Our optimal phylogenetic reconstruction (T_Optimal; AIC_c = 414160.2536) is a significantly better fit to the mtDNA matrix than a recent consensus topology derived from the literature (T_Consensus; AIC_c = 421460.9166; see the methods section and Figure 2). Nevertheless, the two topologies agree in many instances. For example, several traditional orders identified as having little support for monophyly (e.g. Coraciiformes, Ciconiiformes, Caprimulgiformes and Falconiformes 53) were also polyphyletic in our analyses. However, the two trees also differ in many respects, most notably in the placement of Passeriformes. In T_Consensus, the clade is relatively derived in the tree, whereas in T_Optimal it forms the basal-most clade in Neoaves. Several traditionally hard-to-classify lineages (e.g. Pteroclidae, Opisthocomidae, Phaethontidae, Podargidae and Steatornithidae) are of suspect placement in T_Optimal. These and other uncertainties tend to be localized and do not (as we report below) overly influence date estimates for the basal nodes in the avian tree. Some of the topological differences, however, are of operational importance, as they cause either redundancy or obsolescence of some fossil constraints used in estimating divergence times. Overall, of the 18 total internal fossil calibrations considered, 16 were used on T_Consensus, and 17 on T_Optimal (Figure 2).

A substantial signal was present for both a departure from a molecular clock and a lack of ancestor-descendant autocorrelation of substitution rates. Using penalized likelihood in r8s, both topologies T_Consensus and T_Optimal were found to be unclock-like, with optimal smoothing values (log_10λ) of 1.0 and 0.4, respectively. Analyses in Dating5 clearly rejected the constant-rate Poisson model (-ln L = 63214.8; χ² = 8051.61, df = 271, p = 0.000) but could not reject the stationary (high variance) rate model (-ln L = 2123.01; χ² = 268.352, df = 269, p = 0.482) which produced a large index of dispersion R = 469.782. Bayesian analyses in Multidivtime delivered positive but very small values for the degree of autocorrelation of substitution rates across both topologies (Table 1). Finally, analyses of T_Consensus using BEAST indicated that non-autocorrelated models of rate variation fit the data significantly better than a molecular clock (Table 2). Of the non-autocorrelated models, the lognormal distribution (UCLN) had a much better harmonic mean model likelihood than the exponential distribution (UCED), and relaxation (T_Flexible) of a fixed topology further increased fit. Using each of these uncorrelated models, the covariance of substitution rates between ancestor and descendent branches across the tree was not significantly different from zero.

Given the extensive phylogenetic uncertainty within Neornithes, we focus on divergence times of clades for which monophyly is not contentious (Table 3). Two key trends are recognized. First, for a given dating method, differences in topology tended to have a minor but systematic influence on inferred ages. In general, T_Optimal delivered older average date estimates than T_Consensus using r8s (8.9 MY) and Multidivtime (3.6 MY), but the opposite trend was found with PATHd8 (-8.2 MY). When confidence/credibility intervals are considered, however, topology did not significantly influence most individual date estimates. Overall, in terms of the degree of discrepancy between fossil and molecular dates on a whole-tree basis (average δ_{Realized MA-FA}), topology had a noticeable (> 5 MY) influence on divergence estimates for only the PATHd8 analyses (Table 3).

Second, the choice of the relaxed clock method had a strong influence on inferred ages. R8s, Multidivtime and BEAST tended to deliver similar estimates for most clades of interest (Table 3). In contrast, PATHd8 generated considerably younger dates with much smaller confidence intervals, despite using the same bootstrapped phylograms and fossil constraints as r8s. Dating5 tended to produce the most extreme results, with inferred basal split estimates similar to those from Multidivtime, but some derived split estimates younger than those from PATHd8. Most significantly, PATHd8 and Dating5 together identified five of these major clades as having crown diversification restricted to the Cenozoic (Ratites, Charadriiformes, Procellariiformes, Cuculiformes and Apodiformes), although the remaining methods generate estimates for these same nodes that are on average more than 50% older. In terms of comparing molecular and fossil age estimates (average δ_{Realized MA-FA}), r8s, Multidivtime and BEAST all show considerable discordance between the two sources of data, with the average molecular estimates for the major nodes (Table 3) being 36–45 MY older than corresponding fossil ages. PATHd8 and Dating5, in contrast, exhibit greater agreement between estimates from 'rocks' and 'clocks', with an average discrepancy of 17–25 MY.

