Intuition behind Partial Residual Plots

Question

I've been reading how univariate analysis in data with a lot of variables can be misleading due to "Simpson's paradox". I found the explanation of this phenomenon pretty fascinating but easy to understand. What I am having a hard time wrapping my head around is using "partial residual plots" to combat it. Wikipedia says that the plot should look like

$$\text{Residuals} + \beta_iX_i \text{ vs. } X_i.$$

I've also seen residual plots simply defined as

$$\text{Residuals } \text{ vs. } X_i.$$

In this case, I can see how this plot would show non-linear relationships. But for neither plot can I understand how it would help us see correlation any better than in normal univariate analysis (like Pearson's r).

What is an an intuitive explanation as to why this plot is better than looking at univariate correlation between independent and dependent variables?

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Edit: To further add to my confusion I have now seen the title "Residual Plots" used for the following

1. Residuals vs Predictions
2. Residuals vs Variable
3. Residuals + Variable*(associated coefficient) vs Variable

All of these are advertised as having the same purpose: identify linear or non-linear relationships between independent variables and dependent variables in higher dimension sample sets.

Answer 1

While it is mentioned in a number of regression texts, the plot you have mentioned here does not seem particularly useful to me. A far better alternative is the added variable plot, which correctly represents the relationship between an individual explanatory variable and the response variable conditional on other explanatory variables. For the explanatory variable $x_k$ the plot shows the following variables on the vertical and horizontal axes respectively:

$$\begin{matrix} Y_{\bullet [k]} & & & \text{Residuals from regressing } Y \text{ against } \mathbf{x}_{-k}, \\[6pt] X_{k \bullet [k]} & & & \text{Residuals from regressing } x_k \text{ against } \mathbf{x}_{-k}. \\[6pt] \end{matrix}$$

This latter plot has several useful properties. The line-of-best-fit in the plot will match the estimated regression coefficient for that explanatory variable, and the residuals match the residuals of the overall regression. The plot isolates the relationship between $Y$ and $x_k$ conditional on the other explanatory variables. It allows you to easily diagnose the relationship between the explanatory variable and response, and thereby diagnose errors in the model assumptions (e.g., patterns that vary from the assumed model form).