# Saturated model: Perfect fit, no explanatory power

I have some questions about saturated models. In a nutshell, I have read that a saturated model has the following characteristics:

1. One parameter for evey data point, so perfect fit ($\hat{y} =y$).
2. None degrees of freedom, so none explanatory power of the model. By explaining eveything, ends up explaining nothing at all.

Please, could you explain the above points to me? I keep reading that 'a saturated model has a parameter for each observation which as a consequence results in a perfect fit', but it is not clear to me.

Furthermore, are the following examples of a saturated model?

1. $Y_i \sim \text{Bernoulli}(\pi_i), i=1,2.$
2. $Y_i \sim \text{Bernoulli}(\pi_i),\ \text{logit}(\pi_i) = \beta_0+\beta_1x_i,\ i=1,2.$

P.S. I have read here the answer for question 1, but I am not familiar with contingency table analysis.

• The second point is more an issue of logic rather than statistics (it's true of scientific models more generally, for example -- if every conceivable outcome could be explained by the model "whatever happened, magic fairies did it", it cannot predict anything, so it's useless) although it can be cast in a statistical light (e.g. via that same sense of prediction, for example). – Glen_b Jun 27 '16 at 0:51
• I don't think the first point is strictly true. For example, you could imagine cases where the parameters are constrained such that a perfect fit isn't achievable. A common example is penalized regression, which can work even in cases where there are more parameters than data points (if the penalty/prior is appropriate for the data). – user20160 Jun 27 '16 at 3:55