[QUESTION] PROTEST implementation

[valentynbez]

In original Jackson 1995 the procedure for permutational testing is:

• permute
• gather $M^2$ values
• select a fraction that is < original $M^2$

In the vegan implementation procedure goes as follows:

• permute
• gather $M^2$ values
• select a fraction that is >= original $\sqrt{1 - M^2}$

vegan/R/protest.R

Lines 19 to 38 in 1c98b98

	sol$t0 <- sqrt(1 - sol$ss)
	N <- nrow(X)
	## Permutations: We only need the goodness of fit statistic from
	## Procrustes analysis, and therefore we only have the necessary
	## function here. This avoids a lot of overhead of calling
	## procrustes() for each permutation. The following gives the
	## Procrustes r directly.
	procr <- function(X, Y) sum(svd(crossprod(X, Y), nv=0, nu=0)$d)
	permutations <- getPermuteMatrix(permutations, N)
	if (ncol(permutations) != N)
	stop(gettextf("'permutations' have %d columns, but data have %d observations",
	ncol(permutations), N))
	np <- nrow(permutations)
	perm <- sapply(seq_len(np),
	function(i, ...) procr(X, Y[permutations[i,],]))
	Pval <- (sum(perm >= sol$t0 - EPS) + 1)/(np + 1)

What is the reason for choosing such an approach? (I am not the strongest in maths, it might be that I'm missing simple logic somewhere).

Also, as far as I understand, $m_{12}^2$ refers to the sum of squares or goodness-of-fit statistic from the publication of Peres-Neto & Jackson (2001). The same Jackson in 1995 refers to it as $M^2$.
Do you think the notation could be unified to avoid confusion?
I compared implementations in QIIME2 and vegan both, you can take a look at it here: https://forum.qiime2.org/t/procrustes-and-protest-analysis/20123/4