# Box-Cox Transformation - Explained

What is a Box-Cox Transformation?

# What is a Box-Cox Transformation?

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.

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## How is a Box-Cox Y Transformation Used?

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 , to achieve comparable error sums of squares. The Y Transformation option of the Box-Cox matches transformations that range from = 2 to 2 in increments of 0.2. To select an appropriate value of , 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 2. The value that optimizes the probability is chosen. This value also optimizes the SSE over the value. The 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 value. A line plot signifies 95% confidence a one-sided interval for . 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() < SSE(best) * exp(ChiSquareQuantile(0.95,1) / dfe) where SSE(best) is the SSE computed by using the described Best Chi-Square Quantile (0.95,1) is the 0.95th quantile of a 2 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.