INLA for Skewed Distribution

[Noah Silverman]

Hello,

I have some data where the dependent variable is best described by a skewed Student-T distribution. (It is not centered on zero, has right skew, etc...) 

INLA has the Student-T family.  But, I don't see anything that would handle the skew.  Do I need to "T standardize" my dependent variable first? Or, is there another way to handle this?

I was wondering what you think of the following options:

1. Just fit the model with the T family and skewed dependent variable
   
2. Un-skew the dependent variable, using a form of Box-Cox for T distribution
3. Normalize the dependent variable to Guassian and fit a standard normal mode

Additionally, given the three choices above, what would be the best model summary statistic to determine the "best" one?

Any ideas or suggestions would be greatly appreciate.

Thank You!!!

[Finn Lindgren]

Hi Noah,

since the observation family applies to the _conditional_ distribution
of the response/dependent variable (or to the regression residuals in
the case of additive error observation families) and not to the
marginal distribution, neither of those options would directly help
with your problem.
In some cases, transforming the marginal distribution of the data as
in your option 2 may be sensible, but that's highly problem dependent.

I would suggest trying option 1 and then looking at the residuals to
see how skewed they are; this is not something one can tell from the
raw observation data; it is a model output.
If the residuals aren't very skewed, it's likely ok. If they are, you
would need to consider further options. (I have some in mind, but they
would be research projects, so not available off-the-shelf...)

Finn

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Finn Lindgren
email: finn.l...@gmail.com

[Helpdesk (Haavard Rue)]

On Wed, 2022-11-09 at 17:22 +0000, Finn Lindgren wrote:
> I would suggest trying option 1 and then looking at the residuals to

the most recent testing version actually have deviance residuals
implemented, its for testing only....

inla(..., control.compute=list(residuals=TRUE))



--
Håvard Rue
he...@r-inla.org

[JJ Hubbard]

Are the deviance residuals comparable across data that have been transformed and are on different scales? Or we do need to apply the inverse transform to them?

[Noah Silverman]

Thanks Finn,

I appreciate it.

[Helpdesk (Haavard Rue)]

deviance residuals, in the frequentist sense, are defined as

sign * sqrt(saturated.deviance)

where 'sign' is usually taken as the sign of y-y.predicted

all this is pr observation, so I avoid the _i notation.

In the Bayesian framework, there is no natural definition of residuals,
but we can define it as

sign * sqrt(E(saturated.deviance))

where the E() is over the posterior, like integrating out
hyperparameter(s) and the uncertainty of the linear predictor.

the 'sign' is a different story, as there is no definition of the 'sign'
now. it is defined as attached, which is 'very likely' in correspondance
with the freq.definition. Another issue, is that I do not know, in this
part of the code, if data are discrete or not, so...

well, I know all this is a little 'loose', but it is anyhow useful to
have some kind of 'residuals' available, although they should not be
used for any kind of ''testing''-purpose.

of'course, for some models, like binary data, then all this make less
sense, but this is well know


If anyone has other options or other comments, please let us know

Best
H

[JJ Hubbard]

Thank you very much, I will look into the deviance residuals.

Finn, you said that observation family applies to the _conditional_ distribution, so the family argument in inla() is not determined by the distribution of your target variable, but by the conditional distribution of Y given the predictors?

[Finn Lindgren]

Yes. That’s how virtually all regression models work (it’s not an inla specific thing)

Finn

On 11 Nov 2022, at 01:20, JJ Hubbard <jordanj...@gmail.com> wrote:

Thank you very much, I will look into the deviance residuals.

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