Re: Calculating standard error of nonlinear regression coefficients
- From: "rudnyi" <rudnyi@xxxxxxxxxxxxxxxxxxxxx>
- Date: 21 May 2005 02:40:43 -0700
Jack Tomsky wrote:
> > I have implemented a simple nonlinear regression in
> > Excel using the solver to minimize the squared error
> > and obtain least squares estimates of the
> > coefficients. Can anyone tell me how to compute the
> > standard error of the coefficients, and/or suggest a
> > good reference?
> > Thanks,
> > Eric
>
>
> By nonlinear regression, I'm asuming that you mean nonlinear in the
regression coefficients.
>
> Let f(y_j) = f(b_1, ..., b_k; x_j1, ..., x_jk)
>
> The covariance matrix of the least-squares estimates of the b's is
approximately that when the model is linearized by
>
> f(y_j) = Sum[(df/db_i)*b_i] + const.
>
> The derivatives are then plugged into an n by k design matrix,
forming A, and the covariance matrix of the parameter estimates is
approximated by
>
> SIG(b) = sig^2*(A'A)^(-1).
>
> Excel has matrix multiplication and matrix inversion functions.
>
> Jack
Well, the linerization may introduce its own error. You can use
directly sum of squares to estimate error bounds for parameters - note
that the error bounds can be unsymmetric. There is a nice book where
you can find more detail
Seber, G.A.F. and Wild, C.J.
Nonlinear regression, (John Wiley & Sons. New York, 1989)
The computational procedure becomes more complicated though. I am not
sure that it can be done in Excel.
Evgenii Rudnyi
http://Evgenii.Rudnyi.Ru/
.
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