One can take different approaches to fully study the effect of staining. One
approach would be statistical (or combinatorial) as suggested by Wayde
although he also pointed out that if there is more than 3 or 4 variables, the
amount of tests would not be practically feasible. The solution (or at least
attempt to minimize the number of tests) would be the Taguchi method as
described by Robert Chapman in one of the two darkroom magazines (I can't
never remember which is which) although that might get to theoretical or
mathematical for some.
Another approach is to study the physical property of the system and come up
with a physical model that explain things better. This is usually my preferred
approach, but the difficulty of this approach is, of course, sometimes you
cannot come up with a good explanation or model before you fully study the
effect, so there is a catch 22 situation.
But applying this approach to pigment staining, I would like to point out why
the gum-pigment ratio *without* the dicrhomate is meaningless. The thing is,
the pigment staining depends on how much pigment is *in touch* with the
surface of the paper. Although the mix might look dark, only one thin layer of
pigment physically touch the paper because the pigment is suspended in gum
because of the thickness of gum. Suppose you have 0.5 g of pigment in 5 ml of
gum, the pigment is well suspended there. Now lets say you add a very small
amount of dichromate (say 2 drops for theoretical discussion), now the mix is
*very* slightly diluted, so you have less concentration of pigment in the gum
mix, so less pigment touch the paper, so theoretically you should have less
pigment staining.
Logic would say then, that the more you dilute, the less staining it would be,
but I am sure you have read, heard, or experienced that this is not the case.
The reason is when you add dichromate, you also dilute the thickness or
viscosity of the gum. At some point the gum becomes not viscous enough to hold
the pigment in suspention (think about trying to hold sand in suspension in
water), so the pigment starts to settle to the bottom, so more pigment touch
the surface of the paper, and you start to get *more* staining. You cannot see
this with your eye because they still look the same, and watercolorists don't
have to worry about this because they don't have to deal with lifting the
pigment later as gummists do.
So with just a small amount of dichromate added, you might get less staining,
but with more, you might get more staining. I do have some tests that shows
evidence of this physical properties. This is the reason why I also think that
the gum-pigment ratio test without dichromate means nothing practically.