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I'm a rookie with statistics, and I'm struggling to understand this:

  • it is well known that a confounding factor can cause a spurious association, leading to rejecting a true null hypothesis (i.e. due to the confounding factor Z, I could conclude that there is a causal relationship between X and Y, while one is not there)
  • the question is: can the opposite also be true? I.e. can a confounding factor lead to failing to reject a false null hypothesis? (i.e. somehow 'masking' a possibly existent causal association.) If yes, what would be a convincing example?
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    $\begingroup$ A related concept is called "suppression". See Kim (2019) for a discussion of suppression in causal terms. $\endgroup$ – Noah May 1 at 3:51
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Yes

Rephrasing the opposite of a confounder: It is definitely possible that an unobserved variable yields the impression that there is no relationship, when there is one.


Confounding usually refers to a situation where an unobserved variable yields the illusion that there exists a relationship between two variables where there is none:

confounding

This is a special case of omitted-variable bias, which more generally refers to any situation wherein an unobserved variable biases the observed relationship:

OVB

It's easy to imagine a scenario where this would have a canceling effect on the estimate instead:

rhozero

(I wrote $\rho=0$ for the illustration, but the unobserved relationship does not have to be linear.)

You could call this phenomenon omitted-variable bias, cancellation, or masking. Confounding usually refers to the kind of causal relationship shown in the first figure.

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  • $\begingroup$ Thanks Frans. In the second and third figures, should I assume that there is a (positive) association between O and X? (It seems that this is implied in the Wikipedia reference on OVB). $\endgroup$ – Franco Apr 27 at 10:30
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    $\begingroup$ @Franco Yes you're right! I included this in the figures $\endgroup$ – Frans Rodenburg Apr 27 at 11:04
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    $\begingroup$ Surely the second situation is a priori much less likely, because O has to exert a specific (equal but opposite) effect to the true causal relationship in order to cancel $\endgroup$ – benxyzzy Apr 27 at 20:04
  • $\begingroup$ In this answer, does an arrow with a little plus or minus sign denote a causal relationship when black and a correlation when red? $\endgroup$ – eric_kernfeld Apr 27 at 22:07
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    $\begingroup$ @benxyzzy The cancellation as shown in the figure is perfect, but it does not have to be to result in a false negative. Also, even if the cancellation is only partial, you would still have a biased estimate. $\endgroup$ – Frans Rodenburg Apr 27 at 23:24
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Following on existing answers, I wanted to give a concrete example. Imagine trying to figure out if the gas pedal affects the speed of a car. You observe how far the gas pedal is pressed and how fast the car is going at various times and see no correlations, so we conclude there's no causal effect between them. However, what we are missing is the fact that the car is going up and down hills and the gas often has to be floored when the car is going slowly up a hill. If we knew the slope of the road, we could control for that and find the true causal relationship.

This is an example of the last diagram of Frans's answer.

This example is even clearer if you try to associate the gas pedal to acceleration, rather than speed. The car's total acceleration will be (gas pedal) - (hill slope). Supposing you have cruise control on, then the cruise control will try to keep acceleration right around zero. So the gas will be set to cancel out the slope of the hill and will be entirely uncorrelated to the acceleration (which will be dominated by the changes in slope that the cruise control has yet to compensate for).

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    $\begingroup$ Fantastic example! $\endgroup$ – PLL Apr 29 at 8:45
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First, I think you are mixing the usage of "correlation" and "causal relationship". They are different things. To discuss the differences, and how to find "causal relationship", we need a lot of efforts.

Here I will only answer if a confounding variable can hide correlation.


Yes, here is an intuitive example (data is generated by y = c(runif(100), runif(100)+2), x = seq_along(y) in R):

We have x, y and the group variable. The group information is represented as a color of the points.

  • If we do not know the group / build a regression model using all data, we can say, x and y are positively correlated.
  • If we use group information / build a regression model on each group. we will say x and y have almost no correlation.

enter image description here

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    $\begingroup$ Thanks Haitao. Surely for my inexperience, I have difficulty in correctly understanding the answer: it seems to me that if I omit the group variable (say, z) I see a correlation between x and y, while when I introduce it, such correlation disappears. Isn't z a confounding factor in the usual sense, then? (i.e. creating a spurious association, when omitted) $\endgroup$ – Franco Apr 27 at 10:27
  • $\begingroup$ +1 for a nice graphic, but the example could be much clearer. Here, the code is the true causal model: if you intervene on it, you know exactly how it will respond. If the variable on the X axis and the group variable were explicitly present in the code, then it would be easier to understand what you mean. $\endgroup$ – eric_kernfeld Apr 27 at 22:03
  • $\begingroup$ Just as there are multiple causal models compatible with a given dataset, there are multiple ways you could write the code. Maybe including them both would make for an even better answer. $\endgroup$ – eric_kernfeld Apr 27 at 22:05

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