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| Colliders are one possible failure of causal inference, although in an atypical manner. Generically a collider ''Z'' is associated with two independent variables, ''X'' and ''Y''. Therefore when controlling for the collider ''X'' and ''Y'' become associated. | Colliders are closely related to [[Statistics/Confounders|confounders]], but in the 'opposite' way. Generically, consider a variable ''Z'' that is associated with two independent variables, ''X'' and ''Y''. If ''Z'' is controlled for, ''X'' and ''Y'' become associated. |
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| In this example, controlling for ''Z'' 'creates' a correlation. (Note that in [[R]], the [[Analysis/BernoulliDistribution|Bernoulli distribution]] is handled as a special case of the [[Analysis/BinomialDistribution|Binomial distribution]]. Comparing the sum of ''X'' and ''Y'' to 0 is effectively checking if either is 1.) | The presence of a collider causes '''collider bias''' or '''Berkson's paradox'''. [[https://statmodeling.stat.columbia.edu/2026/05/13/recent-discoveries-on-the-persistence-of-statistical-fallacies/|Alex Dimakis]] provides a more concrete example: "Assume that to be a successful actor you have to be either extremely good looking or extremely talented. Assume also that talent and looks are independent in the population. However, among sucessful [sic] actors you will observe a negative correlation between looks and talent." See in the following demo that controlling for ''Z'' 'creates' a correlation. (Note that in [[R]], the [[Analysis/BernoulliDistribution|Bernoulli distribution]] is handled as a special case of the [[Analysis/BinomialDistribution|Binomial distribution]]. Comparing the sum of ''X'' and ''Y'' to 0 is effectively checking if either is 1.) |
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| The important consequence is that colliders should not be controlled for. |
Collider
A collider is a variable that is caused by multiple variables.
Contents
Description
Colliders are closely related to confounders, but in the 'opposite' way. Generically, consider a variable Z that is associated with two independent variables, X and Y. If Z is controlled for, X and Y become associated.
The presence of a collider causes collider bias or Berkson's paradox.
Alex Dimakis provides a more concrete example: "Assume that to be a successful actor you have to be either extremely good looking or extremely talented. Assume also that talent and looks are independent in the population. However, among sucessful [sic] actors you will observe a negative correlation between looks and talent."
See in the following demo that controlling for Z 'creates' a correlation. (Note that in R, the Bernoulli distribution is handled as a special case of the Binomial distribution. Comparing the sum of X and Y to 0 is effectively checking if either is 1.)
> X <- rbinom(1000, 1, 0.5) > Y <- rbinom(1000, 1, 0.5) > Z <- rbinom(1000, 1, ifelse(X+Y>0, 0.9, 0.2)) > cor(X,Y) [1] -0.02387166 > cor(X[Z==1], Y[Z==1]) [1] -0.3387377 > cor(X[Z==0], Y[Z==0]) [1] 0.2764379