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Suppose two variables X1 and X2 are correlated and we know that X1 causes X2 and X2 causes X1.

For example, leg strength and an interest in cycling interest might be correlated. And (suppose) we know that having stronger legs makes you more likely to take up cycling and cycling causes your legs to become stronger.

How would you describe this type of bidirectional causal influence? Is there a word for it?

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2 Answers 2

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I've seen the terms mutual causation, bidirectional causation, and reciprocal causation used. I'm not sure whether there is a standard term in statistics or whether these are different terms meaning the same thing that have arisen in different fields.

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    $\begingroup$ Also feedback in the classic sense, positive for self-amplifying or negative for self-damping (not to be confused with feedback as commentary on or evaluation of performance in any sense) $\endgroup$
    – Nick Cox
    Commented Apr 27 at 14:56
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    $\begingroup$ Wow. I thought for sure I had simply forgotten a known term for something so common. It's fascinating that there is not a single well-known term to describe this type of relationship. $\endgroup$ Commented Apr 28 at 0:04
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    $\begingroup$ @EndAnti-SemiticHate That is very common. Different communities of practice will have different vocabularies. $\endgroup$ Commented Apr 28 at 0:32
  • $\begingroup$ Bear in mind also that these two variables may cause one another both directly, and indirectly via paths through other variables (which would also be causes and effects of both variables). See Puccia, C. J., & Levins, R. (1986). Qualitative Modeling of Complex Systems: An Introduction to Loop Analysis and Time Averaging (C. J. Puccia & R. Levins, Eds.). Harvard University Press, for a radically different analytic framework for such complex causal systems as compared to structural causal models. $\endgroup$
    – Alexis
    Commented Apr 28 at 17:49
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On a macro scale, this concept is often discussed in terms of mutual causation, causal cycle, and feedback loop within the framework of structural causal models (SCM) by Dr. Judea Pearl. However, SCM theory does not cover such phenomena, because the developed graphical criteria apply only to systems described by directed acyclic graphs (DAGs). A simple justification is time constrains: for instance, if variables $X_1$ and $X_2$ occur at different time points $t_1$ and $t_2>t_1$, it is logically inconsistent to have both $X_1\rightarrow X_2$ and $X_1\leftarrow X_2$ simultaneously, as the future cannot influence the past.

On a micro level, this phenomenon can generally be referred and understood as a dynamical system. Here, the causal cycle $\{X_1\rightarrow X_2, X_1\leftarrow X_2\}$ may actually represent the aggregation or equilibrium state of a dynamic system, such as:

$$ \begin{aligned} X_{1,t} &\leftarrow f_1(X_{1,t-1},X_{2,t-1},U_{1,t})\\ X_{2,t} &\leftarrow f_2(X_{1,t},X_{2,t-1},U_{2,t})\\ \end{aligned} $$

In your example: increased leg strength today leads to a desire to bike tomorrow, biking tomorrow leads to immediate enhanced leg strength, which in turn increases the inclination to bike the following day, and so forth.

However, it's crucial to note that a causal loop is distinct from a bidirected arrow, $X_1\leftrightarrow X_2$, often indicating a latent confounding association, where both $X_1$ and $X_2$ are influenced by an unobserved factor $U$. In this case, $X_1$ and $X_2$ do not directly cause each other, but are both influenced by the common factor $U$. For instance, overall physical fitness level can influence both leg strength and interest for biking.

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