Analytical results are presented in the electronic supplementary material, table S2. illustrates the effects of changing parameters in the calibration densities. The divergence time estimates calculated in both BEAST and MCMCTree using minimum and maximum constraints on every node are highly congruent, although the BEAST estimates are consistently slightly younger because hard maximum constraints were employed.
Figure 1. (a) Posterior time estimates when uniform and (f) lognormal priors are used to constrain node ages with fossil-based minima and maxima in BEAST. (b–e) Posterior mean estimates showing the impact of changing the parameters of the lognormal distribution (more ...)
When non-uniform distributions, such as lognormal or truncated Cauchy, are employed to describe the calibration density relative to the minimum constraint alone, the results show that divergence estimates are extremely sensitive to parameter choice. For instance, changing the mean or the standard deviation of the lognormal distribution can cause the mean divergence estimates, as well as lower and upper 95% posterior intervals, to differ by hundreds of millions of years (b–d; electronic supplementary material, table S2). The effect of increasing the mean of the lognormal distribution is more apparent with smaller values of the standard deviation (s). As the mean increases, the upper range of probable ages is truncated by the inferred ages of the deepest nodes within the topology and, ultimately, the maximum constraint on the root, which permits estimates from becoming unreasonably ancient. The effect of increasing the standard deviation of the lognormal distribution is that the mode of the distribution is shifted towards the minimum bound and exhibits increased kurtosis; younger ages and narrower intervals result.
Similar outcomes are achieved when permuting the parameters of the truncated Cauchy distribution, applied to the minimum bounds in MCMCTree
]. When the location parameter (p
) is increased, the peak in the prior is shifted away from the minimum constraint towards older ages, and when the scale parameter (c
) is increased, the prior distribution becomes flatter and the 95% limit of the soft maximum constraint becomes older. Increasing both parameters tends to produce older and more diffuse estimates until they are constrained by the prior at the root. Evidently, the results are sensitive to the maximum constraint placed at the root. In representing the oldest possible divergence time in the tree, this constraint dictates the maximum envelope of time during which all speciation events can occur. When increasingly diffuse arbitrary densities are applied, divergence times can only become as ancient as this constraint will allow.
When the prior density is constrained by both minimum and maximum constraints, rather than just minimum constraints, analyses that employ a lognormal distribution in BEAST, a skew-t distribution in MCMCTree or a uniform distribution in either program, calculate posterior estimates and 95% highest posterior density intervals that are closely comparable (a,f; electronic supplementary material, table S2).
Analyses without molecular data show that, regardless of which prior density function is employed, the specified calibration densities are not those implemented in the estimation of divergence times. For example, the initial uniform densities specified for Apocrita (node 9) and Aculeata (node 10) are transformed into non-uniform effective priors (a) while the common uniform initial prior for Lepidoptera–Diptera (node 12), Diptera (node 13) and Drosophila–Mayetiola (node 14) is transformed into three distinct non-uniform effective priors (b).
Figure 2. The effect of truncation in the establishment of the joint time prior. The dashed line represents the user-specified uniform prior, the red line represents the effective priors, and the solid black lines represent the marginal posteriors. (a) Interaction (more ...)