What is meant when we say we have a saturated model?
A saturated model is one in which there are as many estimated parameters as data points. By definition, this will lead to a perfect fit, but will be of little use statistically, as you have no data left to estimate variance.

For example, if you have 6 data points and fit a 5th-order polynomial to the data, you would have a saturated model (one parameter for each of the 5 powers of your independant variable plus one for the constant term).
