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How do you obtain predicted probabilities for the one-inflated component (nu model) of a one-inflated beta regression in gamlss?

I have built the following model

zib <- gamlss(prop.abun.max ~ season + time + temp + 
                 last.rain.bom + rain + wind + cloud + 
                 re(random = ~1|site), 
                 sigma.formula = ~1, 
                nu.formula = ~ season + time + temp + 
                last.rain.bom + rain + wind + cloud + 
                re(random = ~1|site), 
                family = BEINF1, 
                data = na.omit(subset2))

I obtain predicted probabilities for the beta distribution component of my model (the mu model) using

  head(predict(zib, what = "mu", type = "response"))
[1] 0.7519171 0.7366541 0.7605794 0.6904190 0.7578658 0.7280828

This produces values in the range 0-1 which I assume are predicted probabilities.

However, similar code that references the one-inflated component of my model (the nu model) obtains values that are on the range 0.3-4.1. These values are clearly not predicted probabilities as many values are greater than 1.

  head(predict(zib, what = "nu", type = "response"), n = 10)
 [1] 0.6079466 0.9698540 0.7028005 0.6680394 0.6896672 0.6375064 0.6461947 0.6620159 1.2440965 0.7722830

The best post I have found on this is here. This posts asks a similar question for a zero- and one-inflated beta model in gamlss. However, this answer is not applicable to a one-inflated beta model only (i.e without the zero-inflation) and the reference text suggested by the post no longer seems to be available.

Any advice/assistance to obtain the correct predicted probabilities for the one-inflated component (nu model) of a one-inflated beta regression in gamlss would be very much appreciated?

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    $\begingroup$ Are you sure they aren't on the logit scale? $\endgroup$
    – Demetri Pananos
    Commented Sep 9, 2021 at 20:46
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    $\begingroup$ $\nu$ is parametrized as $\nu = p_1/(1 - p_1)$, the odds of a $1$. Fitting is done on the log-odds (logit) scale. So in order to calculate $p_1$, you have to take $\nu/(1 + \nu)$, as explained in the post you linked to. $\endgroup$
    – COOLSerdash
    Commented Sep 9, 2021 at 21:15

1 Answer 1

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The predicted probabilities that Y=1 are given by

p1 = nu/(1+nu)

So just predict nu and then transformation nu to p1.

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