I have a case study where I use a multi-state (actually multi-event with incomplete states’ classification) analysis to estimate and testing hypotheses on the state-specific survival probabilities and their chances of changing from one state to another.

I have three alive states (A, B, and C). The model selection indicates that some probabilities in the state transition are equal. These:

A ->B = A->C; B->A = C->A; B->C = C->B

Therefore, my probabilistic matrix (Matrix 1), for the state-transition part, looks like this (speaking of which: is there a way of showing a table in the forum? I have not been able to make the "table" BBCode tag work):

https://docs.google.com/document/d/e/2PACX-1vS85cs7k-T3GJqQfCmwRsEBbA-PGC4m02oeNGmH25Ue7-NIU3-0Hg2VjLMMJJemT4d8wc2Ccq5ULDq-/pub

Now, I want to test the effect of a time-varying covariate X (25 intervals) on some of these probabilities.

Without entering in too many details of what I have done in E-SURGE - this wants to be a general question - I have re-parameterized the above matrix so that the B->B and C->C were no longer the complementary probabilities (because I wanted to test the effect of X also on the probability of remaining in the same B or C state). This is the new probabilistic matrix (Matrix 2):

https://docs.google.com/document/d/e/2PACX-1vQCFA8aP9N-VfGuxhP2xZDOqDHx2c_4sD0q3ietPJKa_jZwjkYeabR4leJ0lBU35uLK7_o8UvwQXZka/pub

I have used Matrix 1 to test the effect of X on:

A->B|C (|means “or” here)

B|C-> A

And Matrix 2 to test the effect of X on:

B->B (same as C->C)

B->C (same as C->B)

My doubts are due to the fact that the multinomial (or generalized) link function is used here to force the parameters in a row to add 1. Recently, I read an article by Guery et al. (2019)* that made me doubt this approach. They suggest that is not always possible to use a covariate with a parameter constrained by a multinomial link (like it uses to be the case in state transitions with > 2 states) and they show how to do a post-hoc analysis correctly using the Delta Method. The reason for that, they explain, is because there is no a one-to-one correspondence between the demographic (biological) parameters and the mathematical parameters which "...renders obscure, if at all possible, the way to implement a desired constraint on demographic parameters".

While I see they are right when they say there is no a one-to-one correspondence between the demographical and mathematical parameters**, I fear that in a case like this the remedy may be worse than the disease.

To get an idea of how the lack of such a one-to-one correspondence affected my hypothesis-testing, I have tested the same model, where X affected the prob of B-> B, using two different parameterizations of the probability matrix (maintaining the equalities between some probabilities as I had found before) and I have seen that I get almost the same identical return (and QAICc). The estimates of p B-> B in the two models differ from less than 0.0001. All this makes me think that the effect I find is real and that, even recognizing the problem, this can be a reasonable approximation. Any opinion is welcome.

* Guéry, L., Rouan, L., Descamps, S., Bêty, J., Fernández‐Chacón, A., Gilchrist, G., & Pradel, R. (2019). Covariate and multinomial: Accounting for distance in movement in capture–recapture analyses. Ecology and evolution, 9(2), 818-824.

**

Matrix 1:

Pr(B->C) = exp(betaBC) / 1 + exp(betaBC) + exp(betaBA)

Note that here the B->B is the complementary probability

Pr(B->C) is a demographic parameter

betas are mathematical parameters

If you have only two alive states (say, A and B, in which case you are probably using the logit link), there is a one-to-one correspondence between the demographic and mathematical parameter because:

Pr(A->B) = exp(betaAB) / 1 + exp(betaAB)

Matrix 2:

Pr(B->C) = exp(betaBC) / 1 + exp(betaBC) + exp(betaBB)

Note that here the B->A is the complementary probability