4 - Correlation and association

Summary

It is difficult to understand why statisticians commonly limit their inquiries to Averages, and do not revel in more comprehensive views.

(Francis Galton, 1889)

When we make a set of measurements, it is instinct to try to correlate the observations with other results. One or more motives may be involved in this instinct. For instance we might wish (a) to check that other observers' measurements are reasonable, (b) to check that our measurements are reasonable, (c) to test a hypothesis, perhaps one for which the observations were explicitly made, or (d) in the absence of any hypothesis, any knowledge or anything better to do with the data, to find if they are correlated with other results in the hope of discovering some new and universal truth.

The fishing trip

Take the last point first. Suppose that we have plotted something against something, on a fishing expedition of this type. There are grave dangers on this expedition, and we must ask ourselves the following questions.

1. Does the eye see much correlation? If not, calculation of a formal correlation statistic is probably a waste of time.

2. Could the apparent correlation be due to selection effects? Consider, for instance, the beautiful correlation in Figure 4.1, in which Sandage (1972) plotted radio luminosities of sources in the 3CR catalogue as a function of distance modulus. […]