Estimating Transformations for Regression via Additivity and Variance Stabilization

Abstract I propose a method for the nonparametric estimation of transformations for regression. It is much more flexible than the familiar Box-Cox procedure, allowing general smooth transformations of the variables, and is similar to the ACE (alternating conditional expectation) algorithm of Breiman and Friedman (1985). The ACE procedure uses scatterplot smoothers in an iterative fashion to find the maximally correlated transformations of the variables. Like ACE, my proposal can incorporate continuous, categorical, or periodic variables, or any mixture of these types. The method differs from ACE in that it uses a (nonparametric) variance-stabilizing transformation for the response variable. The technique seems to alleviate many of the anomalies that ACE suffers with regression data, including the inability to reproduce model transformations and sensitivity to the marginal distribution of the predictors. I provide several examples, including an analysis of the “brain and body weight” data and some data on ...

[1]  D. Cox,et al.  An Analysis of Transformations , 1964 .

[2]  J. Kruskal Analysis of Factorial Experiments by Estimating Monotone Transformations of the Data , 1965 .

[3]  K. Ruben Gabriel,et al.  The Biplot as a Diagnostic Tool for Models of Two-Way Tables , 1978 .

[4]  T. Hassard,et al.  Applied Linear Regression , 2005 .

[5]  Bradley Efron Transformation Theory: How Normal is a Family of Distributions? , 1982 .

[6]  Jerome H. Friedman,et al.  Smoothing of Scatterplots , 1982 .

[7]  S. Weisberg,et al.  Residuals and Influence in Regression , 1982 .

[8]  Comment: Distinguishing between the Scale of the Estimand and the Transformation to Normality , 1984 .

[9]  The Analysis of Transformed Data: Comment , 1984 .

[10]  D. Hinkley,et al.  The Analysis of Transformed Data , 1984 .

[11]  E. B. Fowlkes,et al.  Estimating Optimal Transformations for Multiple Regression and Correlation: Comment , 1985 .

[12]  Robert E. Kass,et al.  Estimating Optimal Transformations for Multiple Regression and Correlation: Comment , 1985 .

[13]  J. Friedman,et al.  Estimating Optimal Transformations for Multiple Regression and Correlation. , 1985 .

[14]  Daryl Pregibon,et al.  Estimating Optimal Transformations for Multiple Regression and Correlation: Comment , 1985 .

[15]  Robert E. Kass,et al.  Comment: Some Observations on ACE Methodology , 1985 .

[16]  R. Tibshirani,et al.  Generalized additive models for medical research , 1986, Statistical methods in medical research.

[17]  B. Efron Better Bootstrap Confidence Intervals , 1987 .

[18]  William H. Press,et al.  Numerical recipes , 1990 .