Solved – Understanding ‘predictor’ residual plots in multiple regression

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Suppose I have two standardized (i.e., z-scored) predictors Age at Marriage and Rate of Marriage, and a dependent variable Divorce and I fit a linear regression model to predict Divorce from the two predictors.

Predictor residual plot

First, we predict Age at Marriage (now as dep. variable) from Rate of Marriage, obtain the residuals from this prediction. Second, we plot these residuals against Divorce .

Picture below is the resultant Predictor residual plot.

Updated Question

This technique can be extended to several-predictor situations. Imagine, for example, we had $3$ predictors predicting $Divorce$. Then, we had to pick one of the predictors, and regress that predictor on the other $2$ predictors, and obtain the residuals and then plot the residual against Divorce.

My question is what would be the interpretation of such plots in 3-or more predictor situations?

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Best Answer

When you regress Age of marriage on Rate of marriage, each of your residuals is computed as:

residual = Age of marriage – (Estimated) Expected Age of marriage

where (Estimated) Expected Age of marriage = b0 + b1xRate of marriage" and b0 and b1 are the estimated intercept and slope of the simple linear regression of *Age of marriage" on Rate of marriage.

When is a residual positive? When:

Age of marriage – (Estimated) Expected Age of marriage > 0

or, equivalently, when

Age of marriage > (Estimated) Expected Age of marriage. In other words, when the *Age of marriage" is older than the (estimated) expected *Age of marriage".

When is a residual negative? When:

Age of marriage – (Estimated) Expected Age of marriage < 0

or, equivalently, when

Age of marriage < (Estimated) Expected Age of marriage. In other words, when the *Age of marriage" is younger than the (estimated) expected *Age of marriage".

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