Confusion over multi-state occupancy parameterization

[darryl]

Hi All,
There's been a couple of recent posts on phidot and off-list questions about the different parameterizations floating around for the recently developed multi-state occupancy model. So I thought I'd post this quick note to try and clarify things.

As background, the multi-state occupancy model was developed for situations where rather than just species presence/absence (occupied/unoccupied) you may be able to classify 'presence' into multiple additional states. For example, non-breeding/breeding, disease presence/absence or low/high abundance. It doesn't have to be limited to just 2 categories, but that's all that MARK and PRESENCE have been geared up for so far. The key is that there is ambiguity about one of the states, but not the other (e.g., if you see evidence of breeding then there has to be breeding, but not seeing evidence of breeding does not negate the possibility that breeding may be occurring there, you may have just missed seeing the required evidence).

Right, now in chapter 10 of our book and also in the online exercises put together by Terri Donovan and Jim Hines, the following parameterization for the psi parameters has been used:

Book and Online:
psi1(BO) = Pr(occupied and no breeding)
psi2(BO) = Pr(occupied and breeding)
1-psi1(BO)-psi2(BO) = Pr(unoccupied)

But MARK and PRESENCE use a different definition based upon the recent paper Nichols et al. (2007) Ecology 88: 1395-1400

MARK/PRESENCE:
psi1(MP) = Pr(occupied)
psi2(MP) = Pr(breeding given site is occupied)
1-psi1(MP) = Pr(unoccupied)

The reason for the difference is that the former is using a multinomial probability structure (eg, tossing a 3-sided coin) while the latter is using a conditional binomial probability structure (eg toss 1 2-sided coin and if that is a 'heads' then toss the second 2-sided coin, but if it was tails do nothing). Importantly it's the same model, just a different way of parameterizing it. Which one to use depends on the biological question of interest, but the latter one has some numerical niceties which is why that has been implemented in MARK and PRESENCE so far.

The relationship between the parameterizations is:
psi1(MP) = psi1(BO)+psi2(BO)
psi2(MP) = psi2(BO)/psi1(MP) or equivalently psi(BO) = psi1(MP)*psi2(MP)

I hope this helps.

Darryl

[jlaake]

Thanks for pointing this out more clearly. One small typo that should be obvious

psi2(MP) = psi2(BO)/psi1(MP) or equivalently psi(BO) = psi1(MP)*psi2(MP)

should be

psi2(MP) = psi2(BO)/psi1(MP) or equivalently psi2(BO) = psi1(MP)*psi2(MP)

A more important point to consider is that the type of model/design matrix you create will depend on the parameterization. With the BO parameterization, you would expect that Psi1 and Psi2 would have covariates that shared parameters because they are both modelling the occupancy process. For example, in the breeding example Darryl used, let's say you had a covariate temperature. In the BO example, the following would be a very plausible model/design:

Psi1(intercept+temperature)Psi2(intercept+breeding +temperature)
Psi1 1 0 temperature
Psi2 1 1 temperature

In the MP model this could be

Psi1(intercept+temperature)Psi2(breeding)
Psi1 1 temperature 0
Psi2 0 0 1

where Psi1 determines occupancy and Psi2 determines breeding given the site is occupied. You could also suppose that temperature affected breeding as well as occupancy but if it did then it would be unlikely to have the same slope and the MP design would be

Psi1(intercept+temperature)Psi2(breeding+temperature)
Psi1 1 temperature 0 0
Psi2 0 0 1 temperature

Notice that in these cases the design matrix is composed of 2 non-overlapping design matrices and none of the columns are shared among the parameters. That is the normal circumstance with most models in MARK/PRESENCE with some exceptions like p1,p2 in MSOccupancy and p,c in Closed capture models in MARK. In RMark these exceptions are handled with a the share argument which can only be used for specific parameters.

As Darryl pointed out the mlogit link could be used if the BO parameterization was used. In fact, the BO parameterization may be the better more flexible choice if models are developed to allow more than 2 states because the conditional probability structure becomes more difficult to interpret and use. If in fact the BO parameterization was used it could be handled like Psi in the Multistratum models which would be a single parameter with a variable number of indices depending on the number of states defined in the data using alphabetic characters. Obviously the definition of delta would also have to be expanded.