Often a plot of a response versus a predictor variable shows the relation to be nonlinear. Two approaches to fitting a regression function to such data are: (1) fit a nonlinear function, or (2) transform the response and/or predictor to make the relationship linear and then fit a linear function.
In the first approach, the nonlinearity can often be handled in a simple linear regression by specifying as the regressor an appropriate nonlinear function of the predictor. This was the case in Example 7.7, pg 371, of Applied Statistics for Engineers and Scientists, where fuel consumption seemed to be linearly related to the fifth power of the equivalence ratio. Often, however, fitting an appropriate nonlinear function of the predictor can entail serious computational difficulties and is beyond the capabilities of many computer packages.
This lab lets you investigate the second approach: Transforming the response and/or predictor variable. The transformations considered will be power transformations: transformations that raise the variable to a power. In the lab, there will be a scatter plot with two slidebars; one oriented along the horizontal axis, which specifies the power of the transformation for the predictor (the x variable), and one along the vertical axis, which specifies the power of the transformation for the response (the y variable). To move the bars, there are three options: (1) click and drag, (2) press the arrows at the end of each bar, OR (3) place the curser where you want the bar to be and click once. As you move the bar, the transformation power will be adjusted. When the power zero is chosen, the natural logarithm is computed. For any other value l, the transformation y^l or x^l, is chosen. The transformation value of -1 is called the reciprocal transformation. The values are transformed by the function, . The square root transformation corresponds to the power .5, and takes the form . To gauge the strength of linear association of the transformed data, the value of the Pearson correlation coefficient will be displayed.
At the left of this window, you will see a menu bar. This can be used to access any segment of this lab. To start the lab, select Applet. Once open, there will be instructions at the bottom that can be navigated by slide bar or pressing Next and Previous. Once you have completed the lab, you may be instructed to answer the summary questions. Even if you are not, have a look: they will be a good test of your understanding.