Saturated model

In the statistics one is saturated model , including full model or saturated model (of Latin saturare German saturate ) called a model in which the same number of parameters such as observation pairs occur. For each data point there is a parameter (since there are data points, there are also parameters) and thus a saturated model does not impose any restrictions on the data. While there is no error with this model (there are no residual degrees of freedom ), the model is not informative. Nevertheless, the model can be saturated as a basis for calculating a model with parameters are used ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle p<n}$

Plausibility function

Since a saturated model has as many free parameters as there are observations , the saturated model can be interpreted as a model with freely selectable parameters. The saturated model is the most general model and thus has the highest likelihood (plausibility). For the likelihood function (plausibility function) for a saturated model with unknown parameters to be estimated, this results ${\displaystyle \vartheta _{i}}$ ${\displaystyle y_{i}}$ ${\displaystyle n}$ ${\displaystyle \vartheta _{i}}$

${\displaystyle L(\vartheta _{i};y_{i})=\prod _{i=1}^{n}f_{Y}(y_{i};\vartheta _{i})\quad {\text{mit}}\;{\hat {\vartheta }}_{i}=y_{i}\mathrm {\;f{\ddot {u}}r\;alle\;\;} i=1,\ldots ,n}$.

Hence, in a saturated model, a total number of parameters must be estimated. Since the fit to the data is perfect in a saturated model, there are no residual degrees of freedom and no residual deviation . A saturated model has zero deviance . ${\displaystyle n}$

Individual evidence

1. ↑ Lothar Sachs , Jürgen Hedderich: Applied Statistics: Collection of Methods with R. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2018, ISBN 978-3-662-56657-2 , p. 834
2. ↑ Michael J. Crawley: Statistics with R. , p. 327
3. ↑ David Collett : Modeling survival data in medical research . Chapman and Hall / CRC, 2015. pp. 154 ff.