Let's say I have some machine that measures some real-world phenomenon and outputs some raw data. The raw data need some (elementary) processing before it can be converted to a quantitative output, say $Y_1$.
It is thought that $Y_1$ is a linear response to some specific real-world quantity, say $X_1$, so that:
$Y_1 = a_1 \cdot X_1 + b_1 \tag 1$
As a side note, I have exact values for $X_1$, which can be measured directly, but the direct method is normally impractical.
I would like to propose that the raw data measured by the machine is more meaningful to process in a different way to produce a new readout, say $Y_2$, so that:
$Y_2 = a_2 \cdot X_1 + b_2 \tag 2$
Furthermore, I want to show, that when another variable, representing a measure of noise, is added, say $X_2$, the model is enhanced:
$Y_2 = a_2 \cdot X_1 + X_2 + b_2 \tag 3$
Ideally I want to test the superiority of model 3 against model 1 - is this possible?
If not, can I at least test 2 against 1?