Q1 when is strong correlation sufficient to prove cause and

Q1: when is strong correlation sufficient to prove cause and effect?

Q2: Under what circumstances is it acceptable to use the regression equation for predicting, even when there is statistically insufficient evidence to suggest correlation?

Solution

1. Strength - The larger the absolute value of the coefficient, the stronger the linear relationship between the variables. An value of one indicates a perfect linear relationship, and a value of zero indicates the complete absence of a linear relationship.

Correlation is not sufficient for causation. One can get around the Wikipedia example by imagining that those twins always cheated in their tests by having a device that gives them the answers. The twin that goes to the amusement park loses the device, hence the low grade.

A good way to get this stuff straight is to think of the structure of Bayesian network that may be generating the measured quantities, as done by Pearl in his book Causality. His basic point is to look for hidden variables. If there is a hidden variable that happens not to vary in the measured sample, then the correlation would not imply causation. Expose all hidden variables and you have causation.

2.

The correlation coefficient, r, tells us about the strength of the linear relationship between x and y. However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient r and the sample size n, together.

We perform a hypothesis test of the \"significance of the correlation coefficient\" to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data is used to compute r, the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we only have sample data, we can not calculate the population correlation coefficient. The sample correlation coefficient, r, is our estimate of the unknown population correlation coefficient.