[GSCA] Harmonization of Weight Scaling Procedures

[emstruong]

My understanding is that the original code for GSCA-M in cSEM scales the estimated weights, such that the variance of the construct scores is equal to $1$.

Importantly, this procedure is only relevant to GSCA models that are explicitly fitted to common factors (meaning that this applies to both GSCA-M and IGSCA).

The original code is linked here:

cSEM/R/00_foreman.R

Lines 115 to 129 in c27461a

	if(.disattenuate & all(csem_model$construct_type == "Common factor")) {
	W <- calculateWeightsGSCAm(
	.X = X,
	.csem_model = csem_model,
	.conv_criterion = .conv_criterion,
	.iter_max = .iter_max,
	.tolerance = .tolerance,
	.starting_values = .starting_values
	)
	# Weights need to be scaled s.t. the composite build using .X has
	# variance of one. Note that scaled GSCAm weights are identical
	# to PLS ModeA weights.
	W$W <- scaleWeights(S, W$W)

I'm wondering why this is---or whether it is really---necessary given that both the indicators and the construct scores should be standardized and hence have a variance of $1$, anyways. Therefore, it's not clear to me why scaleWeights() should be necessary. In fact, I wonder if it might be a mistake to try to scale the weights.

In the past, the scaled weights for GSCA-M would then be used to compute the construct scores here:

cSEM/R/00_foreman.R

Lines 197 to 198 in c27461a

	## Calculate proxies/scores
	H <- X %*% t(Weights)

But AFAICT, this is probably not compatible with GSCA-M or IGSCA's assumptions, because in GSCA-M, the construct scores $\mathbf \eta$ is equivalent to $\mathbf \eta = (\mathbf{X - UD}) \mathbf{W}$ and not $\mathbf \eta = (\mathbf{X}) \mathbf{W}$.

...So then perhaps this scaleWeights() procedure should not be applied to GSCA type models at all?