transition patterns matrix

[Sébastien]

Hello,
I have some trouble with the "transition patterns" matrix when modelling the transition probabilities.

First, why is the AIC and rank of the model Phi(from.t) Psi(from.to+t) p(i), for example, function of the transition patterns used ?

Then, if the AIC of the model is indeed a function of the transition patterns, we should use always the same transition patterns during all the model selection process? (i.e., use the same transition patterns for all models)?
If yes, this could be problematic when “pooling” some states together. For example, when considering a model like Psi(from(1, 2 3 4).to+t), one has to set complementaries at the same place from rows 2 to 4 but this won’t be necessarily the case for other models?

Thanks for any help,
Sebastien

[CHOQUET]

Hello Sebastien,

In reply to

>First, why is the AIC and rank of the model Phi(from.t) Psi(from.to+t)
>p(i), for example, function of the transition patterns used ?

Let's take two simple examples to explain what happen.
a) First, the model Phi(from.t) Psi(from.to) p(i) using the identity link will not change if the transition patterns change. It can be easily checked. It is the same for the multinomial (or generalized) link because no equality is set between parameter of different rows(example: Psi(f)).

b) The model "Phi(from.t) Psi(from(1 2).to(1)+others) p(i)" may depend on the patterns transition matrix. If psi(11) is the complementary of the first row, psi(11) is not constraint and psi(21) a part of the constraint parameter, then the equality between psi(11) and psi(21)(set by the sentence from(1 2).to(1) ) will not be satisfied.

Suppose now that the last row is the set of the complementary then
psi(11)=psi(21) and the other transitions are equal(exept complementaries) using the identity link.
However, this is not anymore the case with the multinomial link because we have
logit(psi(12)/(psi(11)+psi(12)+psi(13))=alpha
logit(psi(32)/(psi(31)+psi(32)+psi(33))=alpha
as psi(31) is not equal to psi(11) then psi(12) is not equal to psi(32) .

Thus, the multinomial logit (in general) destroys constraints between rows.

c) For the model Phi(from.t) Psi(from.to+t) p(i), a change of the pattern will change the constraint. Furthermore, when the additive term with time is back transform with the multinomial logit then additivity will be not be fully satisfied(see b)).

>Then, if the AIC of the model is indeed a function of the transition >patterns, we should use always the same transition patterns during all >the model selection process? (i.e., use the same transition patterns for >all models)?

Yes, in general.

>If yes, this could be problematic when “pooling” some states together. >For example, when considering a model like Psi(from(1, 2 3 4).to+t), >one has to set complementaries at the same place from rows 2 to 4 but >this won’t be necessarily the case for other models?

One solution is to use some other formulation that the separate formulation. One example is described in
Grosbois, V., Tavecchia, G(2003). Modeling dispersal with capture-recapture data: Disentangling decisions of leaving and settlement. Ecology. 84(5)
but others are possible and can be easily implemented in E-SURGE.

To summarize, special attention should be paid by constraining parameters inside the transition matrix. People have to think about before setting equalities, covariates, additivity inside the transition matrix.

Sincerely,

Rémi