Solved – How are Multiple Linear Regression Lines Plotted

Linear regression fits the best line across a scatter plot. If we have several predictor variables, is there then a best fit line for every predictor variable? Meaning that there'll be a best fit line for y and x1, and y and x2 — and it becomes a 3D scatter plot.

And how about multioutput regression lines where there are several target variables as well — do we then have a line for y1 and x1, y2 and x1, y1 and x2, y2 and x2?

A multiple regression line is a line in a p+1 dimensional space, where p is the number of predictors (or independent variables). With p > 2 this will be hard to visualize, but we statisticians don't let that stop us.

You could make a line relating each predictor to the DV, controlling for the other predictors, but you have to decide what levels of all the other predictors to use. If there are no interactions, then choosing different levels will shift the lines, but not change their slope, but if there are interactions, then it will change the slopes as well. So, with no interactions you might choose the mean of each continuous predictor and the most common value of other predictors.

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