Parameter counting

[cooch]

So... the moral of the story is
- don't rely on automatic counting of parameters but do a systematic count based on how the PIM or DESIGN matrix is constructed.
- write out the likelihood terms near the end of the study for CJS (and related models) and similar terms near the start of the study for JS (and related models)
- use "experience" to try and figure out how many estimable functions should exist. For most CJS (and related models) these will be product terms and ratios as noted above. With experience, you will be able to spot these fairly easily.

This, plus other approaches, are also detailed throughout chapter 4.

Moral - any method that relies on numerical approaches will be 'twitchy'. There are some analytical (symbolic) approaches you can apply, but the learning curve is fairly steep.

[cschwarz@stat.sfu.ca]

Hmmm...

Model 2: phi(stock:time) p(release:time)

My understanding is that the stocks are geographically separated and that for each stock, there are 2 release groups (e.g. week 1 and week 2). The capture occasions are "receivers" at dams downstream so there is no "time" per se (time being the receivers). In fact, the arrivals of each stock at the same receiver could be temporally separated as well, i.e. stock 1 arrives at receiver 5 in july while stock 2 arrives at receiver 5 in august.

SO there a total of 6 groups of fish released, 2 from each stock.

Likely a "better" notation would be
phi(stock:time) p(release(stock):time)
i.e. releases are nested within stocks so there is NO connection between release 1 of stock 1 and release 1 of stock 2 etc.

But, if the release groups are "connected", e.g. males and females so that a sex effect is being estimated for recovery probabilities, then we get the six products as being:

1. phi(s1, k-1) p(r1, k)
2. phi(s1, k-1) p(r2, k)
3. phi(s2, k-1) p(r1, k)
4. phi(s2, k-1) p(r2, k)
5. phi(s3, k-1) p(r1, k)
6. phi(s3, k-1) p(r2, k)

with 3 phis and 2 p's for 5 "raw" parameters.

There is still confounding with estimable parameters being:
(a) phi(s1, k-1) p(r1, k)
(b) p(r2,k)/p(r1,k) => hence (2 above) = (a)*(b)
(c) phi(s2,k-1)/phi(s1,k-1) => hence (3 above) = (a)*(c)
and (4 above) = (a)*(c)*(b)
(d) phi(s3,k-1)/phi(s1,k-1) => hence (5 above) = (a)*(d) and
(6 above) = (a)*(d)*(b).

Rather than being 5 raw parameters there are 4 estimable functions.


"Regardless of the previous issue, did you mean estimable parameters 3...6 here?" - oops - yes.

[aswea]

Wonderful!

I think I understand most of these posts. The point is to figure out how many 'fundamental' parameters are needed to arithmetically solve the the others-- the remainder are counted as estimable. I was near the right track but not sure what the likelihood terms near the end of the study would look like for model two especially.

Dave (hoping I have that dhewitt!) is right about the way I have model two structured. The release groups are timed so that each stock hits the river mouth around the same time. The hypothesis is that recapture probablity is influenced by water conditions and thus by release timing.

Evan, I will try the math in the Chapter 4 addendum once more. It's involuntary-- my eyes keep slipping past to the examples.

Thanks so much!

~Aswea

[dhewitt]

Thanks Carl for explaining that so clearly. And yes, Aswea, I'm Dave.