Choosing one parameterization over another

Question

Suppose I have a random response Y and a fixed predictor X. My model is

\begin{align*} Y &= \frac{aX}{b + aX} \end{align*}

where $X, Y, a, b > 0$. I have a sample $Y_1, ..., Y_n$ and n is very large. In my problem domain most people fit a nonlinear least squares model on

\begin{align*} Y &= \frac{\beta X}{1 + \beta X} \end{align*} where $\beta = \frac{a}{b}$. This is multiplying the numerator and denominator by $\frac{1}{b}$, but it reduces the parameter space from 2 dimensions to 1.

Is there a reason why this parameterization might be preferred if there are enough degrees of freedom to estimate two parameters instead of one?

Answer 1

You don't really have two parameters that can be identified separately and uniquely in your first formula, regardless of the number of degrees of freedom. Say that you had particular values for $a$ and $b$; the formula would give the same prediction if you used any common multiple $m$ of both parameter values, of the form $m a$ and $mb$. So all you can get for a unique solution is the ratio of the two parameters in your formula, which in your second formula appears as $\beta$.