Bayesian Statistics: An Introduction, 4th Edition

6.2 Examples on the use of the correlation coefficient

6.2.1 Use of the hyperbolic tangent transformation

The following data is a small subset of a much larger quantity of data on the length and breadth (in mm) of the eggs of cuckoos (C. canorus).

Here n = 9, , , Sxx=12.816, Syy=6.842, Sxy=7.581, r=0.810 and so z=tanh–10.810=1.127 and 1/n=1/9. We can conclude that with 95% posterior probability ζ is in the interval , that is, (0.474, 1.780), giving rise to (0.441, 0.945) as a corresponding interval for ρ, using Lindley and Scott (1995, Table 17) or Neave (1978, Table 6.3).

6.2.2 Combination of several correlation coefficients

One of the important uses of the hyperbolic tangent transformation lies in the way in which it makes it possible to combine different observations of the correlation coefficient. Suppose, for example, that on one occasion we observe that r=0.7 on the basis of 19 observations and on another we observe that r=0.9 on the basis of 25 observations. Then after the first set of observations, our posterior for ζ is N(tanh–10.7, 1/19). The second set of observations now puts ...