Interpretation of predicted values in model comparisons (regression) I am interested in understanding how to interpret the difference in predicted values across two partially related linear multiple regression models.
Let's assume that the first model is (in Wilkinson notation) " $ y_{full} \approx 1 + x_1 + x_2 + x_3 $ ", so that the variable $y$ is modeled using the "full set" of regressors available. We call this the Full model. ("1" is the intercept)
Let's assume that the second model is " $ y_{partial} \approx 1 + x_2 + x_3 $ " , so that the variable $y$ is modeled using the "full set" of regressors available, excluding $x_1$. We call this the Partial model.
Now, let's pretend that we have some new values of the independent variables ( $ \tilde x_1, \tilde x_2, \tilde x_3 $ ) and use these values to predict y. Of course, all the independent variables are used in the full model, while the partial model only uses the last two, exluding $x_1$. We would obtain $ \hat y_{Full}$ and $ \hat y_{Partial} $, that is, the two predicted values (or the two predicted vectors).
I am interested in the meaning of the difference between the two predicted values ($\Delta$). Mathematically, given $\beta$ as the slopes and $\alpha$ as the intercepts, such difference can be formalised as:
$$
\Delta_{Full,Partial} = \hat y_{Full} - \hat y_{Partial}
= \tilde x_{2} ( \beta_{2,Full} -  \beta_{2,Partial} ) +  \tilde x_{3} ( \beta_{3,Full} -  \beta_{3,Partial} ) +  \tilde x_{1} (\beta_{1,Full}) + \alpha_{Full} - \alpha_{Partial}
$$
(note that $\alpha$ = intercept)
However, I would like to understand the practical meaning of such difference. For example, it is good to say that $\Delta$ indicates the "contribution of $x_1$ to $y$ given the existence/effect of $x_2$ and $x_3$"?
I hope that the question is clear. I am very curious about how to interpret such difference. If you also have example of applications in experimental studies, I would like to read them.
Thanks in advance.

I’d call $\Delta$ “the change from adding $x_1$ as a predictor”.
As an example, suppose $y$ is a final exam grade, $x_1$ is a midterm grade, $x_2$ is a participation grade from the first half of a class, and $x_3$ is whether the student had taken a related class beforehand, and the regressions come from last semester’s data. Then with new data on this semester’s students, $\Delta$ tells you the change in predictions for the final given the performance on the midterm.
Calling $\Delta$ “the contribution of the midterm grade to the prediction of the final grade” is awkward. You can make the phrase “the contribution of $x_1$” precise with a formula, and there may be areas where that phrase has a well-established meaning, but in general misunderstandings are so likely that I’d avoid talk of statistical “contributions” entirely.
