Mallows's C_{p}
In statistics, Mallows’s C_{p},^{[1]}^{[2]} 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.^{[3]}
Contents
Definition and properties
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 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 σ^{2} 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.
If P regressors are selected from a set of K > P, the C_{p} statistic for that particular set of regressors is defined as:
where
- is the error sum of squares for the model with P regressors,
- Y_{pi} is the predicted value of the ith observation of Y from the P regressors,
- S^{2} is the residual mean square after regression on the complete set of K regressors and can be estimated by mean square error MSE,
- and N is the sample size.
Alternative definition
Given a linear model such as:
where:
- are coefficients for predictor variables
- represents error
An alternate version of C_{p} can also be defined as:^{[4]}
where
- RSS is the residual sum of squares on a training set of data
- d is the number of predictors
- and refers to an estimate of the variance associated with each response in the linear model (estimated on a model containing all predictors)
Note that this version of the C_{p} does not give equivalent values to the earlier version, but the model with the smallest C_{p} from this definition will also be the same model with the smallest C_{p} from the earlier definition.
Limitations
The C_{p} criterion suffers from two main limitations^{[5]}
- the C_{p} approximation is only valid for large sample size;
- the C_{p} cannot handle complex collections of models as in the variable selection (or feature selection) problem.^{[5]}
Practical use
The C_{p} statistic is often used as a stopping rule for various forms of stepwise regression. Mallows proposed the statistic as a criterion for selecting among many alternative subset regressions. Under a model not suffering from appreciable lack of fit (bias), C_{p} has expectation nearly equal to P; otherwise the expectation is roughly P plus a positive bias term. Nevertheless, even though it has expectation greater than or equal to P, there is nothing to prevent C_{p} < P or even C_{p} < 0 in extreme cases. It is suggested that one should choose a subset that has C_{p} approaching P,^{[6]} from above, for a list of subsets ordered by increasing P. In practice, the positive bias can be adjusted for by selecting a model from the ordered list of subsets, such that C_{p} < 2P.
Since the sample-based C_{p} statistic is an estimate of the MSPE, using C_{p} for model selection does not completely guard against overfitting. For instance, it is possible that the selected model will be one in which the sample C_{p} was a particularly severe underestimate of the MSPE.
Model selection statistics such as C_{p} are generally not used blindly, but rather information about the field of application, the intended use of the model, and any known biases in the data are taken into account in the process of model selection.
References
- ^ Mallows, C. L. (1973). "Some Comments on C_{P}". Technometrics. 15 (4): 661–675. doi:10.2307/1267380. JSTOR 1267380.
- ^ Gilmour, Steven G. (1996). "The interpretation of Mallows's C_{p}-statistic". Journal of the Royal Statistical Society, Series D. 45 (1): 49–56. JSTOR 2348411.
- ^ Boisbunon, Aurélie; Canu, Stephane; Fourdrinier, Dominique; Strawderman, William; Wells, Martin T. (2013). "AIC, C_{p} and estimators of loss for elliptically symmetric distributions". arXiv:1308.2766.
- ^ James, Gareth; Witten; Hastie; Tibshirani. An Introduction to Statistical Learning. http://www-bcf.usc.edu/~gareth/ISL/ISLR%20Sixth%20Printing.pdf: Springer. p. 211. ISBN 978-1-4614-7138-7.
- ^ ^{a} ^{b} Giraud, C. (2015), Introduction to high-dimensional statistics, Chapman & Hall/CRC, ISBN 9781482237948
- ^ Daniel, C.; Wood, F. (1980). Fitting Equations to Data (Rev. ed.). New York: Wiley & Sons, Inc.
Further reading
- Chow, Gregory C. (1983). Econometrics. New York: McGraw-Hill. pp. 291–293. ISBN 0-07-010847-1.
- Hocking, R. R. (1976). "The analysis and selection of variables in linear regression". Biometrics. 32 (1): 1–50. doi:10.2307/2529336. JSTOR 2529336.
- Judge, George G.; Griffiths, William E.; Hill, R. Carter; Lee, Tsoung-Chao (1980). The Theory and Practice of Econometrics. New York: Wiley. pp. 417–423. ISBN 0-471-05938-2.