why use profile likelihood for Box Cox transformation?

"Paul Livingstone" <paul.livingstone at aerostructures.com.au> writes:

The alternative model, Y^lambda = a + bX + e, has been explored
before by non-statistician colleagues. But instead of using boxcox
and maximising the profile likelihood, the model has been twisted,
shuffled, differenced and logged, to get

ln(dY/dX) = A + B.ln(Y) + E

and lambda ( =f(B) ) estimated via LS regression. Note: RHS contains
Y, not X. This relationship has some physical justification.


I assume that these two approaches are not equivalent, is this correct?  

Correct.

I assume the Box Cox approach (profile likelihood) is better, is
this correct and why?

This is sort of similar to the issue of output least squares vs.
system least squares in inverse problems theory. 

If what you have is a relation between Y and x and (only) Y is
measured with errors, you'd be getting a bias towards zero in the
estimated B by using the "shuffled" equation.

Then again, it's not really obvious that Box-Cox is right either
because it mixes up the functional relation and the error
chacteristics. Y^lambda should be linear in X _and_ have normally
distributed errors with a constant variance. You might need one lambda
to linearize and another to stabilize the variance.
 
If the errors really enter at the systems level (you have a stochastic
differential equation), it's a different story altogether!

O__  ---- Peter Dalgaard             Blegdamsvej 3  
  c/ /'_ --- Dept. of Biostatistics     2200 Cph. N   
 (*) \(*) -- University of Copenhagen   Denmark      Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)             FAX: (+45) 35327907