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# Box Cox Transformation – Definition

### Box-Cox Transformation Defined

A Box-Cox power transformation refers to a way of transforming response to satisfy the usual regression assumption of homogeneity and normality of variance.  The regression model is therefore used to fit the transformed response. The Box-Cox power transformation can be used to transform a variable for other various purposes.

### A Little More on the Box-Cox Y Transformation

Box-Cox Transformation is effective when the response Y provides a positive result.

The transformation that is commonly used increases the response to some power.

Box and Cox (1964) developed and provided an in-depth explanation of power transformation. The constructed transformation formula provides a continuous definition regarding the parameter λ, to achieve comparable error sums of squares.

The Y Transformation option of the Box-Cox matches transformations that range from λ = –2 to 2 in increments of 0.2. To select an appropriate value of λ, the likelihood function for every transformation is determined through computation. The computation is based on the assumption that errors are normal and independent with mean zero and variance σ2. The λ value that optimizes the probability is chosen.  This value also optimizes the SSE over the λ value. The λ value that optimizes the probability is determined using a quadratic interpolation between the two grid points that surround the grid point with the lowest SSE.

The report of Box-Cox Transformations indicates a pilot demonstrating the sum of squared errors (SSE) values against the λ value. A line plot signifies 95% confidence a one-sided interval for λ. The confidence interval is determined by the confidence area as defined in the  Box and Cox (1964, p. 216).

The following inequality defines the confidence region:

SSE(λ) < SSE(λbest) * exp(ChiSquareQuantile(0.95,1) / dfe)

where

SSE(λbest) is the SSE computed by using the described Best λ

Chi-Square Quantile (0.95,1) is the 0.95th quantile of a χ2 distribution with 1 degree of freedom. The dfe refers to the error in the degree of freedom in the variance analysis table for the regression model.

The Box-Cox Transformations description has the following options:

Refit with Transform

This option allows for the specification of a value for lambda and helps to define transformed Y variable the presents the fit for list square to the transformed variable.

Replace with Transform

This option helps to specify the lambda value the define the variable of Y transformed and used the transformed variable to replace the existing list square fit. When the multiple responses occur, it only replaces the report of the response being transformed.

Save Best Transformation

This option focuses on creating a new column in the table of data and store the formula for the best transformation.

Save Specific Transformation

This option focuses on creating a new column in the table of data and store the formula for the specified transformation.

### Academic Research on Box-Cox Transformation

• The BoxCox transformation technique: a review, Sakia, R. M. (1992). The statistician, 169-178.  Box & Cox (1964) proposed a parametric power transformation technique in order to reduce anomalies such as non-additivity, non-normality, and heteroscedasticity.  This paper explores the Box-Cox transformation techniques that are used for power transformation. According to the author, although the transformation has been extensively studied, no bibliography of the published research exists at present. The author further suggests that an attempt is made here to review the work relating to this transformation.
• Improving your data transformations: Applying the BoxCox transformation, Osborne, J. W. (2010). Practical Assessment, Research & Evaluation, 15(12), 2.
• This paper presents a brief discussion and overview of traditional normalizing transformation and how Box-Cox integrates, encompasses, and improves on these traditional approaches to normalizing data. The author presents various examples of the applications and provides details concerning the automation and use of Box-Cox in SAS and SPSS. According to the author, many people in the social science interacts with data that do not match the normality assumptions of variance. Therefore, the author presents ways through which the confirmation to the normality can be achieved.
• The use of the BoxCox transformation in limited dependent variable models, Poirier, D. J. (1978). Journal of the American Statistical Association, 73(362), 284-287. This article discusses the general limited dependent variables and their roles in the transformation of power.  The author provides by-products of the general model provide an effective nesting framework that distinguishes statistically various limited dependent variables. The study further suggests and illustrate the ability of the general model to differentiate between competing specifications
• Quantile regression, BoxCox transformation model, and the US wage structure, 1963–1987, Buchinsky, M. (1995). Journal of Econometrics, 65(1), 109-154. This study explores and discusses the changes in the increments (‘returns’) to teaching and knowledge at various points of the wage distribution. The study also explores the changes that occur within the inequality of group wage. According to the study, the newly generated quantile regression techniques used together with the Box-Cox transformation model to survey the current population data.
• Variance estimates in models with the BoxCox transformation: implications for estimation and hypothesis testing, Spitzer, J. J. (1984). The Review of Economics and Statistics, 645-652. This paper examines the variance-covariance matrix parameters within the estimation model. The study presents that breaking the variance-covariance matrix into components play important role in understanding various concepts such as the efficiency of the algorithm, the estimation of OLS and the impacts of lack of scale invariance in the t-ratios for the linear coefficients on hypothesis testing.
• Experience with using the BoxCox transformation when forecasting economic time series, Nelson Jr, H. L., & Granger, C. W. J. (1979). Journal of Econometrics, 10(1), 57-69. This study is based on forecasting economic series using the Box-Cox transform.  The author presents that despite the wide use of Box-Cox transform, it sometimes does not produce superior forecasts. The procedure used in this study was based on considering transformations xci) = (xi – 1)/L.
• Extending the Box-Cox transformation to the linear mixed model, Gurka, M. J., Edwards, L. J., Muller, K. E., & Kupper, L. L. (2006). Journal of the Royal Statistical Society: Series A (Statistics in Society), 169(2), 273-288. This paper focuses on discussing the extension of Box-Cox transformation in the linear mixed model.  According to the author, the Box-Cox method based on a univariate linear model helps in the selection of a response transformation to ensure the validity of related assumptions. The author further presents that the need to extend models to linear mix method raises many questions of concerns.  The most important is to determine how the distribution of two random sources impacts the validity assumptions.
• A note on the multivariate BoxCox transformation to normality, Velilla, S. (1992).  This study focuses on examining some aspects of the multivariate Box-Cox transformation to normality that has been a matter of concern for some time. The investigation determined the optimal value of the quantile index in the list median square model. The study discovered that the quantile index must be upwardly adjusted to achieve the desired bound
• A Monte Carlo investigation of the Box-Cox transformation in small samples, Spitzer, J. J. (1978). Journal of the American Statistical Association, 73(363), 488-495. This paper investigates the small-sample characteristics of models that transform both independent and dependent and variables by using the similar Box-Cox transformation. The bias that occurs in the model tends to be the major problem. Nonetheless, the size and sign of the parameter used for transformation depend on the dependent variable.  The hypothesis tests of this study revealed that t statistics often results in inappropriate decisions because the normal sampling contains heavier tail areas than the t distribution.
• The estimation of economic depreciation using vintage asset prices: An application of the BoxCox power transformation, Hulten, C. R., & Wykoff, F. C. (1981). Journal of Econometrics, 15(3), 367-396. This paper explores the use of Box-Cox in evaluating and interpreting economic data. The study uses the Box-Cox power transformation to mitigate the problem that arises from economic depreciation and rate of estimation.  The study discovered that the use of the Box-Cox model is analogous to the use of CES production function.
• A primer on BoxCox estimation, Spitzer, J. J. (1982). The Review of Economics and Statistics, 307-313. This paper examines the use of Box & Cox technique in reducing anomalies such as non-normality, non-additivity, and heteroscedasticity. The author also suggests that although the model is a matter of concern, no valid literature has been developed to provide an in-depth investigation into the matter.  Therefore, he seeks to provide a deep understanding of the concept.