y-axis in plot(gam) , difference between trans = plogis and trans = exp

Question

I saw a comment here by @gavin-simpson y-axis values in plot(gam) , I don't understand why trans = exp is used instead of trans = plogis , how do u decide which one to use?

The code I am using

library(mgcv)
b <- gam(outcome ~ s(week, k = 4, fx = TRUE, by = food) + food, data = df1, family = betar(link="logit"), method = "REML")
summary(b)

My dependent variable is in proportion from 0 to 1. Week is a form of timeline measure, and food is a categorical variable.

When I do trans = plogis, I get the plot below

plot(b, pages = 1, trans = plogis, shift = coef(b)[1])

How shall I interpret this plot here with trans = plogis ?

When I do trans = exp, I get the plot below

plot(b, pages = 1, trans = exp, shift = coef(b)[1])

My question is why am I getting values larger than 1 when I do trans = exp ?

I am new to GAM and still learning, any guidance is appreciated.

Answer 1

In the linked question they were talking about negative binomial models, which by default use the $\log$ as the link-function, since it deals with counts and frequencies just like Poisson-models. Beta-models deal with stuff like probabilities which want a link-function that deals with the limits at 0 and 1, the default being $\text{logit}$.

What the default link-function is for your family of models in R can be found with help, e.g.: help(nb), or help(betar). Other link-functions can be specified, but this is advanced.

The inverse link-function is generally assumed to be easy to find. plogis being R for $\text{logit}^{-1}$ is a little weird, but since it rises strictly monotonically from 0 to 1 it works as a CDF and so R implemented it as one. Probit models pull the same trick in the other direction: https://en.wikipedia.org/wiki/Probit_model