# Correlation matrix

Is it good practice to consistently use a correlation matrix when the direction of the relationship between dependent and independent variables is unknown (that is, when we’re uncertain about an inverse or direct relationship between variables)? Or, should established research be used as the basis for determining the type of relationship between variables?

## 2 Answers

Yes and Yes. Correlation matrices are extremely useful, and as part of any multivariate analysis they should always be extracted, studied, and pondered upon. Yet, they won't tell you the direction of a relationship. Correlation is not Causation must be the number 1 truism in statistics. The challenge is that correlation is very easy to establish while causation is a lot more challenging to confirm. There are interesting basic statistical methods supporting causality such as Granger Causality, and Path Analysis. There are more complicated ones presented by Judea Pearl in "Causality" published by Cambridge University Press. But, as you suggest logic, established research, judgment, empirical evidence should support causality just as much if not more than any statistical models. When logic and empirical evidence on one side support or converge with the model results you are on reasonably solid ground. Occasionally, a model can uncover a counter-intuitive result that will make a field move forward. But, you better question such a model at length before believing it. Confounding variables can remain a stealthy challenge in such situations.

The correlation matrix will remain the same no matter which is taken to be dependent and which the independent variable (ignoring which number comes first in the matrix). There is no suggestion of causality. See xkcd on the subject.

What does usually change with the assumption is the form of a typical regression result. For example, with classical least-squares regression the aim is to minimise the sum of the squares of the residuals in the dependent variable, and so the choice of which one this is affects the result.