# Significance of regression coefficients in two different linear models

Suppose that I have data $$\left\{ (x_i, y_i, z_i ) : i=1, 2, \dots, N\right\}$$. I have fitted two linear models: $$\left[\begin{matrix} z_1\\ z_2\\ \vdots\\ z_N \end{matrix}\right]=\left[\begin{matrix} 1&x_1\\ 1&x_2\\ \vdots&\vdots\\ 1&x_N \end{matrix}\right]\left[\begin{matrix} a_1\\ b_1 \end{matrix}\right]$$ and $$\left[\begin{matrix} z_1\\ z_2\\ \vdots\\ z_N \end{matrix}\right]=\left[\begin{matrix} 1&x_1&y_1\\ 1&x_2&y_2\\ \vdots&\vdots&\vdots\\ 1&x_N&y_N \end{matrix}\right]\left[\begin{matrix} a_2\\ b_2\\ c_2 \end{matrix}\right].$$ That is, the first model uses $$x_i$$ values and the second one uses both $$x_i$$ and $$y_i$$ values to explain $$z_i$$.

Now I am considering whether the estimated coefficients $$b_1$$ and $$b_2$$ ('slopes') are statistically significant (that is, are the $$x_i$$:s significant to the model). First of all, I am not sure how to get started with that that problem. Secondly, is it possible that $$b_1$$ would be significant and $$b_2$$ would not, or vice versa?

• Sure they can, if you search our site, you'll find many examples of people asking about cases when exactly this happened.
– Tim
Commented Nov 2, 2022 at 19:00
• It seems like this is a really trivial question. Testing the significance of a regression coefficient is covered in most elementary stats books - i.e. Kutner, Nachtsheim, Wasserman, Neter "Applied Linear Regression Models". Commented Nov 2, 2022 at 20:15

If the columns of the design matrix $$X_{N\times(p+1)}$$ are linearly independent, $$b_{1}$$ and $$b_{2}$$ coefficients indicate the amount of change in the responses $$z_{i}$$ while other predictors are held fixed. In the case where there is only a single predictor, this amount of change can be interpreted as the slope of the 2D regression line. In the case where there are 2 predictors, $$a_{2}$$ and $$b_{2}$$ in your coefficient vector together represent the 3D regression plane. Thus, $$b_{1}$$ and $$b_{2}$$ are surely significant and both help to explain the response vector $$z$$.
• what is $p$ ? Here there are only 2 or three covariates... Besides, your conclusion "b1 and b2 are surely significant" seems to disagree with @Tim's comment under the post. Can you elaborate on this? Commented Nov 2, 2022 at 20:25
• $p$ is the number of columns aside from the 1 vector. I am not sure what he means by statistically "significant" but changing the values of $b_{1}$ and $b_{2}$ will change the response, indeed. Commented Nov 2, 2022 at 20:28