linear model nested in time-dependent?

[Jochen]

Dear all
Being a MARK-user for some time but not a statistician I would like to be sure whether the following test is justified:

suppose fitting first a simple time-dependent model using the PIMs like Phi(something)p(t) and then a linear model where time-depency is replaced with some covariate, e.g. Phi(something)p(rainfall) using the DM - would it be correct to test these two models using LRT?
MARK does compute the test and I found one example in the literature (Warnock et al. 1997. Condor 99: 906-915) but no particular advice in "the book"

Thank you!

[cooch]

Dear all
Being a MARK-user for some time but not a statistician I would like to be sure whether the following test is justified:

suppose fitting first a simple time-dependent model using the PIMs like Phi(something)p(t) and then a linear model where time-depency is replaced with some covariate, e.g. Phi(something)p(rainfall) using the DM - would it be correct to test these two models using LRT?
MARK does compute the test and I found one example in the literature (Warnock et al. 1997. Condor 99: 906-915) but no particular advice in "the book"

Thank you!

I'll defer asking why you want an LRT in the first place - obviously, at one level, this *is* a fundamental consideration.

But, speaking to your question - strictly speaking, no (in my opinion). The LRT is only valid if used to compare hierarchically nested models. Meaning, the more complex (general) model of the pair must differ from the simple model only by the addition of one or more parameters. Adding additional parameters will always result in a higher likelihood score.

So, if your model is

theta = intcpt + beta(covariate)

versus

theta = incpt + beta(time 1) + beta (time 2) + ... + beta(time n)

they are clearly not hierarchically nested, since to get to the second model from the first, you'd need to (i) drop a term (for the covariate) and then (ii) add terms for the time intervals. I should probably resurrect it, and add some discussion of it to the linear models chapter (as if it wasn't long enough already).

I ran a simulation of this sort of thing once a long time ago, and the test statistic between the two models is was anything but Chi-sq (however, having said that, there *might* be an argument that you could generate the appropriate null for the test statistic by using such a Monte Carlo approach, to derive your nominal P-value, but that would require some work - even if conceptually acceptable to you).

[gwhite]

Actually, the models Evan described are nested, as is the dot model. So the triplet:

theta = incpt + beta(time 1) + beta (time 2) + ... + beta(time n)
versus
theta = intcpt + beta(covariate)
versus
theta = intcpt

are all nested. To see that the beta(covariate) model is nested in the beta(time) model, just run the covariate model with a different beta for each of the n rainfall values. Nothing sacred about using the value of 1 for a dummy variable value as we commonly do -- just makes interpretation of the beta values easier.

Now, when is it appropriate to do a LRT? In theory, only before you have inspected the model selection results produced by MARK using AICc.

However, with that said, I would advocate a still different approach: ANODEV, also implemented in MARK. What ANODEV does is tell you what percent of the deviance between the dot and t model is explained by the covariate, and provides an F test as to whether this amount is statistically significant. You make the decision on whether the amount explained is biologically important. To run the ANODEV procedure, you have to specify the set of 3 nested models, i.e., the t, covariate, and dot model. The MARK help file explains the procedure crudely, and gives you at least one reference on the procedure (as I remember).

Gary

[cooch]

Actually, the models Evan described are nested, as is the dot model. So the triplet:

theta = incpt + beta(time 1) + beta (time 2) + ... + beta(time n)
versus
theta = intcpt + beta(covariate)
versus
theta = intcpt

are all nested. To see that the beta(covariate) model is nested in the beta(time) model, just run the covariate model with a different beta for each of the n rainfall values. Nothing sacred about using the value of 1 for a dummy variable value as we commonly do -- just makes interpretation of the beta values easier.

While that seems reasonable, it differs from some of the classic linear models literature (which is what based my earlier response on). I'll dig further. As per my earlier note, though, my concern is not so much as to whether or not they are nested (Gary's argument seems solid), but whether or not the difference in deviances is chi-sq distributed in such cases. My earlier simulation attempts at pretty much exactly this problem strongly suggested otherwise.

Stay tuned...

[Jochen]

Thanks, Gary and Evan!
The ANODEV approach appears to be a nice solution for my particular problem.
Jochen

[cooch]

While that seems reasonable, it differs from some of the classic linear models literature (which is what based my earlier response on). I'll dig further. As per my earlier note, though, my concern is not so much as to whether or not they are nested (Gary's argument seems solid), but whether or not the difference in deviances is chi-sq distributed in such cases. My earlier simulation attempts at pretty much exactly this problem strongly suggested otherwise.

Stay tuned...

Added a bunch of text to chapter 4 (sidebar beginning on p. 54) and chapter 6 (sidebar beginning on p. 36), on the determination of which models are nested, and which are not. Might be of some help...