What is an empirical proof that two concepts are not independent/orthogonal?

Given there are two psychological concepts A and B that are considered to be independent/orthogonal.

Would the following empirical results proof the opposite? If not why?

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.

References to papers that use similiar arguments would be very helpful.

• This might be more suitable on statistics SE? I see no direct necessary relation to this site. Sep 24 '18 at 11:17
• The questions is about the interpretation of such empirical results in the context of cognitive science research. Sep 24 '18 at 14:20

Since the data show a two-way dependence (low A -> low B, hi A -> hi B and the reverse) this would prove the variables are not independent, as

Linearly independent, orthogonal, and uncorrelated are three terms used to indicate lack of relationship between variables.

Reference
- Rodgers et al., The American Statistician (1984); 38(2): 133-4

• Thank you for your answer! One follow up question: What can be derived from such a two-way dependence? Might one even say A and B are essentially the same? Sep 24 '18 at 14:17
• @thando My pleasure. They are likely not the same, as no numerical values are given
– AliceD
Sep 24 '18 at 14:21
• You need to be very careful with the interpretation of non-independence though. Ultimately everything is non-independent (because you can intervene with a brick and send performance on every task to zero, followed by very highly correlated recovery of all abilities). [non]orthogonality is a stronger claim and correspondingly harder to demonstrate. You might want to explore "systems factorial technology" or "Signed difference analysis". Sep 24 '18 at 16:32
• Thank you @steveLangsford, also for the hints! What would be reasonable hypotheses about the relation of the two concepts, given the experimental results in the question above? Sep 24 '18 at 18:48
• From a strict logic perspective the description leaves a pretty wide field of possible relationships still feasible. But presumably there's a lot more constraining information contained in how A and B are supposed to work and the manipulation that was used to fix their levels? Sep 24 '18 at 19:20