Despite these differences, all methods agree that the basal splits within Neornithes occurred deep within the Cretaceous (Table 3, nodes A-E). The youngest estimate for the root of Neornithes (PATHd8, T_Optimal) is of Early Cretaceous age, 37 MY older than the oldest undisputed neornithean fossil 10. Conflict among methods instead involves the diversification of derived lineages (Figures 3 and 4). Two patterns can be discerned. First, PATHd8 and Dating5 support bursts of speciation (many lineages arising almost simultaneously), whereas the remaining methods generally support more gradual diversification. Second, and more germane to the 'rock-clock' problem, PATHd8 alone supports an extensive post-K-Pg radiation of Neoaves. For example, from results of the non-autocorrelated rate models in BEAST allowing topological uncertainty (T_Flexible; see Figure 4), not only are the basal splits inferred to have occurred in the Cretaceous, but 37 credibility intervals (green bars) do not overlap the K-Pg boundary. Finally, no support is shown for episodic evolution, either correlated with speciation events (52; no effect) or an increase in inferred substitution rate either during early diversification or following the K-Pg mass extinction (Figure 5).

Whether using fossil or molecular data, a daunting impediment to divergence time estimation in birds is that resolution of many inter-ordinal phylogenetic relationships has proven obstinate, despite large data matrices with multiple character types 53. Although our reconstruction T_Optimal is overly optimistic in being fully resolved, it provides a useful alternative to the conservative T_Consensus (Figure 2).

T_Optimal recovers several traditional orders as polyphyletic (Caprimulgiformes, Coraciiformes, Falconiformes, Ciconiiformes), consistent with expectations 53 (but see 54). Although T_Optimal has caprimulgiform (nightbirds) families much more distantly related to one another than previous morphological 55 and molecular 56 investigations, differences in taxon sampling confounds direct comparison across studies. While Coraciiformes (kingfishers and relatives) is not found to be monophyletic, the two recovered sub-groupings both fall within a larger clade containing owls (Strigiformes), parrots (Psittaciformes) and woodpeckers and relatives (Piciformes). The monophyletic status of the order Falconiformes has received mixed support in previous analyses 545758596061. Placement of Falconidae in T_Optimal is suspect and likely stems from insufficient taxon sampling from this family 62. Regardless, no support was found for hypotheses uniting falconiform taxa with owls (Strigiformes) 54 or New World vultures (Cathartidae) with storks (Ciconiiformes) 61.

Several monotypic avian families have traditionally proved difficult to classify. The enigmatic hoatzin (Opisthocomidae) has variously been allied with at least four distantly related orders (Galliformes, Cuculiformes, Musophagiformes and Columbiformes; see 6163). We find here an alliance with doves (Columbiformes), similar to joint analyses of mitochondrial and nuclear DNA sequences 63. The taxonomically problematic sandgrouse (family Pteroclidae) has alternatively been considered a member of Charadriiformes (shorebirds and allies 6164) or Columbiformes 545760. Our reconstruction has the sandgrouse distantly related to both orders, and instead allied with Falconiformes. This relationship is unsupported elsewhere and we have little confidence in this placement. The novel placement of the tropicbird (family Phaethontidae) as sister to Sphenisciformes is similarly suspect.

Finally, we find no support in our mtDNA analyses for the neoavian clades 'Metaves' and 'Coronaves' identified from nuclear β-fibrinogen intron analyses 58, although our constraint tree allowed for this possibility (T_Constraint; see additional file 1). A major difference between these trees involves the phylogenetic position of the perching birds (Passeriformes); while nuclear DNA analyses recover Passeriformes as a relatively derived clade within 'Coronaves' 5758, in T_Optimal they instead comprise the basal-most lineage of Neoaves. This may be indicative of different phylogenetic signals in nuclear versus mtDNA sequences, as other mtDNA studies have also inferred a basal phylogenetic position of Passeriformes in Neoaves 65.

While T_Optimal yields interesting hypotheses about avian relationships, the focus of this study is on estimating basal divergence times in Neornithes and we might regard topology as a nuisance parameter (and explicitly so in the BEAST T_Flexible analyses). Topological error is usually not considered in divergence time estimation, but potentially could systematically bias age estimates through: (i) incorrect placement of fossil calibrations; and (ii) improper estimation of branch lengths. Through our joint consideration of T_Consensus and T_Optimal, we find that topology does have a systematic influence on inferred divergence times for nodes of interest (Table 3), but that for the present data set this influence differed in direction across methods and was generally insignificant when confidence/credibility intervals were considered. Removal of the restriction of a fixed topology in BEAST (T_Flexible) yielded age estimates similar to those from Multidivtime analyses assuming T_Optimal. Although yielding diffuse estimates, this 'relaxed topology' approach better reflects uncertainty in the underlying data.

An interesting result reported here is that rates of molecular evolution are found to be non-autocorrelated across the Neornithes tree (Tables 1 and 2), a result previously noted for virus and marsupial data sets 47. An autocorrelation of rates would imply an inheritance of the trait 'rate of evolution'. This makes intuitive sense when considering that ancestor and descendant lineages are likely similar in body size, generation time, DNA repair efficiency, population size and other traits influencing rates of sequence evolution, and the most popular molecular dating methods available indeed implicitly assume that rates are autocorrelated across a tree 6667. However, even if 'rate of evolution' is heritable, nodes separated by long periods of time may accumulate sufficient rate variation that autocorrelation in this trait will break down. Alternatively, 'rate of evolution' may simply be more labile than we expect. Regardless, violation of an implicit autocorrelation assumption did not have significant effects on inferred dates for nodes of interest (Table 3). For example, r8s and Multidivtime, which each deal with rate variation in an ancestor-descendant fashion, deliver age estimates that match quite closely to those generated by BEAST, which does not make such an assumption.

