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

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.

Scale-dependence of Cope's rule in body size evolution of Paleozoic brachiopods

“…The magnitude of size increase between periods is statistically significant. We evaluated within-phylum dynamics by using maximum-likelihood comparisons among three evolutionary models: directional (driven, biased, general random walk) change (DRW), unbiased (passive) random walk (URW), and stasis (22), with DRW generally resulting in a pattern of Cope's rule when there is a positive directionality parameter (a maximum-likelihood estimate of the magnitude of the rate of size change). The brachiopod phylum-level size trend is overwhelmingly supported by the directional model (SI Appendix, Table 3), with a constant and positive rate of size increase of 0.013 log 10 ml/Myr Ϯ 0.005.…”

“…The first test applies the maximumlikelihood approach used above to differentiate DRW, URW, and stasis dynamics manifested in all genus occurrences within these clades of varying nestedness. Although this test is robust to errors in bin ages and sampling heterogeneities and does not require explicit genus-level phylogenies within each clade (22), it loses resolving power for shorter time series, and model selection can only be conducted on clades spanning a minimum of five time intervals. In addition to estimation of the directionality parameter for each clade, this method also allows estimation of the joint directionality parameter: the maximumlikelihood estimate of a single directionality parameter held constant across all constituent clades (22).…”

“…Although this test is robust to errors in bin ages and sampling heterogeneities and does not require explicit genus-level phylogenies within each clade (22), it loses resolving power for shorter time series, and model selection can only be conducted on clades spanning a minimum of five time intervals. In addition to estimation of the directionality parameter for each clade, this method also allows estimation of the joint directionality parameter: the maximumlikelihood estimate of a single directionality parameter held constant across all constituent clades (22). The second test assesses the magnitudes and directions of change in the minimum and maximum sizes from the first to the last interval within clades (10) and can be conducted on clades spanning as few as two intervals.…”

The Breakdown of Static and Evolutionary Allometries during Climatic Upheaval

“…The R package 'paleoTS' (Hunt 2006) allows analysis of paleontological time series using 217 maximum likelihood models, which we use to test whether univariate trait evolution in 218 the different climatic states is best described by stasis, directional evolution or a random 219 walk. To test whether trait evolution occurred more slowly than would be consistent 220 with genetic drift, we calculated Lynch's delta metric (Lynch 1990 Evolutionary allometries were calculated in the same way using the sample means of 236 size and shape to reconstruct the variance-covariance matrix over the entire study 237 interval (total evolutionary allometries) as well as from separate climate phases (phase-238 specific evolutionary allometries).…”

Competition between cryptic species explains variations in rates of lineage evolution

“…Lande (13) postulated that the gradual patterns could represent random genetic drift, but increasingly sophisticated statistical analyses of the Contusotrucana lineage (10) (Fig. 1), for example, indicated a significantly directional component deviating from random null models (14,15). An explanation involving the tracking of a gradually shifting optimum by the evolving lineages is equally illusory: on geological time scales, the variance in surface ocean properties is dominated by orbitally driven insolation changes with periods between 20 and 400 kyr, as was the case for the late Cretaceous habitat of the Contusotrucana lineage (16).…”