# 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?

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.

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