapter from now on but yes I was referring to pgs 20-21 witin chapter 7.

No worries. The tome has gotten sufficiently ponderous (I mean, 'lengthy') that I/we have to use various things to keep track of which example, on which page, in which chapter, etc. Re-starting page numbering within each chapter was a bit of a challenge (especially at the indexing stage), but seems to work well for the intended purpose.

OK I see that the take home exercise is referring to making age 1 the reference

Correct. Point of that exercise was to show that, in principle, the choice of the reference level in a classification variable should not affect overall model fit (but it might affect estimability of certain parameters in some cases; see - sidebar - starting on p. 81 of current version of Chapter 6), but it does change in interpretation of the beta estimates (as discussed at length in Chapter 6)

so perhaps I can rather reference page 15 earlier in the chapter but in the context of a TSM marking (to account for transients):

In that model there are 5 time dummy variables (T1-T5) and 4 interaction variables since Age*T1 falls away ie a total of 10 parameters. My question then refers to how one would approach this if one wanted to use the first time period as a ref - since there would not be an Age*T1 interaction term in the first place, does one not get 5 time dummy variables (T2-T6) and 5 interaction terms for a total of 11 parameters? This is the essence of my confusion about there being a redundant parameter somewhere.

Question asked, question answered. You can't. You can see why if you consider other time intervals as the reference. For example, if you set the second interval as the reference, the DM for apparent survival would be



For the third time interval as the reference,



Play close attention to the column labels in the respective DM's - they indicate which interaction terms are plausible, for a given reference interval. For each of these models, you'll get the same model deviance (meaning, they're equivalent in terms of fit, and the reconstituted estimates on the normal probability scale will be the same), but interpretation of the beta terms differs in obvious ways.

Key - note that in all cases, the reference interval is logical-plausible for both age classes. If it isn't, then you run into the problem you describe.

I should probably add this to the book, at some point.

[GDistiller]

Ok great, thanks very much for the input, much appreciated...