What does Tukey Test's p-value means in residual plot analysis? After having called the residual plots command in order to graphically analyse residuals, I obtain also a written output where I can find the p-value and Test stat associated to each independent variable, each predictor, and a final voice (after all the variables) which is "Tukey test" associated with a Test stat and a p-value. What does it stands for? What information can I draw from that voice?
 A: In the context of residual plots, Tukey's test is checking for curvature as a function of the mean.  It's adding a quadratic term to the model and the p-value is the probability that the coefficient of the quadratic term is 0.  Some documentation (not much detail, though) is here:
http://rstudio-pubs-static.s3.amazonaws.com/6214_e745e9c39bdc42b095964ca64c3ae05a.html
A: The table output of the residualPlots() function from the car package shows the results of a "lack-of-fit test" and/or** "Tukey’s test for nonadditivity" (Fox & Weisberg, 2018, p. 289). 
It adds a quadratic term to the model for each numeric independent variable in the regression model. Here's how I interpret the tests:


*

*Null hypothesis: Coefficient for quadratic term is 0. 

*Alternate hypothesis: Coefficient for quadratic term is not 0. 


The p-value should be the probability that the coefficient of the quadratic term is NOT 0. 
Interpretation:


*

*If p-value is high, we fail to reject null, meaning no evidence for lack of fit, meaning coefficient of quadratic term might be 0. 

*If p-value is low, reject null hypothesis, meaning the test finds evidence that the quadratic term might not be 0, meaning there is strong evidence of non-linearity in the model and it needs to be re-specified before we can trust the results of the model.


For the Tukey test at the very bottom of the table:


*

*Null hypothesis: relationship between residuals and fitted values is 0.

*Alternate hypothesis: relationship between residuals and fitted values is NOT 0.


Interpretation:


*

*If p-value is high, we fail to reject the null, meaning we didn't find evidence of a relationship between residuals and fitted values. 

*If p-value is low, we did find evidence of relationship between residuals and fitted values, and we need to re-specify the model before we can trust the results of the model. 


Everything above, I learned and hopefully interpreted correctly from Chapter 6, section 6.2.1 "PLOTTING RESIDUALS" (specifically pp. 288-290) of:
Fox, J., & Weisberg, S. (2018). An R companion to applied regression. Sage publications.
The relevant chapter is currently available here: https://www.sagepub.com/sites/default/files/upm-binaries/38503_Chapter6.pdf
** I still don't understand if the tests in every row of the output are results of Tukey’s test for nonadditivity or just the one at the bottom.
