Mallows's Cp

In statistics, Mallows's C_p, named for Colin Lingwood Mallows, is used to assess the fit of a regression model that has been estimated using ordinary least squares. It is applied in the context of model selection, where a number of predictor variables are available for predicting some outcome, and the goal is to find the best model involving a subset of these predictors. A small value of C_p means that the model is relatively precise.

Mallows's C_p has been shown to be equivalent to Akaike information criterion in the special case of Gaussian linear regression.

Mallows's C_p addresses the issue of overfitting, in which model selection statistics such as the residual sum of squares always get smaller as more variables are added to a model. Thus, if we aim to select the model giving the smallest residual sum of squares, the model including all variables would always be selected. Instead, the C_p statistic calculated on a sample of data estimates the mean squared prediction error (MSPE) as its population target

where ${\displaystyle {\hat {Y}}_{j}}$ is the fitted value from the regression model for the jth case, E(Y_j | X_j) is the expected value for the jth case, and σ² is the error variance (assumed constant across the cases). The MSPE will not automatically get smaller as more variables are added. The optimum model under this criterion is a compromise influenced by the sample size, the effect sizes of the different predictors, and the degree of collinearity between them.

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