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I have a biological relation: Y/m = (X * b) / (1 + X * b)

where Y and X are variables, m and b are parameters. m is greater than Y, and everything is greater than 0.

I have some training data with X and Y values and would like to estimate the parameters. Currently using nonlinear least squares. Is there any way I can reparameterize this into a glm model?

What if I took an assumed value for m, so that it became a constant?

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The model can be stated as

$\frac{m}{Y} = \frac{1+XB}{Xb}$

or

$\frac{m}{Y} = \frac{1}{Xb} + 1$.

I would expect linear regression to be a solution with the independent variable being $\frac{1}{X}$ and the dependent variable being $\frac{1}{Y}$.

In fact, you can get away with making no assumptions about m. As long as $m>0$, your equation is

$\frac{1}{Y} = c\frac{1}{X} + \frac{1}{m}$.

Linear regression will be able to solve this problem without explicitly enforcing any $m$

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  • $\begingroup$ Intesting! Could run into problems if m/Y=1 as this would force estimates of b to infinity. $\endgroup$
    – conjectures
    Commented May 7, 2015 at 23:01
  • $\begingroup$ That is not possible because $\frac{m}{Y} =1$ implies $Xb = 1 + Xb$ so the solution ought to force something to $\infty$ $\endgroup$
    – Sid
    Commented May 7, 2015 at 23:05
  • $\begingroup$ So this would require an assumption of Gaussian errors to fit this as a glm right? @conjectures and Sid, I was thinking of just making m = max of the Yi in my data. To solve this infinity problem, I could make it max(Yi) + e, where e is some small offset? $\endgroup$
    – Count Zero
    Commented May 7, 2015 at 23:13
  • $\begingroup$ Unless you know/expect m to be something based on a theorem of sorts, I recommend not fiddling with it $\endgroup$
    – Sid
    Commented May 8, 2015 at 1:00

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