1.3.3.5. Box-Cox Linearity Plot

1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.3. Graphical Techniques: Alphabetic

1.3.3.5. Box-Cox Linearity Plot

Purpose:
Find transformation to maximize correlation between two variables

When performing a linear fit of Y against X, an appropriate transformation of X can often significantly improve the fit. The Box-Cox transformation (Box and Cox, 1964) is a particularly useful family of transformations. It is defined as:

where X is the variable being transformed and

is the transformation parameter. For

= 0, the log of the data is taken instead of using the above formula.

The Box-Cox linearity plot is a plot of the correlation between the transformed Y and X for given values of . The value of corresponding to the maximum correlation (or minimum for negative correlation) on the plot is then the optimal choice for .

Sample Plot

The plot of the original data with the predicted values from a linear fit indicate that a quadratic fit might be preferrable. The Box-Cox linearity plot shows a value of = 2.0. The plot of the transformed data with the predicted values from a linear fit with the transformed data shows a better fit (verified by the significant reduction in the residual standard deviation).

Definition

Box-Cox linearity plots are formed by

Vertical axis: Correlation coefficient from the transformed X and Y
Horizontal axis: Value for

Questions

The Box-Cox linearity plot can provide answers to the following questions:

Would a suitable transformation improve my fit?
What is the optimal value of the transformation parameter?

Importance:
Find a suitable transformation

Tranformations can often significantly improve a fit. The Box-Cox linearity plot provides a convenient way to find a suitable tranformation without engaging in a lot of trial and error fitting.

Related Techniques

Linear Regression
Box-Cox Normality Plot

Case Study

The Box-Cox linearity plot is demonstrated in the Alaska pipline data case study.

Software

Box-Cox linearity plots are not a standard part of most general purpose statistical software programs. However, the underlying technique is based on a transformation and computing a correlation coefficient. So if a statistical program supports these capabilities, writing a macro for a Box-Cox linearity plot should be feasible. Dataplot supports a Box-Cox linearity plot directly.