*Bounty: 50*

*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.

Thanks in advance.