Need GLM ideas for nonlinear biochemical model I have a function from molecular bio where I am trying to estimate the parameters $\alpha$ and $\beta$.  
$\frac{Y}{M} = f(\alpha + \beta X)$ where $0 \leq Y \leq M$, and $f(a) = \frac{a}{1+a}$.  
Both Y and X correspond to abundance of a biochemical analyte, that is to say they are strictly positive.  I'd like to fit a GLM to data with X and Y and do inference on estimates of these parameters from data.  
Can have suggestions for GLM formulations that I can try to fit this model?  
 A: The functional form of the relationship is not linear;
since $f(a) = \frac{a}{1+a}$, it looks somewhat similar to a logistic form (it's logistic in $\ln a$), which suggests you might be able to try a quasibinomial model with logit link... but because the argument to $f$ is $\alpha+\beta X$ that won't work - we can't take logs of the two terms separately.
That is, for a quasibinomial GLM you can fit a model of the form $E(\frac{Y}{M}) = \frac{\exp(\alpha+\beta x)}{1+\exp(\alpha+\beta x)}$, but your specification indicates you want it without the "$\exp$": 
$E(\frac{Y}{M}) = \frac{\alpha+\beta x}{1+\alpha+\beta x}$
I don't think you can do that with a GLM with the response in that form.
However, I do see two possibilities: 


*

*Use nonlinear least squares on $E(\frac{Y}{M}) = \frac{\alpha+\beta x}{1+\alpha+\beta x}$

*Let $m=\mu_Y(x)=E(Y|x)$. Since you can transform the relationship $m = \frac{\alpha+\beta x}{1+\alpha+\beta x}$ to $\frac{m}{1-m} = \alpha+\beta x$, you might consider whether a model of the form $E[\frac{y/M}{1\:-\:y/M}]=E[\frac{y}{M-y}] = \alpha+\beta x$ could be fitted as a linear regression  -- or indeed as a (possibly quasi-) GLM, which could allow you to choose a variance function.
Whether you choose 1 or 2 or something else depends on how you think the error term behaves - you need to consider the question "how does the error term enter the relationship between $y$ and $x$ and how is its variance related either to $x$ or to the mean?". Further, if the you're interested in unbiased estimation on the scale of $y$ you may need to correct the predictions of the backtransformed model, say via a Taylor expansion.
