Hi Mick,

If I understand correctly, you use a non-linear transformation on Y and then use MLR. These MLR beta coefficients will minimize the MSE between \widehat{Y}^{1/4} and Y but will not be optimal to minimize the MSE between \widehat{Y} and Y, and this latter point seems precisely to be your issue.

What I suggest you to do is to use Newton-Raphson algorithm. The criterion to minimize is :  $ \sum\limits_{i = 1}^{n} \left(y_i \left[\sum\limits_{j=1}^{p} \beta_j x_{ij}\right]^{1/a}\right)^2 $ if I consider you use p-independent variables and that you have n observations. a equals to 0.4 for your specific issue. You can compute the first and second order derivatives of this criterion (for each $\beta_j$, do not forget the cross-terms). Thus, you will get an expression for both gradient and hessian matrix and will be able to run a Newton-Raphson algorithm. Due to the nature of your problem, this algorithm will converge and the result will be the set of coefficients you are looking for. The only thing that you know have to do is to compute these derivatives (you can either do this with a computer or take a pen get explicit formulas). (Initial conditions for this problem should not matter)

I hope this helps.

Note that this is a usual way to operate and that this method can be use to solve a broader class of problems.

Best regards,

Geoffrey

Hi Geoffrey - You helped my lateral thinking powers to run some iterations around the 0.4 that was suggested by the Box-Cox option in my JMP software.  I found one nearby that allowed the resulting model to satisfy all the assumptions of linear regression, and also give the graph of dependent variable observation on predicted to have a slope of 1.00 & intercept near enough to 0!  So many thanks. The Newton-Raphson algorithm is new to me and I will study this for the next, new improved model.  Best Regards, Michael.