R help - Dec 2004 - why use profile likelihood for Box Cox transformation?

Hi All,

I'm analysing some data that is conventionally modelled as log(Y) = a + bX +
e.  However, using the boxcox function, it appears that the optimum value of
lambda is approx 0.05.  I have 40 data sets of differing sizes and for about
half of these, lambda is significantly non-zero.  So, it is worth looking into.

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?


Any help would be greatly appreciated.

Thanks,
Paul.



Paul Livingstone
Statistical Analyst
AeroStructures®
Level 14, 222 Kings Way
South Melbourne, Vic, 3205
Phone: 03 9694 1083
Mobile: 0418 121 530
Fax: 03 9696 8195
Email: paul.livingstone@aerostructures.com.au
Web: www.aerostructures.com.au


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"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