How do we know if a correlation is significant or not?

Answer 1

There are several statistical measures of correlation: some require only a nominal scale, that is, data classified according to two criteria; others require an ordinal scale, which is the ability to determine whether one measurement is bigger or smaller than another; others require an interval scale, which allows you to determine the difference in values but not the ratio between them. [A good example of the latter is temperature measured in any scale other than Kelvin: the difference between 10 degrees C and 15 degrees C is 5 C degrees, but 15 C is not 1.5 times as warm as 10 C.]

• The contingency coefficient, which is suitable for nominal data, has a chi-squared distribution.
• The Spearman rank correlation, requiring ordinal data, has its own distribution for small data sets but as the number of units increases to n, the distribution approaches Student's t-distribution with n-2 degrees of freedom.
• The Kendall rank correlation coefficient can be used in identical situations and gives the same measure of significance. However, the Kendall coefficient can also be used to test partial correlation - whether the correlation between two variables is "genuine" or whether it arises because both variables are actually correlated to a third variable.
• The Pearson's product moment correlation coefficient (PMCC) is the most powerful but requires measurement on an interval scale as well as an underlying bivariate Normal distribution.

The significance levels of these correlation measures are tabulated for testing.

A simple "rule of thumb" for testing the significance of PMCC is that values below -0.7 or above 0.7 are highly significant. Values in the ranges (-0.7, -0.3) and (0.3, 0.7) are moderate, and values between -0.3 and +0.3 are not significant.