# How to describe or visualize a multiple linear regression model

I'm trying to fit a multiple linear regression model to my data with couple of input parameters, say 3.

\begin{align} F(x) &= Ax_1 + Bx_2 + Cx_3 + d \tag{i} \\ &\text{or} \\ F(x) &= (A\ B\ C)^T (x_1\ x_2\ x_3) + d \tag{ii} \end{align}

How do I explain and visualize this model? I could think of the following options:

1. Mention the regression equation as described in $(i)$ (coefficients, constant) along with standard deviation and then a residual error plot to show the accuracy of this model.

2. Pairwise plots of independent and dependent variables, like this:

3. Once the coefficients are known, can the data points used to obtain equation $(i)$ be condensed to their real values. That is, the training data have new values, in the form $x$ instead of $x_1$, $x_2$, $x_3$, $\ldots$ where each of independent variable is multiplied by its respective coefficient. Then this simplified version can be visually shown as a simple regression as this:

I'm confused on this in spite of going through appropriate material on this topic. Can someone please explain to me how to "explain" a multiple linear regression model and how to visually show it.

• What is the purpose of your document and who are the audiences? I'd start from getting similar articles and look for some examples on how they are done in your own field. I am more familiar with biomedical literature and most of the time, we just use a table. Illustrations are more often seen when the authors try to explain an interaction. – Penguin_Knight Mar 12 '14 at 15:01
• @Penguin_Knight, this is in computer science domain, however I think this is a generic rather than restricted to a particular domain. Please correct me if I'm wrong. – kris Mar 12 '14 at 15:18
• Hmm... though question. I'd say the only generic part, for me, is don't show more than you should, and make sure the components to be emphasized really get emphasized. Even in just my field, I have seen all three options. 1) tabulating the results is the most common, followed by 3), but mostly the form of plotting predicted outcome, and then 2). But for 2), I'd use what @gregory_britten suggested: use adjusted X instead of each individual X. – Penguin_Knight Mar 12 '14 at 15:34
• use distribution plot.... look at the distribution of the fitted values that result from the model and compare it to the distribution of the actual values. – owais qureshi Jan 30 at 4:47
• I know this is from years ago, but if you revisit here, could you post the data? Then people would have something to work with to show different possibilities. – gung - Reinstate Monica Jan 30 at 4:57

My favorite way of showing the results of a basic multiple linear regression is to first fit the model to normalized (continuous) variables. That is, z-transform the $X$s by subtracting the mean and dividing by the standard deviation, then fit the model and estimate the parameters. When the variables are transformed in this way, the estimated coefficients are 'standardized' to have unit $\Delta Y/\Delta sd(X)$. In this way, the distance the coefficients are from zero ranks their relative 'importance' and their CI gives the precision. I think it sums up the relationships rather well and offers a lot more information than the coefficients and p.values on their natural and often disparate numerical scales. An example is below:

EDIT: Another possibility is to use an 'added variable plot' (i.e. plot the partial regressions). This gives another perspective in that it shows the bivariate relations between $Y$ and $X_i$ AFTER THE OTHER VARIABLES ARE ACCOUNTED FOR. For example, the partial regressions of $Y \sim X_1 + X_2 + X_3$ would give bivariate relations between $X_i$ against the the residuals of $Y$ after regressing against the other two terms. You would go on to do this for each variable. Function avPlots() from library car gives these plots from a fitted lm object. An example is below:

• Thanks @gregory_britten for this information. The problem I have at hand has 8 independent variables. Do you think the 'added variable plots' would be reasonable for large number of input variables? – kris Mar 12 '14 at 15:16
• In line with the idea of the first plot, if working in R, I suggest looking at the RMS package which makes all of this easy. The nice thing is that one can ask for meaningful step changes in the covariance, thus avoiding the need to standardize. – Thomas Speidel Mar 12 '14 at 15:22
• @suzanne Yes definitely. The added variable plot gives you two dimensional perspectives for any number of variables. It may be particularly revealing in higher dimensions. One often finds revealing patterns in the residuals which were not at all obvious in the observed Y. – gregory_britten Mar 12 '14 at 17:21
• I don't quite understand the X1|X2&X3 notation in this context.I know how it is used in regards to probabilities, but I can't quite understand what it is saying here – Casebash Mar 13 '14 at 2:25
• @Casebash It is the partial regression on X1, given X2 and X3 are in the model – gregory_britten Mar 13 '14 at 13:27

Since they all have to do with explaining the contributors for cirrhosis, have you tried doing a bubble/circle chart and use color to indicate the different regressors and circle radius to indicate relative impact upon cirrhosis?

I'm referring here to a Google chart type that looks like this:

And on an unrelated note, unless I'm reading your plots wrong, I think you have some redundant regressors in there. Wine is already a liquor so if those two are separate regressors it doesn't make sense to keep both of them, if your goal is to explain the incidence of cirrhosis.