[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

[jarioksa]

vegan::protest gathers and compares Procrustes correlation r instead of symmetric Procrustes sum of squares m². These are inversely related as $r = \sqrt{1-m^2}$: when m² goes down, r goes up and therefore the direction of inequality was changed in comparisons. However, same results will be obtained with both.

There is no special reason for this approach: there are several approaches that give the same results, and I just picked one of equal alternatives.

As to notation: vegan mentions $m_{12}^2$ in help page, but does not use that symbol in printed output or calculations. I don't know why Jackson opted for using uppercase M in 1995 and lowercase m in 2001 (with Peres-Neto), but I have no way to retroactively unify his notation. Vegan bases protest on the 2001 paper and therefore uses that notation. The Procrustes rotation (function procrustes) was based on Mardia et al (1979) and the basic notation is taken from that book; protest was added later, and therefore it also basically uses the same notation as procrustes.

[valentynbez]

Thanks for the clarification!
Another question I have is that vegan limits the number of available permutations by the number of samples. This helps to avoid identical permutations in both groups.
Does this have any implications on a p-value? I assume if Nperm > Nobs p-value becomes unreliable, as we obtain repeated samples, is that right?

[jarioksa]

What do you mean with this? Vegan does not limit the number of permutations, except in the case when the user requests a higher number of permutations than the maximum number of possible permutations (and this limitation is made in the permute package, not in vegan). In that case permute switches to complete enumeration of all possible permutations and to so-called exact test.