# Re: [S] SE's for ratio of two parameters

Douglas Bates (bates@stat.wisc.edu)
25 Jun 1998 10:31:03 -0500

Pedro de Barros <pbarros@ualg.pt> writes:

> Dear all, this is in fact more like a statistical question, but I hope
> someone from the list will know the answer.
> I have been fitting the Michaelis-Menten model (simple hyperbola) to several
> data sets. I have been able to get approximate SE's for the two parameters
> thanks to Prof. Ripley, but I am unsure of how to calculate the SE for a
> derived parameter which is A=B1/B2. I have the (approximate) var-covar
> matrix for B1, B2, so I thought I would use it to get an approximation to
> the SE of A.
> Any good hints? (I was thinking of treating A as a ratio estimator, but I
> could not get at bibliography on this specific problem...)

If you are fitting the Michaelis-Menten model by nonlinear least
squares, you can re-define the model so the ratio you see is one of
the parameters in the model, then re-fit.

The Michaelis-Menten is often defined as
velocity ~ Vm * concentration / (K + concentration)
where Vm and K are the parameters. If you would prefer to write it as
velocity ~ concentration/ (R1 + R2 * concentration)
so that R1 = K / Vm, then do so and re-fit the data. This will
provide estimates of the standard errors of the R1 and R2 parameters.
As with any nonlinear model, these are approximate standard errors.

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