First, the log transformation is equivalent to lambda = 0 for the Box-Cox, which is not too different from lambda = 1/3 for the cube root. The correlation between the data transformed by these two values for lambda will be very high.
Second, in the absence of outliers, the loglikelihood profile method would seem to dominate other criteria. It calculates the likelihood of the transformed data assuming normality with constant variance. Departures from normality with constant variance of both types you mention--nonnormality or nonconstant variance--are penalized the theoretically appropriate amount by this approach.
Finally, the Box-Cox transformation to normality, although often serviceable, is an aging technology. It is built on the somewhat optimistic premise that the transformation of the response that improves its adherence to normality (and constant variance) will also lead to (or at least allow) a simple model of the expected value of this transformed response. (V&R2, page 216 repeat the suggestion of Box and Cox of using "the largest linear model" on the right hand side when using boxcox() to find a suitable value for lambda.) More modern approaches allow separate models for the expected value of the response and for its variance.
A lovely reference on all of this is R. J. Carroll and D. Ruppert's Transformation and Weighting in Regression, London and New York: Chapman and Hall. My edition (the first) is dated 1988, but it may have been updated.
Terry Elrod
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Prof. Terry Elrod; 3-23 Fac. of Business; U. of Alberta; Edmonton AB; Canada T6G 2R6
email: Terry.Elrod@Ualberta.ca; tel: (403) 492-5884; fax: (403) 492-3325
Web page: http://www.ualberta.ca/~telrod/
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