Homework 8.1: Heredity in Darwin’s finches (35 pts)

We continue working with Peter and Rosemary Grant’s wonderful data set about Darwin’s finches.

Introgressive hybridization occurs when a G. scandens bird mates with a G. fortis bird, and then the offspring mates again with pure bred G. scandens bird. This brings traits from G. fortis into the G. scandens genome. As this may be a mode by which beak geometries of G. scandens change over time, it is useful to know how heritable a trait is. Heritability is defined as the ratio of the covariance between parents and offsprings to the variance of the parents alone. To be clear, the heritability is defined as follows.

Compute the average value of a trait in a pair of parents.
Compute the average value of that trait among the offspring of those parents.
Do this for each set of parents/offspring. Using this data set, compute the covariance between the average offspring and average parents and the variance among all average parents.
The heritability, \(h^2\), is the ratio of the covariance between parents and offspring to the variance of the parents, \(h^2 = \sigma_{po}/\sigma_p^2\).

This is a more apt definition than, say, the bivariate correlation \(\rho\), because it is a direct comparison between parents and offspring.

Heritability data for beak depth for G. fortis and G. scandens can be found here and here, respectively. (Be sure to look at the files before reading them in; they do have different formats.) From these data, obtain plug-in estimates for the heritability of beak depth in the two species, with confidence intervals. How do they differ, and what consequences might this have for introgressive hybridization?