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I have a vague memory that this relationship is possible, but I can't think of any examples off the top of my head. Is there a canonical or common example(s)?

I am imagining that in two independent experiments, increasing input B leads to an increase in output A, and the same for input C. But if both inputs B and C are increased together, that this causes a reduction in output A. I think this kind of relationship would be most likely due to an interaction of B and C, i.e. a B*C term in the equation for A. It could be caused by an additional hidden variable.

What would be the correct terminology for this relationship? Is it simply non-linear? Or multivariate non-linear? Or...?

craq
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    Since your considerations are only of correlation, you are asking about inherently *linear* regression models. I changed the tags to reflect this. – whuber Sep 19 '18 at 16:11
  • @whuber maybe that's where I'm getting confused - I wasn't aware that correlation only applied to linear regression models. I think I might actually be looking for examples of "multiple nonlinear regression" models? – craq Sep 20 '18 at 04:47
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    Correlation applies to all multivariate random variables and all multivariate datasets. One's choice to use this to describe relationships is a choice to analyze *linear* associations among the variables. I am using "correlation" in the narrow but very well known and well defined sense of statistics. If you are trying to use it to refer to a broader form of *association* among data or variables, then please state that explicitly in your question. – whuber Sep 20 '18 at 13:50
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    @whuber yes, I think I am trying to (mis-)use the term correlation to apply to a broader form of association. I have tried to rewrite the question to reflect this. – craq Sep 20 '18 at 22:21

3 Answers3

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Aaron is married to Cat, but also has an affair with Betty, who thinks he's single. He wants to maintain this situation for foreseeable future.

A = 1 when Aaron is in the the bar, A = 0 when he's not there. B = 1, when Betty's in the same bar, and C = 1 when Cat's in the same bar. Aaron likes going to this bar, especially if either of the ladies are there, but certainly not when both of ladies are there.

This can be summarized in logical equation: A = B NAND C or algebraic equation: $A = 1-B*C$

Clearly cor[B*C,A]<0 while cor[A,B]>0 and cor[A,C]>0

P.S. You can substitute "is in the bar" with "makes love" to drive a point home

Aksakal
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  • I had trouble following this at first, probably because "being in the bar" is a boolean state, and I was thinking about continuous variables. I also wasn't sure you were serious. But... I think it actually does satisfy the criteria I was looking for, so +1. – craq Sep 20 '18 at 04:56
  • Correlations can be applied to discrete ordinal variables too. I was just trying to make it a simple example. Also you can make it a continuous case by measuring how many hours spent in the bar. It’s all the same idea of interactions between variables – Aksakal Sep 20 '18 at 12:10
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The key idea is to remember that a correlation is essentially a measure of linear association, so if you want to come up with a counterexample that suits your case, think of nonlinear relationships.

This is an example quite common in Economics: Think of the following three random variables: $Wage$, $Education$, $Age$. It's easy to argue that $Wage$ is positively correlated with years of $Education$, and the same may be said about $Wages$ and $Age$, possibly because of an experience or seniority effect.

However, one might argue that Wages are negatively correlated with the interaction $Age \times Education$: for a certain level of education, wages might be negatively correlated with age.

Hope this helps.

MauOlivares
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  • I think I agree with you that I should be thinking of nonlinear relationships. I'm not sure I see how age and education combine non-linearly? I can see how each could be nonlinear by themselves: Age, once someone approaches retirement age. For education it could be argued that being overqualified reduces wages - or at least earning potential because you spend so much time in class that there's no time to work. – craq Sep 20 '18 at 04:50
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After some helpful comments by whuber, it seems that the term I am after is "interaction". Reading up on interaction here and elsewhere brought up the following examples supported by real data (all from the wikipedia page on statistical interaction. If you're interested, that was a good read):

  1. body temperature dependent on ambient temperature and species
  2. stroke recovery dependent on stroke severity and treatment
  3. cookie quality dependent on oven temperature and baking time
  4. sweetness of coffee dependent on amount of sugar added and amount of stirring
  5. strength of steel dependent on amount of carbon added and quenching
  6. lung carcinoma risk dependent on smoking and asbestos exposure
  7. diabetes risk dependent on genetics and diet
  8. perception of climate change, dependent on political orientation and level of education

and some contrived examples:

  1. enjoyment of food dependent on a main component (hot dogs vs ice cream) and sauce (chocolate vs tomato)
  2. weight loss dependent on exercise and diet
  3. test performance dependent on practice and stress levels
  4. employee performance dependent on training and autonomy
  5. health dependent on sleep and diet
  6. effect of headache treatment on dose and gender
  7. psychoactive effect of barbiturates and alcohol

Specifically for my question, the following interactions cause an inversion of their independent effects (numbering from the list above): 3, 8, 9, 10, 11, 12, 14. The others amplified the independent effects, or there was no effect for one of the independent variables when considered, er, independently.

craq
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