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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?

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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:

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

  2. 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.

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  • $\begingroup$ I suppose if I used nonlinear least squares with a Gaussian assumption on the error (as in y/m = f(xb) + e, e ~ N(0, v^2), I couldn't do inference on b, except perhaps using bootstrap. $\endgroup$ – Count Zero Aug 5 '15 at 12:46
  • $\begingroup$ Certainly you can do inference on $b$, at least approximately. $\endgroup$ – Glen_b -Reinstate Monica Aug 5 '15 at 12:50
  • $\begingroup$ I should expand on that comment, I suppose; there are approximate t- and F- tests that are often used in nonlinear least squares. You can of course use bootstrapping and in large samples that should work well. In small to moderate samples I've come to be quite cautious about using the bootstrap without checking its coverage properties in situations similar to the one I'm in. Sometimes in simulations the actual coverage proportion across many simulations has been a good deal lower than the nominal coverage probability for the interval. $\endgroup$ – Glen_b -Reinstate Monica Aug 5 '15 at 21:38

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