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 usually don't look at (nested) model comparison this way, i.e. in the perspective of the difference of predictions. When I have multiple predictors the question in general is whether to add a predictor in the regression or not. In this I follow Gelman's advise (for example, Gelman et al. Data Analysis using Multilevel Regression) to include predictors if they make sense and if they contribute some information. That is if the weight of a predictor is practically zero you may exclude it but if it makes sense to have it in the regression you can leave it and this will change your interpretation of the model.
To address more directly the question I prefer to interpret the difference of coefficients (or actually even better the distribution of the coefficient with and without the new predictor) more than the difference of predictions itself. For example if adding $x_1$ makes the distribution of coefficient $\beta_{2, Full}$ centred in zero with a small variance while the distribution $\beta_{2, Partial}$ has most density on the positive side I would interpret that the effect of $x_2$ indicated by the partial model is not relevant when you take into account $x_1$.
Hope this helps.
