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Concept
Mallows's Cp
Summary
In statistics, Mallows's Cp, 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 Cp means that the model is relatively precise.
Mallows's Cp has been shown to be equivalent to Akaike information criterion in the special case of Gaussian linear regression.
Mallows's Cp 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 Cp statistic calculated on a sample of data estimates the sum squared prediction error (SSPE) as its population target
where is the fitted value from the regression model for the ith case, E(Yi | Xi) is the expected value for the ith case, and σ2 is the error variance (assumed constant across the cases). The mean squared prediction error (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.
If P regressors are selected from a set of K > P, the Cp statistic for that particular set of regressors is defined as:
where
is the error sum of squares for the model with P regressors,
Ypi is the predicted value of the ith observation of Y from the P regressors,
S2 is the estimation of residuals variance after regression on the complete set of K regressors and can be estimated by ,
and N is the sample size.
Official source
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