In my recent efforts to understand Phylogenetic Independent Contrasts (PIC), I came across a lecture that was done by the man who proposed the method himself, Joe Felsenstein. So who better to explain the method than the creator himself? PIC is a comparative method that statistically compares traits between species or clades by using phylogenies. However, this poses a problem: how can one compare traits in an independent manner when phylogenies are hierarchical? The answer: you cannot. As Felsenstein explains, it creates biases in the statistical analyses and although it seems like one is comparing traits individually, you are not. As a response to these faulty comparative methods, Felsenstein created a method that takes into account weighted differences of branch lengths to calculate a variance of each trait. Because the branch lengths are taken into account, non-independence of species is circumvented. The phylogenies are assumed to work under Brownian Motion i.e. the probability of a trait to move upward or downward in a tree is the same and traits evolve in a linear manner.
Figure 1: PIC explained graphically. The differences in distances between branches of nodes are calculated using the Brownian Motion model. These values are then used to plot the differences between traits and a regression can be calculated. (http://www.life.umd.edu/classroom/zool360/L5_evolution/img023.gif)
The algebraic side to PIC is a little more complicate than what has come across in my post, but the idea is "fairly" simple. PIC's are highly useful in evolutionary biology and one can find correlations between traits and the environment through evolutionary time. Its kind of ... magical.
To understand PNC in a more in depth manner, I have attached the link to the lecture conducted by Felsentein. The lecture is 59 minutes long, but it is worth the watch.
Prepare to be fascinated.
Felsenstein, J., 1985. Phylogenies and the comparative method. American Naturalist, 1-15.