Mallows' C _p statistic

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Mallows' C _p statistic , named after Colin Lingwood Mallows, is a global measure of goodness that assesses the goodness of a regression fit . It is mainly used in the context of a model selection or a variable selection , in which the aim is to find the best subset of the total predictors that deliver the best prediction. A small value of means that the model is relatively precise. ${\displaystyle C_{p}}$

In the special case of a linear regression , Mallows' statistic is equivalent to the AIC ( Akaike information criterion ). ${\displaystyle C_{p}}$

Definition and characteristics

Mallows' statistic addresses the problem of overfitting a model, in which the residual square sum becomes smaller and smaller as more variables are added to the model. So if you want to choose the model that has the smallest residual squared sum, you will always choose the model with all variables. ${\displaystyle C_{p}}$

Mallows' instead statistic uses the mean squared error ( english mean squared prediction error , in short, MSPE ): ${\displaystyle C_{p}}$

${\displaystyle \operatorname {E} \sum _{j}\left({\hat {Y}}_{j}-\operatorname {E} (Y_{j}\mid X_{j})\right)^{2}/\sigma ^{2}}$,

where is the fitted value from a regression model with j variables, the expected value of this case and the variance of the error terms . The mean squared forecast error does not automatically decrease as more variables are added to the model. ${\displaystyle {\hat {Y}}_{j}}$ ${\displaystyle \operatorname {E} (Y_{j}\mid X_{j})}$ ${\displaystyle \sigma ^{2}}$

When predictors are selected from a total of , the statistics for those predictors are usually defined as: ${\displaystyle p}$ ${\displaystyle K>p}$ ${\displaystyle C_{p}}$

${\displaystyle C_{p}={SQR_{p} \over S^{2}}-N+2p}$,

in which

• ${\displaystyle SQR_{p}=\sum _{i=1}^{N}(Y_{i}-{\hat {Y}}_{p})^{2}}$the residual sum of squares for a model with predictors, ${\displaystyle p}$

• ${\displaystyle {\hat {Y}}_{p}}$the predicted value of the -th observation with predictors, ${\displaystyle i}$ ${\displaystyle p}$

• ${\displaystyle S^{2}}$the mean square deviation after a regression analysis on the complete data set with predictors, ${\displaystyle K}$

• and is the number of observations. ${\displaystyle N}$