# #StackBounty: #regression #multiple-regression #predictive-models #model Interpretation of predicted values in model comparisons (regre…

### Bounty: 50

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) $$’ y_{full} = 1 + x_1 + x_2 + x_3 ‘$$, 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 $$’ y_{partial} = 1 + x_2 + x_3 ‘$$, 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:

$$Delta_{Full,Partial} = hat y_{Full} – hat y_{Partial} = tilde x_{2} ( beta_{2,Full} – beta_{2,Partial} ) + tilde x_{3} ( beta_{3,Full} – beta_{3,Partial} ) + tilde x_{1} (beta_{1,Full}) + alpha_{Full} – alpha_{Partial}$$

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.