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A GIC rule for assessing data transformation in regression. (English) Zbl 1058.62056

Summary: The functional form used in regression may be generalized by the Box-Cox transformation. We adopt the generalized information criterion (GIC) approach to determine a need for the G. E. P. Box and D. R. Cox [J. R. Stat. Soc., Ser. B 26, 211–243 (1964; Zbl 0156.40104)] transformation of the response variable. The utilization of the constructed variable reduces the problem to one of variable selection based on GIC. Our method leads to comparing the partial correlation coefficient between the dependent variable and the constructed variable of an artificial regression, with critical values depending on a penalty parameter. The method is illustrated with simulation examples and several well-known examples from the literature in regression diagnostics.

MSC:

62J05 Linear regression; mixed models
65C60 Computational problems in statistics (MSC2010)

Citations:

Zbl 0156.40104

Software:

MINITAB
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References:

[1] Akaike, H., Statistical predictor identification, Ann. Inst. Statist. Math., 22, 203-217 (1970) · Zbl 0259.62076
[2] Allen, D. M., The relationship between variable selection and data augmentation and a method for prediction, Technometrics, 16, 125-127 (1974) · Zbl 0286.62044
[3] Andrews, D. F., A note on the selection of data transformations, Biometrika, 58, 249-254 (1971) · Zbl 0226.62108
[4] Atkinson, A. C., Regression diagnostics, transformations and constructed variables (with discussion), J. Roy. Statist. Soc. Ser. B, 44, 1-36 (1982) · Zbl 0508.62058
[5] Atkinson, A. C., Diagnostic tests for transformations, Technometrics, 28, 29-37 (1986) · Zbl 0586.62097
[6] Bickel, P.; Zhang, P., Variable selection in nonparametric regression with categorical covariates, J. Amer. Statist. Assoc., 87, 90-97 (1992) · Zbl 0763.62019
[7] Box, G. E.P.; Cox, D. R., An analysis of transformations (with discussion), J. Roy. Statist. Soc. Ser. B, 26, 211-252 (1964) · Zbl 0156.40104
[8] Cho, K.; Yeo, I. K.; Johnson, R. A.; Loh, W. Y., Asymptotic theory for Box-Cox transformations in linear models, Statist. Probab. Lett., 51, 337-343 (2001) · Zbl 1036.62014
[9] Cook, R. D.; Wang, P. C., Transformations and influential cases in regression, Technometrics, 25, 337-343 (1983) · Zbl 0526.62059
[10] Cook, R. D.; Weisberg, S., Residuals and Influence in Regression (1982), Chapman & Hall: Chapman & Hall New York · Zbl 0564.62054
[11] Hinkley, D. V.; Runger, G., The analysis of transformed data (with discussion), J. Amer. Statist. Assoc., 79, 302-320 (1984) · Zbl 0553.62051
[12] Mallows, C. L., Some comments on \(C_p\), Technometrics, 15, 661-675 (1973) · Zbl 0269.62061
[13] Nishii, R., Asymptotic properties of criteria for selection of variables in multiple regression, Ann. Statist., 12, 758-765 (1984) · Zbl 0544.62063
[14] Potscher, B. M., Model selection under nonstationaryautoregressive models and stochastic linear regression models, Ann. Statist., 17, 1257-1274 (1989) · Zbl 0683.62049
[15] Ryan, T. A.; Joiner, B. L.; Ryan, B. F., Minitab Student Handbook (1976), Duxbury Press: Duxbury Press North Scituate, MA
[16] Schoukens, J.; Rolain, Y.; Pintelon, R., Modified AIC rule for model selection in combination with prior estimated noise models, Automatica, 38, 903-906 (2002) · Zbl 1011.93504
[17] Schwartz, G., Estimating the dimension of a model, Ann. Statist., 6, 461-464 (1978) · Zbl 0379.62005
[18] Shao, J., Linear model selection by cross-validation, J. Amer. Statist. Assoc., 88, 486-494 (1993) · Zbl 0773.62051
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