On measurements of phenotypic parallel evolution

Several metrics have been proposed to measure phenotypic parallel evolution.  All of these metrics stem from a geometric definition of parallel evolution in which two evolutionary trajectories are, literally, parallel or non-parallel to each other.  Two metrics fit this definition: the interaction term between population and habitat in a two-factor ANOVA and a measure of the angle between two multivariate trajectories of evolution.  A third metric is derived from the general direction of multivariate trajectories; although this might fit our intuition about parallel evolution, it does not fit the geometric definition.  A fourth metric is based on the amount of variation explained by the habitat variable in a one-factor ANOVA (i.e., the R2).  We show here that the R2 metric does not reliably measure any aspect of parallelism and should be avoided.  We also discuss the importance of establishing proper ancestor-descendent relationships in attempting to use any of the valid metrics to quantify parallel evolution.  Finally, because different metrics measure different aspects of evolutionary trajectories, we recommend being explicit about what one is trying to measure (angle, direction, or length of trajectories).

Methods
Data from the paper 

Dial, T. R., D. N. Reznick, and E. L. Brainerd. 2016. Effects of neonatal size on maturity and escape performance in the Trinidadian guppy. Functional Ecology 30:943-952, were downloaded from Dryad and trimmed to the first three columns, and rows with missing data.