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This article gives a short introduction to inferring causal relations from multivariate experimental data. This introduction will not cover all issues involved, but provides a starting point for the non-expert. It is important to note that even for many experts from statistics, mathematics, data analysts or related professions, the inference of causality appears to contradict intuition. However, there is a good mathematical foundation for this collection of principals. The basic concept of causality in a mathematical framework revolves around deciding whether or not some variable influences something in another variable. If such influence is detected, one can speak of a causal influence on that variable. The concept of causation in combination with statistics easily sparks heated debates. Numerical experts often feel uncomfortable when statistical results are coupled claims about causal influence. This is the case in situations that are backed up by or stem from designed experiments. However, work by Judea Pearl and others, have elegantly shown that some form of claims about cause and effect can be substantiated from numerical analysis alone. How does it work? We need at least 3 variables to do this causality-dance. In this framework we can then hypothesize about which variables influence the other variables. In the situation where one has the variables A, B and C, and use a notational scheme that displays A causing B as A -> B. Now, realize that when A causes B, knowing something about B gives you some information on A as well. In a statistical Bayesian framework one can often find this expression in this notation: P(a|b), meaning the probability of observing A being a knowing (given) B is b. This also means that one could use regression analysis for removing the information, variation in the correct jargon, that B shares with A from A. In doing so, the rest of A is uncorrelated to B. Returning to our toy example of three variables, we can postulate four different causal models. Model
| A independent of B given C?
| | A->C->B | yes | | A<-C<-B | yes
| | A<-C->B | yes | | A->C<-B | no |
From this table, the core principle emerges. As soon as one can identify a causal model that gives an independent set A and B that becomes dependent given C, one is able to direct the arrows. The direction of the arrow represents a causal relation. This causal relation needs to be treated with some care, as this model infers a valid causal relation, but this does not prove that this is a direct causal effect. Some intermediate variables may exist that are between the used variables and the end point C. The difference between a direct cause and an indirect cause is simply explained by the following example. In a murder case, what killed the victim? Was it the person that did the shooting, the gun, the bullet, the bullet entering the victims body, the bullet rupturing the victims heart, the ruptured heart that stopped pumping or the lack of oxygen in the brain. It is, in this list, unclear what killed the victim. However, there is a unequivocal cause and effect relation between the murderer and the victim. Even though it was the lack of oxygen that eventually killed the victim. Send me an email if you have a specific question regarding causal inference, see the contact me on the left. |
