Understanding output of powerTransform In the car package, we have the function powerTransform which transforms variables in a regression equation to make the residuals in the transformed equation as normal as possible. I am confused about what this transformation is and further in the following example:
# Box Cox Method, univariate
summary(p1 <- powerTransform(cycles ~ len + amp + load, Wool))

# fit linear model with transformed response:
coef(p1, round=TRUE)
summary(m1 <- lm(bcPower(cycles, p1$roundlam) ~ len + amp + load, Wool))

What I am confused about is what exactly the model p1 is. Is it simply the linear model without a transformation, then it finds the optimal parameter, we then use that to specify m1? So what is the regression equation for p1, m1??
 A: The powerTransform() function in the car package determines the optimal power at which you should raise the outcome variable (in this case, cycles) prior to including it in a linear regression model. The optimal power is denoted by lambda, so outcome^lambda becomes the transformed outcome variable.
For your example, you can type p1$lam in the R Console and you will see that the optimal lambda is lambda =−0.05915814.
In practice, lambda is rounded before being used to transform the outcome, so that we get a nice power.
If you type p1$roundlam in the R Console, you will see that the rounded optimal value for lambda is lambda = 0.
Now that you know what optimal transformation will be used on the outcome variable cycles in the Wool data set, you can type 
bcPower(Wool$cycles,p1$roundlam) 
to see the values of cycles^0 (i.e., the outcome values raised to the power 0). The linear model fitted to these transformed outcome values is:
m1 <- lm( bcPower(cycles, p1$roundlam) ~ len + amp + load, data = Wool )

So the real question is: What does it mean to raise a variable to the power 0? By convention, it means that we are log transforming the variable using the natural log transformation. You can check that this is indeed the case by comparing the results of the following R commands:
bcPower(Wool$cycles,p1$roundlam)

log(Wool$cycles)

In summary, the model being fitted for m1 is simply:
m1.new <- lm( log(cycles) ~ len + amp + load, data = Wool )

You can compare summary(m1) with summary(m1.new) to convince yourself that this is indeed the case.
