Is there a term for dividing out a parameter to make model identifiable?

Question

Suppose I have a nonlinear model of the form: \begin{align}EY|X = \frac{aX}{aX+b}\end{align} where $a, X, b > 0$. I reparameterize the model as \begin{align} EY|X =\frac{\beta X}{1 + \beta X} \end{align} where $\beta = \frac{a}{b}$. $\beta$ is identifiable, so I can estimate it with nonlinear regression or some parametric model.

My question is about terminology. I am working on a complex system where each variable is a similar function of other variables. In each case I have to identify some term to divide by so I have identifiable parameters. For example, in another case I have \begin{align}EY|X_1,X_2 &= \frac{aX_1}{aX_1 +bX_2}\\ &=\frac{\beta X_1}{\beta X_1 + X_2} \end{align} where $\beta = \frac{a}{b}$. Is there a term that desribes the role b plays here, as a parameter that has to be divided out to make the model identifiable? Something like "normalizing constant" except that it is not assumed known and its role is to help identify rather than normalize.

Answer 1

This has a similar flavor to nondimensionalization in physics/engineering. There the "non-identifiability" of the separate parameters is commonly leveraged in design of experiments. This has given rise to many well-known "reduced" parameters in science.

Answer 2

A related kind of parameter rewriting strategy, used as an aid to optimization rather than than identification, is what the economists call concentrating and statisticians calling profiling. See this example.

Answer 3

The process is reparameterization, where, in the example, there is one supernumerary parameter that makes the system unnecessarily underdetermined. An alternative reparameterization is to $\frac{X_1}{X_1-\alpha X_2}$. The set $\{a,b\}$ has one more parameter than needed, but it is arbitrary to say which one, as it is the set that is overdefined, not $a$ or $b$ taken separately. Thus, $b$ by itself is only one of the two parameters that are problematic, and would likely not have a stand alone name. It is the set that is redundant, and would result in regression instability in a spuriously underdefined system such that reparameterization is not optional, it is indicated.