Given the following model as an example:
$$Y=\beta_0+\beta_A\cdot A+\beta_B\cdot B+\beta_{AB}\cdot A \cdot B+\epsilon$$
In alternative notation:
$$Y\sim A + B + A: B$$
The main question:
When permuting entries of variable $A$ to test its coefficient ($\beta_A$) in a model, should an interaction term that includes it such as $B\cdot A$ be recomputed as well?
Secondary question:
And what about testing the $B\cdot A$ interaction term coefficient ($\beta_{AB}$)? Are its permutations computed regardless of the variables $A$ and $B$?
A bit of context:
I want to perform a test on the coefficients of a model (it's a canonical correlation analysis, but the question is applicable to any linear model including interactions).
I'm trying my hands with permutation tests. While it's fairly straightforward to test the canonical correlation itself, how to do the same with the variable scores, or coefficients, is a bit unclear to me when including an interaction term.
I've read How to test an interaction effect with a non-parametric test (e.g. a permutation test)?, but my question is much more practical.
Best Answer
As I'm just starting with permutation tests, I though a question was a good idea. Indeed, thanks to comments by @Glen_b and @user43849, I perceived many misunderstandings and inconsistencies of the theory from my part. For one, I was thinking about testing the magnitude of the coefficient instead of the effect, which is what actual interest.
So, as I'm learning, an actual answer to be criticized sounded just as good.
To answer this question and appoint a permutation strategy that complies with my requirements, I resorted to Anderson MJ, Legendre P. "An empirical comparison of permutation methods for tests of partial regression coefficients in a linear model." Journal of statistical computation and simulation 62.3 (1999): 271-303.
There, the authors do empirical comparisons between four permutational strategies, in addition to normal theory $t$-statistic tests:
Here I'll quote the description given to the strategy put forward by Manly. Given a model $Y=\mu+\beta_{1\cdot2}X+\beta_{2\cdot1}Z+\epsilon$:
So this strategy conserves the covariance of the independent variables X and Z. Other methods focus on the testing of partial coefficients in isolation, and these are discussed in the text. Also, possible drawbacks of the strategy of permutation of raw data are given both in the text and in the literature.