multistate model with continuous covariates for psi

[lrmont]

Hi
I have recently posted a question regarding confidence intervals for a product of reals in a multistate model, and have a related question, but on a separate topic, so I decided to post as a new topic. I am using a multistate model to infer a tradeoff between survival and reproduction (transition between two states: Breeding and Non-breeding) according to body size (BS) and condition (SMI) in bats. From my reading of the Appendix C.17 in the Mark Book, I understand that the formula to estimate psi parameters without restriction would be Psi.s = list(formula=~-1+stratum:tostratum). I have not found examples where continuous individual covariates are added to the model. I want to have the linear part of the models nested within the stratum:tostratum (mainly interested in N-B) levels, so was wondering whether the model Psi.s = list(formula=~-1+stratum:tostratum:(SMI+BS)) makes sense. Thanks a lot for your time.
Best,
Leandro

[awan]

Hi Leonardo,

I am also using continuous covariates in the transition structure of the multi-state model. I am not sure exactly what stratum:tostratum:(SMI+BS) does (I haven't used that notation myself before), but for example to include the effects of two separate covariates on the transitions, I use -1 + stratum:tostratum + stratum:tostratum:BS + stratum:tostratum:SMI). This will then estimate separate parameters for how BS and SMI influence each of the transitions. If you only want to look at N->B, then you could include a dummy variable coded 1 for N-B and 0 for B-N which would then also be included in the interaction.

Hope that helps

Andy

[lrmont]

Thanks for your reply Andy
I was hoping that the notation I used (stratum:tostratum:(SMI+BS)) would estimate BS parameters that are partial (independent from) to SMI. and vice versa, as in a multiple regression. I wonder if the notation you suggest estimates separate parameters for each variable, as if they were separate regressions, not considering the partial contribution. I hope we can sort this out. Thanks again.
Leandro

[jlaake]

I must have missed the earlier post. What Andy wrote will give you the equivalent algebraic formula

Alpha_st + beta_1st*BS + beta_2st*SMI

for transition from stratum s to stratum t with an intercept alpha_st for BS=0 and SMI=0 and slopes beta_1st and beta_2st. This is equivalent to a multiple regression with 2 continuous covariates with the intercept and slopes being dependent on st pairing. So not sure what you meant with your comment regarding regression. The equation you wrote only contains the beta slope terms but assumes a constant intercept for each transition.

Jeff

[lrmont]

Thanks Jeff
Can you please just help clarify one more thing? I compared the results using the two equations and want to understand the difference in interpretation. Using (stratum:tostratum:(SMI + BS)) and looking at the N-B transition, the beta for SMI is positive and the beta for BS is negative. Using (-1 + stratum:tostratum + stratum:tostratum:BS + stratum:tostratum:SMI), the beta for SMI is still positive, but the beta for BS is positive. I am trying to understand the change in signal of the BS variable. Why is it negative only when I assume a constant intercept for each transition? Thanks again and sorry for the long thread.

Leandro

[jlaake]

I took this thread offlist so I could get into the details with Leandro. Here is my response to him that is relevant to his last question

I didn't read your initial post well. It is now clear you are using -1 in your first formula whereas your most recent posts did not include a -1. Using -1 is simply wrong because it is forcing the intercept to be 0. The -1 is fine and needed when you have the separate term stratum:tostratum in the formula because they are factor variables and they create intercepts. You have it with : on numeric variables which create slopes for the numeric variables. Once you include -1 it removes the intercept which forces the intercept to be 0. Would be like regression y=bx. But even if you drop the -1 it would have an intercept but it would be constant for BtoN and NtoB and clearly from the output of the second formula they aren't close to being the same. Intercept for BtoN is 23.8 and for NtoB it is -38.6.

These are not very interpretable because they represent the value when SMI=0 and BS=0 which presumably cannot happen. I suggest you change these variables by subtracting off the mean of each before you do the analysis. Then the intercept is the value for the mean BS and SMI values. Then I suggest using the second formula that Andy suggested.