[R] why use profile likelihood for Box Cox transformation?

Peter Dalgaard p.dalgaard at biostat.ku.dk
Tue Dec 21 00:53:22 CET 2004

"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?  


> 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

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