In statistics, Mallows's ${\textstyle {\boldsymbol {C_{p}}}}$, 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 ${\textstyle C_{p}}$ means that the model is relatively precise.

Mallows's ${\displaystyle C_{p}}$ is 'essentially equivalent' to the Akaike information criterion in the case of linear regression. This equivalence is only asymptotic; Akaike notes that ${\displaystyle C_{p}}$ requires some subjective judgment in the choice of the variance estimate associated with each response in the linear model (typically denoted as ${\displaystyle {\hat {\sigma }}^{2}}$).

Mallows's ${\displaystyle C_{p}}$ 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 ${\displaystyle C_{p}}$ statistic calculated on a sample of data estimates the sum squared prediction error (SSPE) as its population target

where ${\displaystyle {\hat {Y}}_{i}}$ is the fitted value from the regression model for the ith case, E(Y_i | X_i) is the expected value for the ith case, and ${\displaystyle \sigma ^{2}}$ 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 regressors, with k > p, the ${\displaystyle C_{p}}$ statistic for that particular set of regressors is defined as:

where

Given a linear model such as:

where: