Underlying all four models is the assumption that changes in body size can be modeled by a continuous random walk (Brownian motion). Under this process, change is continuous, reversals in direction are frequent, the expected rate of change is the same at all stages and in all lineages, and evolution is unbounded (Felsenstein, 1985). Different theories of biological evolution assume different relationships between the amount of time that passes and the amount of evolutionary change that occurs. The gradual model (Fig. 1A) is the only one that conforms precisely to Brownian motion: the squared differences in body size between any two species should be proportional to the time separating them. For example, closelyrelated species should have similar sizes. This is the pattern expected under conditions of evolution by genetic drift (Lande, 1976; Lynch, 1990). However, natural selection. can lead to a similar outcome if selection pressures vary unpredictably (Felsenstein, 1981).
The other three models are made to correspond to a Brownian motion process by adjusting relative branch lengths on the phylogenetic tree, essentially manipulating time. Under the speciational model (Fig. 1B), the squared differences between speciesspecific body size should be proportional to the number of speciation events that separate them, irrespective of the time elapsed. Under this view (for examples, see Rohlf et al., 1990; Harvey & Purvis, 1991: box 2), change in trait values occurs rapidly at, or shortly after, speciation, and stasis follows until the next speciation event. This pattern is associated with the view that speciation is somehow necessary for change to occur (Eldredge & Gould, 1972; Gould & Eldredge, 1993) and may arise for many reasons – for example, if speciation is associated with niche shifts, which may happen commonly in adaptive radiations. This model is a common default setting for trees used in comparative studies when there is no branch length information.
Under the third model (the pitchfork model, Fig. 1C) squared differences are unrelated to time or speciation events. Closelyrelated species are no more likely to be similar in size than any two species picked at random. We can test it using the Brownian motion process by setting all internode distances to zero, and setting all terminal branches to unit length, creating a star, or pitchfork phylogeny. Success of this scenario would suggest that phylogenetic history is of no importance to the evolution of body size. Another interpretation of this model is that estimated topology used is very wrong: removing any internal structure (making a pitchfork)might then actually be a better representation of the true phylogeny than that used in the other models.
The final model tested (the free model, Fig. 1D) is qualitatively different from the other three, and can be viewed as the null model. Here the topology of the tree is kept fixed, but we allow each branch length to vary freely until the most likely set of branch lengths is found, given the body sizes of all the species in the clade. This set of branch lengths produces the best fit to the Brownian motion process, effectively allowing body size to evolve at any number of rates. The length of each branch is the best descriptor of how size differences among species actually accumulated in that interval. A significant improvement under this model would suggest that the simpler models presented do not reflect the true pattern of bodysize macroevolution. A drawback of this fourth model is that it is too unconstrained: it is hard to imagine that every branch (internode) requires a brandnew parameter governing change. Thus the degrees of freedom (number of parameters to estimate) are inflated, making simpler models harder to reject.