Repeat Exercise 12.9 using X∗ = ln(X) in the linear model. What transformation do you obtain this time and is it satisfactory?
Exercise 12.9
Use the data in Exercise 12.8 and the Box–Cox method to arrive at a transformation on Y. Recall that the Box–Cox method assumes a particular model E(
) = Xβ. For this exercise, use E(Yi) = β0
+ β1Xi. Plot SS[Res(λ)] versus λ, find the minimum, and determine approximate 95% confidence limits on λ. What choice of λ does the Box–Cox method suggest for this model? Fit the resulting regression equation, plot the transformed data and the regression equation, and observe the nature of the residuals. Does the transformation appear to be satisfactory with respect to the straight-line relationship? With respect to the assumption of constant variance? (Note: The purposes are, in addition to demonstrating the use of the Box–Cox transformation, to show the dependence of the method on the assumed model and to illustrate that obtaining the power transformation via the Box–Cox method does not guarantee either that the model fits or that the usual least squares assumptions are automatically satisfied.)
Exercise 12.8
The following growth data (Y = dry weight in grams) were taken on four independent experimental units at each of six different ages (X = age in weeks).
(a) Plot Y versus X. Use the ladder of transformations to determine a power transformation on Y that will straighten the relationship. Determine a power transformation on X that will straighten the relationship. (b) Use the Box–Tidwell method to determine a power transformation on X for the linear model. Does this differ from what you decided using the ladder of transformations? Is there any problem with the behavior of the residuals?
(c) Observe the nature of the dispersion of Y for each level of X. Does there appear to be any problem with respect to the least squares assumption of constant variance? Will either of your transformations in (a) improve the situation? (Do trial transformations on Y for the first, fourth, and sixth levels of X, ages 1, 5, and 9, and observe the change in the dispersion.)