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When testing relationships between variables, we are often trying to establish a finding that Variable A causes (or is connected with) an increase in Variable B, or that Variable A causes a decrease in Variable B, or that Variable A doesn't cause neither an increase nor a decrease in Variable B.

However, I have a vague idea that effects of one variable on another variable can often be bidirectional. More specifically, I think that sometimes Variable A simultaneously causes both an increase and a decrease in Variable B.

For example, how is the openness of the window related to the number of flies in the house? Suppose there are no flies in your house. If your window is open, some flies will fly into your home. After some time, they will fly out, but other flies will fly in. Perhaps the number of the flies will be constant over time because the fly in/fly out rate is stable. However, if we kept our window closed, the number of flies would be zero. This may lead us to believe that the openness of the window causes an increase in the number of flies. However, if the openness of the window is related just to the increase in the number of flies, then no flies would ever fly out of the house, but we know this is not true. My point is that the openness of the window causes an increase in the number of flies that fly in the house, but it also causes an increase in the number of flies that fly out of the house. Or, more simply, openness of the window is related both to the increase and the decrease in the number of flies in the house.

I suspect a similar mechanism may be behind some confusing findings in the literature. For example, some studies link cannabis usage to increased cancer risk, while other studies link it to decreased cancer risk. What if cannabis simultaneously increases and decreases cancer risk?

What I suspect most researchers would do is look for a moderator variable, so that they can say "Cannabis increases cancer risk in those participants, but it decreases cancer risk in these participants". However, I believe that this "bidirectionality" can arise in a standalone manner, with no moderator variables, like in the window/fly example.

What are your thoughts on this topic? Is there any previous literature or statistical techniques related to such bidirectional relationship? I've been trying to find some materials, but I wasn't successful. Any resources, as well as discussion, would be greatly appreciated.

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  • $\begingroup$ I would search here and on Google for "simultaneity bias" and "instrumental variables". $\endgroup$
    – dimitriy
    Apr 12, 2018 at 17:31

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This is a modeling problem. The first thing you need to do is to write down the model of how you think the world works, only after that you may think of estimation techniques.

The case you have described is not of bidirectional effects, it's about non-linear effects. The variable "window" does alter the probability of the variable "flies", just not monotonically or in other ways you think it should alter.

A simple model of what you have described could be something as this:

enter image description here

That is, the number of flies in time $t$ is a function of the number of flies in the previous time $t-1$, the current state of the window and unmodeled factors $U_t$. One such function could be:

$$ \text{Flies t} = \left\{ \begin{array}{ll} \text{Flies t-1}, ~~~~~~~~~~~~~~\text{if Window state t = closed}\\ f_{t}(\text{Flies t-1}, U_t),~~~ \text{if Window state t = open} \end{array} \right. $$

In the sense that, if the window is closed, then you are assured that the current number of flies is equal to the previous number of flies. However, if the window is opened, then the current number of flies is determined by the process $f_t$ which considers the previous numbers of flies and other unmodeled factors.

The example above is just to illustrate that you need to model what you want to measure. This is an essential step for defining the effect you want to estimate and even for deciding whether the data you have is enough for the task.

Regarding you cannabis example: What if cannabis simultaneously increases and decreases cancer risk? Yes, that could happen for sure. You could have heterogenous effects between groups, you could have interaction effects of cannabis with other substances, or other types of non-linearities.

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  • $\begingroup$ Thanks, this really clears things up. Could I just ask you what sort of statistical analysis would you use to answer this sort of a problem? It seems that you are fitting a logical function to the data, and not just a mathematical function (since it contains IFs). $\endgroup$
    – J. Doe
    Apr 22, 2018 at 10:22
  • $\begingroup$ @J.Doe you can model the if statement with an indicator variable. Say $f_t$ is linear and constant over time, then you could write the whole model as $\text{Flies}_t = W\times (\beta ~\text{Flies}_{t-1} + U_t ) + (1-W)\times\text{Flies}_{t-1}$ for example, where $W = 1$ if the window is opened. $\endgroup$ Apr 23, 2018 at 3:18

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