<p>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 </p>

<blockquote>
  <p>Linearly independent, orthogonal, and uncorrelated are three terms used to indicate lack of relationship between variables. </p>
</blockquote>

<p><sub><strong>Reference</strong><br>
<strong>-</strong> <a href="https://www.researchgate.net/publication/254330622_Linearly_Independent_Orthogonal_and_Uncorrelated_Variables" rel="nofollow noreferrer">Rodgers <em>et al.</em>, The American Statistician (1984); <strong>38</strong>(2): 133-4</a></sub></p>

CC BY-SA 4.0
2018-09-24T14:17:08.923Z
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?
thando
CC BY-SA 4.0
2018-09-24T14:21:56.733Z
@thando My pleasure. They are likely not the same, as no numerical values are given
AliceD
CC BY-SA 4.0
2018-09-24T16:32:04.400Z
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".
steveLangsford
CC BY-SA 4.0
2018-09-24T18:48:29.280Z
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?
thando
CC BY-SA 4.0
2018-09-24T19:20:16.123Z
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?
steveLangsford
CC BY-SA 4.0
2018-09-26T05:46:40.823Z
Thank you very much! I posted a [separate question](https://psychology.stackexchange.com/questions/20785/how-can-two-way-dependencies-in-factorial-designs-be-interpreted) about this concern. There is no model about how A and B interact.
thando
CC BY-SA 4.0
2018-09-24T09:02:07.083Z
2018-09-24T09:33:40.943Z
2018-09-24T09:33:40.943Z
AliceD
<p>Given there are two psychological concepts A and B that are considered to be independent/orthogonal.</p>

<p>Would the following empirical results proof the opposite? If not why?</p>

<p>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.</p>

<p>References to papers that use similiar arguments would be very helpful.</p>

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2018-09-24T11:17:53.010Z
This might be [more suitable on statistics SE](https://stats.stackexchange.com)? I see no direct necessary relation to this site.
Steven Jeuris
CC BY-SA 4.0
2018-09-24T14:20:16.337Z
The questions is about the interpretation of such empirical results in the context of cognitive science research.
thando
<terminology><experimental-psychology><experiment-design>
What is an empirical proof that two concepts are not independent/orthogonal?
thando
# What is an empirical proof that two concepts are not independent/orthogonal?

(Asked by: username_0 on 2018-09-23)

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.

**Tags:** terminology, experimental-psychology, experiment-design

> **Comments:**
> **username_1**: This might be [more suitable on statistics SE](https://stats.stackexchange.com)? I see no direct necessary relation to this site.  
> **username_0**: The questions is about the interpretation of such empirical results in the context of cognitive science research.  

---

## 1 answers

### Answer by username_2 on 2018-09-24. Score: 2 | Accepted answer

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](https://www.researchgate.net/publication/254330622_Linearly_Independent_Orthogonal_and_Uncorrelated_Variables)

> **Comments:**
> **username_0**: 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?  
> **username_2**: @username_0 My pleasure. They are likely not the same, as no numerical values are given  
> **username_3**: 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".  
> **username_0**: Thank you @username_3, also for the hints! What would be reasonable hypotheses about the relation of the two concepts, given the experimental results in the question above?  
> **username_3**: 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?  
> **username_0**: Thank you very much! I posted a [separate question](https://psychology.stackexchange.com/questions/20785/how-can-two-way-dependencies-in-factorial-designs-be-interpreted) about this concern. There is no model about how A and B interact.  

---

