# meanBEINF vs predict(model, type = "response') in BEINF GAMLSS. and determining odds of predictor variable coefficient

A variation of this question has been asked, but certain items remain unanswered -

I am modeling the proportion of mortality (Prop) using a single continuous predictor variable which is temperature (Temp). I have three questions.

1.) Should I be using meanBEINF for my model predicted estimates of the response? If so, how would I extract the associated standard errors? You think the way I have it specified currently would give you the response estimates, however, running predict(beinf_mod, type = "response", what = "mu") yields the same results which has me questioning.

2.) If I exponentiate the predictor variable coefficient (contained within the mu parameter) does this give me the odds between (0,1)? nu and tau currently don't have predictor variable coefficients so I'm not sure if those are to be worked in to get odds for the total domain [0,1].

3.) Is my interpretation of the odds correct in this scenario? I am familiar with a regular beta regression or logistic model, but, the uncertainties in question 2 make me wonder if this is appropriate.

Thanks in advance for the help, and it is greatly appreciated.

# generate DB
DB <- data.frame(Prop = c(0.688888889, 0.519230769, 0.378294574, 0.253644315, 0.234200744, 0.156626506,
0.191011236, 0.0625, 0.064516129, 0, 0, 0),
Temp = c(62.90857143, 62.75428571, 60.05428571, 60.23428571, 59.64285714, 57.94571429,
57.71428571, 57.14857143, 54.39714286, 51.87714286, 50.38571429, 49.1))

# beta inflated model. I understand na.omit works on the data, and that na.exclude is not really useful.
# I removed the NA's for this reproduction of the problem
beinf_mod <- gamlss(Prop ~ cs(Temp),
family=BEINF,data=na.omit(DB),na.action=na.exclude)

# obtain predictions for the estimated/expected value of y
predict(beinf_mod, type="response", se.fit=TRUE)

# get the odds of the explanatory variable. exponentiation gets us 1.47,
# so a one unit increase in temperature results in a 47% increase in the odds of mortality within the domain (0,1)
exp(coef(beinf_mod))
$$$$
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