The constructed variable is defined as \(y[\log(y/\widetilde{y}) - 1]\), where
\(\widetilde{y}\) is the geometric mean of y.

The constructed variable is meant to be
added to the right-hand-side of the linear model. The t-test for the
coefficient of the constructed variable is an approximate score test for whether a
transformation is required.

If \(b\) is the coefficient of the constructed variable,
then an estimate of the normalizing power transformation based on the score statistic
is \(1 - b\). An added-variable plot for the constructed
variable shows leverage and influence on the decision to transform y.

References

Atkinson, A. C. (1985)
Plots, Transformations, and Regression. Oxford.

Box, G. E. P. and Cox, D. R. (1964)
An analysis of transformations.
JRSS B26 211--246.

Fox, J. (2016)
Applied Regression Analysis and Generalized Linear Models,
Third Edition. Sage.

Fox, J. and Weisberg, S. (2019)
An R Companion to Applied Regression, Third Edition, Sage.