How can two-way dependencies in factorial designs be interpreted?

Scenario: Given there are results from two factorial experiments that show a two-way dependency between two variables A and B.

Details: In the first experiment A is the independent variable with two conditions: high and low A. B is identically induced in both conditions and measured as the dependent variable. In the high A condition, high B is measured. In the low A condition, low B is measured. The second experiment is analog to the first one, except with A and B reversed: in the high B condition high A is measured, and in the low B condition low A is measured.

Question: What can be reasoned or assumed about the relationship between the two variables A and B, and why?

References to example papers would be excellent.

• This seems more appropriate for Stats SE. Do you want me to migrate it there? Commented Sep 26, 2018 at 9:23

In the full-generality land of arbitrary A and B, the 'details' section of the question gives the grand total of all the conclusions that can be drawn here. What you have is that when A was fixed to be high, B was observed to be high, when B was fixed to be high, A was observed to be high etc etc, and this is the complete total of things you know about A and B. Full-generality land is super harsh. You can observe an arbitrarily large number of points falling on an exact line and full-generality land will insist that the true function might bounce around like a roller-coaster that just happens to pass through the points you've observed. Full generality-land doesn't have to make sense. In full-generality land, fixing A at 0.1 units higher than you did in this experiment might cause the value of B to plummet, or turn into a turkey dinner.

Reality tends to make a lot more sense than full-generality land, except when brains or quantum particles are involved. Some things you might know in reality:

The units A and B are measured in.

Whether A and B are smoothly-varying or can contain discontinuities.

Whether A or B have ceiling/floor caps, and maybe where they are.

Whether the 'fixing' intervention is likely to be monotonic in its effects on A, B, or both (this is a big one, and it's quite likely that you do know this).

The 'causal depth' you plausibly care about (A and B are both caused by X might be interesting, that A and B are both products of the big bang probably not, how far you want to ride that train is domain specific)

That some particular theory of A and B is incompatible with A and B being high simultaneously.

All these things would give you some traction to move beyond the bare-bones statement of the exact results you observed, which is all full-generality land will let you do. Full generality land won't even let you claim that you'd get the same results from the exact same experiment performed tomorrow. Full-generality land has the attitude normally associated with honey badgers.

There's definitely a version of your argument that makes sense. If you have five blind people arguing over whether an elephant is a snake or a wall or a bit of rope, firing a starter's pistol so the elephant runs away is an excellent experiment. Under the theory that an elephant is five different things it is vanishingly unlikely that they will all run in exactly the same direction. Observing that they do all go in the same direction at the same speed should cause the blind scientists to propose a bunch of strongly-connected elephant theories. Only this argument is inductive not deductive, so it requires you to supply a hypothesis space. It's always possible that there were five animals who all ran in the same direction, to make the inductive argument it's critical to guess how likely that scenario is. If you do the experiment in a narrow valley there might acutally only be one direction it's possible to run in. If you do the experiment out in the open plain there's 360 degrees of freedom and coincidences are much more inductively compelling. In the real world, you almost always have some information about the landscape, making results of the kind decribed here strong ones. But you stripped all of that information out to ask the question! Reducing the variables to A and B brings you into full-generality land, where nothing is certain. Maybe you are standing in a narrow valley. Maybe you're underwater. Who knows?

It is possible to operate in full-generality land despite its Dadist tendencies, sometimes you can get away with some impressively weak assumptions. Try these:

Dunn, J. C., & James, R. N. (2003). Signed difference analysis: Theory and application. Journal of Mathematical Psychology, 47(4), 389-416.

Pearl, J (2014). Probabilistic reasoning in intelligent systems: networks of plausible inference.