Fitting and comparing models of phyletic evolution: random walks and beyond

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Abstract
For almost 30 years, paleontologists have analyzed evolutionary sequences in terms of simple null models, most commonly random walks. Despite this long history, there has been little discussion of how model parameters may be estimated from real paleontological data. In this paper, I outline a likelihood-based framework for fitting and comparing models of phyletic evolution. Because of its usefulness and historical importance, I focus on a general form of the random walk model. The long-term dynamics of this model depend on just two parameters: the mean (μstep) and variance (σ2step) of the distribution of evolutionary transitions (or “steps”). The value of μstepdetermines the directionality of a sequence, and σ2stepgoverns its volatility. Simulations show that these two parameters can be inferred reliably from paleontological data regardless of how completely the evolving lineage is sampled.In addition to random walk models, suitable modification of the likelihood function permits consideration of a wide range of alternative evolutionary models. Candidate evolutionary models may be compared on equal footing using information statistics such as the Akaike Information Criterion (AIC). Two extensions to this method are developed: modeling stasis as an evolutionary mode, and assessing the homogeneity of dynamics across multiple evolutionary sequences. Within this framework, I reanalyze two well-known published data sets: tooth measurements from the Eocene mammalCantius, and shell shape in the planktonic foraminiferaContusotruncana. These analyses support previous interpretations about evolutionary mode in size and shape variables inCantius, and confirm the significantly directional nature of shell shape evolution inContusotruncana. In addition, this model-fitting approach leads to a further insight about the geographic structure of evolutionary change in this foraminiferan lineage.