There may be more than one way to express a particular deterministic model (i.e., functional relationship).

Deterministically identical models may be parameterized in alternate formulations depending upon which parameters are of direct interest to the investigator.

Here is an example of three parameterizations of the deterministic model for simple linear regression with two parameters.

The typical formulation is:

where the intercept (a) and slope (b) are the parameters of interest.

One alternate formulation is:

where the investigator may be interested in directly estimating the mean of Y and the slope (b). Note that the mean of X is determined entirely by the data.

A third alternate formulation is:

where the investigator may be interested in estimating two values of Y (y_a and y_b), at two predetermined values of X (x_a and x_b).

The advantage of each parameterization in the above examples is direct estimation of the corresponding parameter values and their SEs.

There is no requirement to use sometimes messy approximation techniques for calculating the means and variances of indirectly estimated parameters of interest.