Partial regression plots – also called added variable plots, among other things – are a type of diagnostic plot for multivariate linear regression models. More specifically, they attempt to show the effect of adding a new variable to an existing model by controlling for the effect of the predictors already in use. They’re useful for spotting points with high influence or leverage, as well as seeing the partial correlation between the response and the new predictor.
The t-test is a common, reliable way to check for differences between two samples. When dealing with multivariate data, one can simply run t-tests on each variable and see if there are differences. This could lead to scenarios where individual t-tests suggest that there is no difference, although looking at all variables jointly will show a difference. When a multivariate test is preferred, the obvious choice is the Hotelling’s \(T^2\) test.
Hotelling’s test has the same overall flexibility that the t-test does, in that it can also work on paired data, or even a single dataset, though this example will only cover the two-sample case.
I recently had to go through some matrix operations in R and then write up the results in LaTeX. Formatting the R output to get it into a form for LaTeX isn’t particularly hard, but it’s tedious and it has a regular structure, so it seemed like it would be easy to code it up. So I decided to try it for R, Python, and Julia.