number of parameters

[jastratford]

Hi everyone, I'm reviewing a paper using MARK and I need some clarification on how to estimate the number of parameters. I asked on ECOLOG and had different answers so I'm coming over to the forum (plus I'm just interested). I don't have the data and I didn't do the analysis but I'm trying to evaluate the table with the AICs and weights.

This is a model using both a continuous variable and a factor with three levels. They didn't specify that they were doing a slope-only analysis (no intercept) so I assume that the intercept is one of the parameters.

From what I understand there are 5 parameters (1 for intercept, 1 for variance, 1 for the slope of the continuous variable, and 2 for the factors [number of levels -1]).

Is this correct?

Thanks so much,

Jeff

[AdamGreen]

Could you clarify what type of MR model they are using? From your description, it sounds like that is the model structure for 1 parameter (e.g., survival or detection), but to assess whether the AIC values are correct, you would also need to include the structure on the other model parameter(s).

For example, a CJS model estimates survival and detection. For a model with the structure you described on survival only with a constant detection probability, you would have 5 parameters - 1 intercept, 2 for the other factor levels, 1 for the continuous variable slope, and 1 intercept for detection. If you had the same structure on both survival and detection, you would have 10 parameters. Obviously with other structures and their combinations between survival and detection, it's hard to say how many parameters should be estimated without more info.

And that's just with a CJS model. Other models have potentially many more higher-level parameters that can be modeled as functions of however many covariates/time/age/etc.

[jastratford]

Hi Adam,
No detection was included (or I would have included it, of course). This is the nest survival model and they included other covariates but I wasn't sure how to enumerate the number of parameters given a factor with X levels. So, from your response, it does look like you have X-1 parameters (given X is the number of levels of a factor) plus the intercept and variance, right? Thanks!

[cooch]

Could you clarify what type of MR model they are using? From your description, it sounds like that is the model structure for 1 parameter (e.g., survival or detection), but to assess whether the AIC values are correct, you would also need to include the structure on the other model parameter(s).

For example, a CJS model estimates survival and detection. For a model with the structure you described on survival only with a constant detection probability, you would have 5 parameters - 1 intercept, 2 for the other factor levels, 1 for the continuous variable slope, and 1 intercept for detection.

This would imply an additive model. Let TRTx be the factor coding variable, and COV be the continuous covariate, then the linear model for survival would be

logit(phi) = beta1 + beta2*(TRT1) + beta3(TRT2) + beta4(COV)

(where TRT1 and TRT2 are the dummy coding for the different treatment levels). So, 4 parameters for phi. If they included the interaction of TRT with COV, then 6 parameters for phi, and so forth.

logit(phi) = beta1 + beta2*(TRT1) + beta3(TRT2) + beta4(COV) + beta5(TRT1*COV) + beta6(TRT2*COV)

Slope only models don't make sense, here, because the x-axis is 'time', and no intercept presumes that everything gets forced through the origin, which doesn't make a lot of sense in this context (IMO - which is why they're not in the MARK book). After all, if y would be 0 when x is 0, then a slope-only model makes sense. That is equivalent to survival being 0 at some point in time, which makes little sense to me.

[cooch]

Hi Adam,
No detection was included (or I would have included it, of course). This is the nest survival model and they included other covariates but I wasn't sure how to enumerate the number of parameters given a factor with X levels. So, from your response, it does look like you have X-1 parameters (given X is the number of levels of a factor) plus the intercept and variance, right? Thanks!

The 'variance' (to which you keep referring) isn't a 'term' in the model. One for intercept, k-1 for k levels of a discrete factor, 1 for each continuous covariate, and (k-1) x number of covariates for interactions.

As an aside, you always learn k-1 degrees of freedom in ANOVA for discrete factors. But, they invariably never teach you why. Simple explanation - df = number of columns in the design matrix needed to code for k levels of a factor, conditional on using only binary dummy variables 1 or 0. The reason they don't teach the underlying logic between the 'k-1' mantra is that you'd need to teach the design matrix, which they should, but don't. While working with the DM in MARK is/can be a pain (sometimes), it is the best tool I know of to fully understand what the structure is of the model you're working with.

[jastratford]

Thanks Evan, that clarifies things. Marc Kery's WinBUGS course used the approach you suggest (showing the DM) and it was very enlightening - but that was years ago.

[cooch]

Thanks Evan, that clarifies things. Marc Kery's WinBUGS course used the approach you suggest (showing the DM) and it was very enlightening - but that was years ago.

The downside is that once you've achieved full understanding, having to use the DM can be a pain (although I think that is overplayed -- you build the DM for your most general model, and most other models are simply nested within that -- which amounts to little more than deleting columns from the DM -- about as painless as you can get). However, in the hands of folks that really understand what they're doing, things like RMark (which was written by Jeff Laake in large part to shield people from the DM) have their place. I think the different perspective Jeff and I tend to have on this sort of thing is that Jeff spends most of his time actually analyzing data (and in his hands, he can do this most efficiently using RMark), whereas I spend most of my time trying to explain to people 'why' it works. And to that end, the DM has considerable value as a 'teaching heuristic'.

This is an old debate. I spent years using SAS for linear models, without really knowing how it worked. I then decided this was not a good way to work, so spent a lot of time after looking at the mechanics of how things like PROC GLM and PROC NLMIXED worked. My general view is that knowing how things work is as important as knowing how to make them work.