  # st: RE: Michaelis-Menten and regression

 From "Nick Cox" To Subject st: RE: Michaelis-Menten and regression Date Wed, 22 Sep 2004 15:23:43 +0100

```Michaelis-Menten function fitting is a can of worms.
The main thing is to be aware of that and to have
looked at the literature on it. There is
a lot, going back decades; do not try to reinvent

_Biometrics_ is a good journal here. You may be able to
exploit stuff on http://www.jstor.org

However, in Stata one attractive route is to reformulate
the problem as a generalised linear model with reciprocal

See the exchange in

Generalized Linear Models for Enzyme-Kinetic Data
J. A. Nelder; D. Ruppert; N. Cressie; R. J. Carroll
Biometrics 47(4) (Dec., 1991), pp. 1605-1615

which I haven't read for some years, so the memories
are hazy. As I recall, the main idea is this. You have

y = ax / (1 + bx)

so

1/y = 1 / ax + b / a

Let us define

X =  1/x

and reparameterise

A = 1 / a

B = b / a

We then have

1/y = AX + B

and the right-hand side is then a piece of cake.
The left-hand side we take care of by
using a reciprocal link, another piece of cake
with -glm-.

In Stata terms

. gen rec_x = 1 / x
. glm y rec_x, link(power -1)

Of course, all this is just algebra with the
deterministic curve and says

Nelder I guess recommends using a gamma
family.

Nick
n.j.cox@durham.ac.uk

Simon Moore

> I have a reasonably simple hypothesis that the form of relationship
> between the independent variable (x) and the dependent variable (y)
> follows the Michaelis-Menten rational function, f(x) = ax/(1+bx).  I
> want to have this in a regression model with the cluster() option: reg
> y f(x) V, cluster().  But I can't see a way of achieving this and
> having reg solve for a and b.  I thought maybe a power expansion of
> f(x) might work, but this does not seem appropriate.
>

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```