Scaling Probabilities in Binomial Regression

[Damien Beecroft]

Hello INLA Community,

I am trying to get the scale parameter to work for a binomial model, but am struggling with the implementation. Here is what I am trying.

```
n = 1000
a = 1
b = 1
z = rnorm(n)
eta = a + b*z
prob = exp(eta)/(1 + exp(eta))
q = 0.5 * runif(n) + 0.5
# q = 0.9
Ntrials = sample(0:10, size=n, replace=TRUE)
y = rbinom(n, size = Ntrials, prob = q*prob)
data = data.frame(y, z, Ntrials)

formula <- y ~ 1 + z
r = inla(formula,
         family="binomial",
         data=data,
         Ntrials=Ntrials,
         scale=q)  # THE IMPORTANT LINE
summary(r)
```

From the documentation of the binomial distribution this seems like it should be correct (link at bottom). However, the regression is totally wrong. I also tried "xbinomial" but that crashed.

Furthermore, the documentation for "scale" that shows up upon invoking `?inla` is the following:

scale    Fixed (optional) scale parameters of the precision for Gaussian and Student-T response models. Default value is rep(1, n.data).

From the results I am getting it seems that the description from `?inla` is correct and the binomial likelihood documentation is false. I also may be reading something wrong.

Thank you in advance!

Best,

Damien

Binomial Likelihood Documentation:

chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://inla.r-inla-download.org/r-inla.org/doc/likelihood/binomial.pdf

[Damien Beecroft]

Also, is there a way to scale probabilities for the cbinomial likelihood as well?

Best,

Damien

[Helpdesk (Haavard Rue)]

There is currently no way to scale the probabilities using 'scale'. for the
binomial likelihood, which is documented here

inla.doc("^binomial$")

suppose we add a scaling, so that

p = scale * prob

where prob depends on the model (like linear predictor), and scale is a constant
> 0, then this would impose a constraint on 'prob' (ir scale > 1), and thereby a
constraint on the lin.predictor, leading to constrained high-dimensional
Gaussians.

it is possible to change the link function to have prob \in [a,b], but not
scaling it in general.

it would be possible to add it for 0 < scale < 1, which almost acts like a
thinning probability, or even let scale depend on covariates (see the model
'tpoisson'

there is a way to defined your own likelihood in 'c', see inla.doc("cloglike")
for which you can add your own scaling, but this would not prevent scale * prob
of going beyond 1 if scale > 1.

Best
H

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Håvard Rue
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[Håvard Rue]

its the same issue. the easy way out is to write your own likelihood function,
where you can do all this

On Mon, 2026-01-26 at 10:06 -0800, Damien Beecroft wrote:

> > chrome-
> > extension://efaidnbmnnnibpcajpcglclefindmkaj/https://inla.r-inla-download.or
> > g/r-inla.org/doc/likelihood/binomial.pdf
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> .

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Håvard Rue

hr...@r-inla.org

[Damien Beecroft]

Thank you for the response. The document invoked from `inla.doc("^binomial$")` is the same one I linked. If you look at the "Expert version" section it states there is a way to scale the probabilities of the binomial regression. So, it seems the binomial documentation is wrong. Also,  `inla.doc("cloglike")`  and  `inla.doc("^cloglike$")` do nothing. I think the correct command is ` vignette("rgeneric", package = "INLA")`. I will try to implement the likelihood function.

Best,

Damien

[Finn Lindgren]

That “expert” part clearly says the scale option is only available in the xbinomial model.

(Whether it works for your case is a separate issue, but it’s clear that trying it with plain binomial is not supposed to work)

Finn

On 26 Jan 2026, at 19:37, Damien Beecroft <dob...@uw.edu> wrote:

Thank you for the response. The document invoked from `inla.doc("^binomial$")` is the same one I linked. If you look at the "Expert version" section it states there is a way to scale the probabilities of the binomial regression. So, it seems the binomial documentation is wrong. Also,  `inla.doc("cloglike")`  and  `inla.doc("^cloglike$")` do nothing. I think the correct command is ` vignette("rgeneric", package = "INLA")`. I will try to implement the likelihood function.

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[Damien Beecroft]

Hello Finn,

Thanks for the quick reply. When I tried xbinomial before it was always failing (even without the scale ooption) and I got confused. However, I noticed your example at the bottom of the binomial documentation and it helped me create a working example. Sorry for the confusion and thank you for the help!

Best,

Damien