# How to visualize a fitted multiple regression model?

I am currently writing a paper with several multiple regression analyses. While visualizing univariate linear regression is easy via scatter plots, I was wondering whether there is any good way to visualize multiple linear regressions?

I am currently just plotting scatter plots like dependent variable vs. 1st independent variable, then vs. 2nd independent variable, etc. I would really appreciate any suggestions.

• One possibility: Added variable plots – Glen_b -Reinstate Monica Oct 20 '13 at 21:49
• Of possible interest as well: Predicted by residual plot in R. – chl Oct 20 '13 at 22:00
• See the effects package in R – Peter Flom Oct 20 '13 at 22:09
• I guess I should have asked for this clarification first: do you mean linear regression with multiple predictors (x's, IVs) - that is multiple regression, or do you mean linear regression with multiple responses (y's, DVs) - that is, multivariate regression? – Glen_b -Reinstate Monica Oct 21 '13 at 3:07

There is nothing wrong with your current strategy. If you have a multiple regression model with only two explanatory variables then you could try to make a 3D-ish plot that displays the predicted regression plane, but most software don't make this easy to do. Another possibility is to use a coplot (see also: coplot in R or this pdf), which can represent three or even four variables, but many people don't know how to read them. Essentially however, if you don't have any interactions, then the predicted marginal relationship between $x_j$ and $y$ will be the same as predicted conditional relationship (plus or minus some vertical shift) at any specific level of your other $x$ variables. Thus, you can simply set all other $x$ variables at their means and find the predicted line $\hat y = \hat\beta_0 + \cdots + \hat\beta_j x_j + \cdots + \hat\beta_p \bar x_p$ and plot that line on a scatterplot of $(x_j, y)$ pairs. Moreover, you will end up with $p$ such plots, although you might not include some of them if you think they are not important. (For example, it is common to have a multiple regression model with a single variable of interest and some control variables, and only present the first such plot).

On the other hand, if you do have interactions, then you should figure out which of the interacting variables you are most interested in and plot the predicted relationship between that variable and the response variable, but with several lines on the same plot. The other interacting variable is set to different levels for each of those lines. Typical values would be the mean and $\pm$ 1 SD of the interacting variable. To make this clearer, imagine you have only two variables, $x_1$ and $x_2$, and you have an interaction between them, and that $x_1$ is the focus of your study, then you might make a single plot with these three lines:
\begin{align} \hat y &= \hat\beta_0 + \hat\beta_1 x_1 + \hat\beta_2 (\bar x_2 - s_{x_2}) + \hat\beta_3 x_1(\bar x_2 - s_{x_2}) \\ \hat y &= \hat\beta_0 + \hat\beta_1 x_1 + \hat\beta_2 \bar x_2 \quad\quad\quad\ + \hat\beta_3 x_1\bar x_2 \\ \hat y &= \hat\beta_0 + \hat\beta_1 x_1 + \hat\beta_2 (\bar x_2 + s_{x_2}) + \hat\beta_3 x_1(\bar x_2 + s_{x_2}) \end{align}

Here is a web-based, interactive tool for plotting regression results in three dimensions.

This 3-D plot works with one dependent variable and two explanatory variables. You can also set the intercept to zero (i.e., remove the intercept from the regression equation).

The graphics require a WebGL-capable browser. The most recent versions of all major desktop browsers support WebGL. • Site is down now -- I get a GoDaddy landing page – spinup Oct 15 '19 at 18:29

To visualize the model, rather than the data, JMP uses an interactive "profiler" plot. Here's a static view. And here's a link to a dynamic view.

It's similar to your scatter plot idea and can be combined with it. The idea is that each frame shows a slice of the model for the corresponding X and Y variables with the other X variables held constant at their indicated values. In the interactive version, the X values can be changed by dragging the red vertical lines.

Disclosure: I'm a JMP developer, so don't take this as an unbiased endorsement.

• Is not crucial that you plot the residuals of the dependent variable with the residuals of the predictors? I thought it should be, as those represent the real relationships between your variables, but that seems rarely reccommended. – Agus Camacho Sep 1 '17 at 15:19
• @AgusCamacho, if you are still interested in that, you should ask a new question. – gung - Reinstate Monica Feb 6 '19 at 1:31