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What is the exact difference between two variables that are multicollinear and two variables that are interacting and two variables that are confounding?

Are they multiple meanings for the same thing?

Or is any one a subset, superset of the other?

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A simple way of thinking about multicolinearity is that some independent/explanatory/right-hand-side/x-variables are (strongly) correlated.

Confounding means that some variable causes both an explanatory variable of interest and the outcome variable. A classic example would be the observation that regions with more storks also have a higher birth-rate, but this is due to a third (confounding) variable: more rural regions have more storks and have a higher birthrate.

So a correlation between two explanatory variables is not enough to conclude that there is confounding (although it can support collinearity): first it needs to be causal, second, the direction of causality must be from the confounding to the explanatory variable of interest, third the confounding variable also needs to cause the outcome variable.

If the explanatory variable of interest causes the third variable, which in turn causes the outcome varialbe, then the third variable is an intervening variable; It represents a mechanism through which the explanatory variable of interest influences the outcome variable. For example, the socioeconomic status of the parents influence the education of the child that in turn influences the socioeconomic status of the child.

An interaction is completely different: it means the effect of the first variable differs depending on the second variable, e.g. is the effect of education on wage the same for men and women? In a linear model the interaction for continuous values will involve the product of the two values with a parameter for the contribution wiki link

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  • $\begingroup$ If you have three variables X,Y, Z , where X and Y predict Z. Then if we make a model predicting Z from X and Y is co-related but not included then it is confounding. When it is included , then X and Y become co-linear. In short the first thing is that the variable is co-related. Then if it's not included, it becomes a confounding variable, if it included ,then it becomes colinear. Is this correct? $\endgroup$
    – ctd2015
    May 20, 2016 at 7:57
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    $\begingroup$ No, a confoundig variable is confounding regardless of whether it is in the model or not. Your results will be biased if you have a model without the confounding variable in it. Whether or not a variable is a confounding variable has to do with the direction of causal relations. The main way that is determined is by theory. $\endgroup$ May 20, 2016 at 9:16
  • $\begingroup$ You can have colinearity without any strong correlations. For instance, let $X_1$ through $X_10$ be independent normals. Let $X_11$ be the sum of those 10 variables. There will be colinearity. (Things like this do happen). $\endgroup$
    – Peter Flom
    Dec 22, 2018 at 11:21

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