How to build a saturated model I am struggling with the understanding of a saturated model.
As far as I know, the saturated model is the model that have as many parameter as the data points. But I don't know how to build it or what is the exact form of the saturated model.
For illustration, I have the example as follow:
$(Y, X_1, X_2, X_3, X_4)$ = $(1, 2, 3, 4,5); (2, 3,4, 5,6); (3,4,5, 6,7), (3, 5, 6,7,8)$
where $Y$ is a Poisson distribution, $X_1, X_2, X_3, X_4$ are the independent variable. The link function is log.
I think the saturated model may have the form:
$log(Y) = coef_1 * X_1 + coef_2 * X_2 + coef_3 * X_3 + coef_4 * ???$
(4 coefficients/ parameters as we have 4 data points)
And to fill in $???$, I think we have many way to choose : $X_1 * X_2, X_1/X_3$ or even $X_1* X_2 * X_3 / X_4$...
So I would like to have 2 question please:

*

*How many saturated model are their given a dataset for a GLM model ? (i.e. we fix the hypothesis that Y follow some distributions and also fix the dataset) and what is its form ? Is there any general principle to construct the saturated model ?


*If there are more than 1 saturated models, what is the "real saturated" model ? (because as I know, the saturated model is defined to be the model that fit perfectly the dataset)
Thank you very much for your help!
 A: The question is not specific to GLMs, or to Poisson models. This applies also to any regression model.
A saturated model is one in which there are as many estimated parameters as observations, as you say. By definition, this will lead to a perfect fit, but will be of little use statistically.


*

*How many saturated model are their given a dataset for a GLM model ?


Given an arbitrary dataset, you can construct as many saturated models as you wish. If there are insufficient variables, then you can add higher order terms, interactions, or other derived variables such a logarithms or fractional powers. Once the number of parameters equals the number of observations, the model will be saturated.



*If there are more than 1 saturated models, what is the "real saturated" model ? (because as I know, the saturated model is defined to be the model that fit perfectly the dataset)


I don't think the term "real saturated" is well-defined. If you have $n$ observations and $p$ variables, then (assuming a model with no intercept) with $n >p$ then you can include all $p$ variables, and just add further terms until you reach saturation. With $n = p$ then you can include all $p$ variables, and the model will be saturated, and with $n<p$ you can choose whatever variables you want (as well as derived terms) to achieve saturation.
