Role of "phi" in multinomial logit models for spatial random effect

[Satoshi Aoki]

Dear INLA-users,

I am trying to estimate rates of choices in a space according to this text (https://inla.r-inla-download.org/r-inla.org/doc/vignettes/multinomial.pdf). In this estimation, I and my collaborator are confused with the role of "phi" in page 2 and 8 of the above text.

The question is: Are these "phi" necessary for Poisson trick? Or are they necessary only for improper models?

When I test models without phi, they work well for besagproper and besagproper2 and do not well for besag, besag2 and SPDE.
When I test a model with phi, it works well for besag and besag2 and does not for besagproper and besagproper2 (not yet tested on SPDE).

The tested models are like these:

Models without phi:
formula<-Y~ -1 + f(alt1, model="besagproper")
models[[1]]<-inla(formula, family = "poisson", data = data1, control.compute=list(config=TRUE,cpo=TRUE),control.predictor = list(compute=TRUE))
formula<-Y~ -1 + f(alt2, model="besagproper")
models[[2]]<-inla(formula, family = "poisson", data = data2, control.compute=list(config=TRUE,cpo=TRUE),control.predictor = list(compute=TRUE))
formula<-Y~ -1 + f(alt3, model="besagproper")
models[[3]]<-inla(formula, family = "poisson", data = data3, control.compute=list(config=TRUE,cpo=TRUE),control.predictor = list(compute=TRUE))

Model with phi:
formula<-Y~ -1 + f(alt1, model="besag") + f(alt2, model="besag") + f(alt3, model="besag") + f(phi,initial=-10,fixed=T)
model<-inla(formula, family = "poisson", data = data, control.compute=list(config=TRUE,cpo=TRUE),control.predictor = list(compute=TRUE), control.inla = list(int.strategy = "eb"))

The goals of these estimation are such as alt1/(alt1+alt2+al3) and their CIs.


Thank you for your time and consideration.

Satoshi Aoki

[Helpdesk (Haavard Rue)]

The multinomial likelihood has dependiencies between the number of counts in the
1..m slots. We can rewrite this as conditional independent poisson counts as
long as we make the linear predictors dependent by introducing \phi_i, which has
one value for each observation vector. The proof is there on page 2-3, and is
'well known'.

we can use a similar trick in Compositional Data Analysis as well

https://link.springer.com/article/10.1007/s11222-024-10427-3

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Håvard Rue
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[Satoshi Aoki]

Dear Rue,

Thanks for your answer!

Satoshi Aoki

2025年3月18日火曜日 16:50:19 UTC+9 Helpdesk (Haavard Rue):