[R-sig-ME] question on unit variance function

[Phillip Alday]

I haven't seen an answer go by yet, so I'll take a small stab. But
first: I don't see anything here related to *mixed* models, which is the
topic of this list; all of these questions seem to be related to more
foundational GLM / regression issues. You might have better luck on
CrossValidated (https://stats.stackexchange.com/).

On the second page of your second reference, there is a claim that the
beta distribution is not a member of the exponential family. This is
half accurate -- the beta distribution is an exponential distribution,
is not in the expoential dispersion family (see e.g.
https://stats.stackexchange.com/questions/435669/can-distributions-that-are-in-the-exponential-family-but-not-the-natural-expone
and
https://stats.stackexchange.com/questions/304538/why-beta-dirichlet-regression-are-not-considered-generalized-linear-models).
This means that it doesn't fit well with the techniques usually used in
the GLM framework (such as glm() and glmer() in R).

The variance of the binomial distribution is a function of its mean
(mu*(1-mu)), which is why the binomial distribution is parameterized
purely by the mean (and number of observations)  and there is no
"residual variance" in a binomial model.

The beta distribution is parameterized either by two shape parameters or
by the mean and precision (which is like variance a scale parameter),
see e.g. the abbreviated discussion here:
https://cran.r-project.org/web/packages/brms/vignettes/brms_customfamilies.html#the-beta-binomial-distribution.
These two very different parameterizations hint at some of the problems
with working with the beta distribution.

For all of these things, the canonical reference is McCullagh and
Nelder's Generalized Linear Models.

That doesn't answer your question directly, but hopefully it gives you
enough to continue your search.

Phillip

On 30/4/20 5:33 pm, HATICE T KUBRA AKDUR wrote:
> Dear Group Members,
>
> I hope you are keeping well during this difficult time.
> Thank you in advance.
> I have some questions below:
>
> In the beta regression model, it is stated V(\mu)= \mu(1-\mu).
>
> It is known that even though beta distribution is in the exponential
> family, logit link is not canonical link for beta regression.
>
> And, V(\mu)= \mu(1-\mu) is unit variance function for binomial
> distribution, logit link is canonical for binomial distribution.
>
>  I am not sure,  V(mu)=mu(1-mu), is it still unit variance function for
> beta regression model with logit link.  Even though, it is not in natural
> exponential family. OR we can say that V(\mu)= \mu(1-\mu) is variance
> function for beta regression with logit link.
>
> 1/g'(mu) (g(mu) is link function) can be used to obtain unit variance
> function for not natural exponential family distribution? or it is only
> true for natural exponential family?
>
> For example, unit variance function of Simplex distribution is
> V(mu)=mu_3(1-mu)_3
> because simplex distribution is in dispersion family. So, to obtain unit
> variance function, the second order derivative of the deviance function is
> used.
>
> I will be very glad, if you enlighten me.
>
> Best,
>
> References:
>
> https://onlinelibrary.wiley.com/doi/abs/10.1111/j.0006-341X.2000.00496.x
>
> https://www.tandfonline.com/doi/pdf/10.1080/0266476042000214501
>
> On Thu, Apr 30, 2020 at 11:10 AM HATICE T KUBRA AKDUR <hkubrasenol using gmail.com>
> wrote:
>
>> Dear Group Members,
>>
>> I hope you are keeping well during this difficult time.
>> Thank you in advance.
>> I have some questions below:
>>
>> In the beta regression model, it is stated V(\mu)= \mu(1-\mu).
>>
>> It is known that even though beta distribution is in the exponential
>> family, logit link is not canonical link for beta regression.
>>
>> And, V(\mu)= \mu(1-\mu) is unit variance function for binomial
>> distribution, logit link is canonical for binomial distribution.
>>
>>  I am not sure,  V(mu)=mu(1-mu), is it still unit variance function for
>> beta regression model with logit link.  Even though, it is not in natural
>> exponential family. OR we can say that V(\mu)= \mu(1-\mu) is variance
>> function for beta regression with logit link.
>>
>> 1/g'(mu) (g(mu) is link function) can be used to obtain unit variance
>> function for not natural exponential family distribution? or it is only
>> true for natural exponential family?
>>
>> For example, unit variance function of Simplex distribution is V(mu)=mu_3(1-mu)_3
>> because simplex distribution is in dispersion family. So, to obtain unit
>> variance function, the second order derivative of the deviance function
>> is used.
>>
>> I will be very glad, if you enlighten me.
>> I attached beta regression and marginal simplex model papers FYI.
>>
>> Best Regards,
>> --
>> Asst. Prof. Dr. Hatice T. Kubra AKDUR
>> Department of Statistics, Faculty of Science
>> Gazi University
>> 06500 Teknikokullar ANKARA, TURKEY
>> Phone: +90 553 324 5380
>> Email: hatice_senol using wsu.edu
>> Homepage: http://websitem.gazi.edu.tr/site/haticesenol
>>
>