A statistic summarizing: Relative timing in trees

1. A statistic summarizing:Relative timing in trees Tanja Gernhard joint work with Erick Matsen, Daniel Ford

2. Tree statistics • Investigate the properties of reconstructed trees • Colless [Colless, 1982] summarizes the information of the shape of the tree (i.e. balanced vs. unbalanced) • Gamma [Pybus, Harvey 2000] summarizes the time between speciation events (graphical representation in LTT plots)

3. Relative Timing Neutrality: Each ranking equally likely

4. How to summarize rank information? Each ranking equally likely equivalent to For each vertex: Each shuffle equally likely RRRRLL RRLRRL shuffles on a vertex

5. 1 1 1 1 ¡ ¡ ¡ ¡ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ m n m n + k k k k 1 1 ¡ ¡ f g k P X 2 1 + = = m n + ¡ ¢ m n ; m 1 1 ¡ ¡ ¡ ¢ ¡ ¢ m n 2 k k 1 1 ¡ ¡ f g k P X 2 = = m n + ¡ ¢ m n ; m Runs summarize shuffle • Runs: Number of sequences in shuffle with same letter • Consider (2,4)-shuffle RRLRRL: 4 runs • Run distribuition of (m,n)-shuffle well-known from statistics:

6. k k i ¡ X X f ( ) g f g f ( ) g f ( ) g k k P P P P R T X R L R R i j i j ¡ ¡ = = = = = m n ; i j 1 0 = = Combining all shuffles in tree • Runs of a tree, R(T): Sum of runs of all shuffles in tree • Distribution for runs in tree T:

7. Application to Ant lineages • We investigated the tree of Ants [Moreau et al, 2007] • LTT plot show substantial increase during Late Cretacceous to Early Eocene

9. Some comments • Testing the posterior of Bayesian methods for neutrality (phylogenies or populations) • Population size might change arbitrary • All programs available from my homepage in the Python package Cass python Stats.dat YourTrees.tre

10. Dankeschön