All methods employed here agree that the basal divergences within Neornithes occurred in the Cretaceous (Table 3, nodes A-E), supporting the refutation of a Cenozoic origin of modern lineages 89 mandated by the discovery of the 66 MY duck Vegavis iaai 10, which minimally forces five basal divergences into the Cretaceous. Moreover, our results are not dependent on this oldest fossil calibration, as analyses in r8s, PATHd8 and Multidivtime without using the Vegavis constraint returned nearly identical results to those reported here (data not shown); indeed, we must paradoxically conclude that this oldest undisputed neornithean fossil was essentially uninformative in our molecular dating analyses. Given the consensus across 'relaxed clock' methods employing very different assumptions about how molecular substitution rate evolves, we regard an Early Cretaceous origin of Neornithes as robustly supported. This inferred Cretaceous origin, and consequent survival of several avian lineages across the K-Pg boundary 68, is consistent with previous molecular studies 41617181920212223242526 and is supported by historical biogeography reconstructions 69.

An explanation occasionally offered for the antiquity of molecular dates is that rates of sequence change may speed up during radiations 33, consistent with a basic tenet of punctuated equilibrium theory where most character change is concomitant with speciation 70, possibly resulting from stochastic effects during founder effect speciation 5271. If true, an elevated number of molecular substitutions recorded during diversification could erroneously be interpreted as a longer time span at a slower background rate, resulting in a systematic overestimation of divergence times for all nodes predating the radiation. Some evidence exists for a correlation between speciation and character evolution 52727374, although a study of island radiations failed to detect such an effect 75. While punctuated molecular evolution may be less frequent in animals (18% of cases versus 44% and 60% in plants and fungi, respectively 52), this effect is nevertheless a prime candidate to explain the strong and persistent discrepancy between molecular- and fossil-based divergence estimates. However, we find no signal for punctuated (speciational) molecular evolution 52 in the present data set. In addition, we fail to detect an accelerated rate associated with either the K-Pg boundary or during the initial diversification of Neornithes (Figure 5). If rates of sequence change were strongly influenced by diversification, we would expect clear departures from the inferred mean standardized substitution rate 76. Although Cenozoic rates tend to be more variable than Mesozoic (ancestral) rates, we find no evidence for an obvious acceleration in substitution rate associated with any of the major nodes for any genetic partition. Although these two approaches to identifying episodic evolution would ideally involve more comprehensive taxon sampling, if punctuated evolution were primarily responsible for inflating inferred molecular dates then we likely would have detected the effect with the present data matrix.

Rather than narrowing the formidable 'rock-clock' gap through application of methods designed to minimize biases and accommodate uncertainty, we have instead considerably reinforced it. In part, the discordant age estimates can be explained through reference to the genuine time-lag (δ_{True MA-FA}; see Figure 1) between the divergence of genetic lineages (predating speciation) and the evolution of diagnostic morphological characters (postdating speciation). However, given the estimated magnitude of δ_{Realized MA-FA} (Table 3), it is unlikely that δ_{True MA-FA} alone explains the dissonance. One the one hand, while the fossil record may be adequate for many evolutionary questions 28, there are clear instances where it is not. The 66 MY Vegavis iaai 10, for example, requires the coexistence of Paleognathae representatives; however, Cretaceous paleognaths are unknown. This may be a result of a geographical bias in fossil sampling favouring the northern hemisphere 21769777879, as many taxa are hypothesized as having southern hemisphere (Gondwana) origins 69. Although fossil gap analysis implies that a Cretaceous origin of several avian orders is unlikely 11, this method improperly assumes that fossils are randomly distributed (uniformly recovered through time) since the origin of a taxon; alternative fossil recovery curves can support very different scenarios, including scenarios that are more consistent with molecular genetic timelines 80, even when the fossil record is particularly sparse 31. Although rightly considered with caution, the increasing number of fragmentary remains of putative neornithean lineages from the Cretaceous 78 lends credence to the ancient origin and diversification of Neornithes. On the other hand, there may yet be some unrecognized biases in the analysis of molecular genetic sequences, and our results suggest new directions for future avian divergence time studies (described below).

Despite broad agreement on the antiquity of basal divergences in Neornithes, arbitration among macroevolutionary models 1234 to best describe the history of more derived lineages is complicated by disparity amongst various 'relaxed clock' results. Two important points of distinction can be recognized (Figures 3 and 4). First, Dating5 (overdispersed clock) and PATHd8 (rate smoothing across sister lineages) both infer bursts of speciation across the avian tree, while r8s (smoothing in an ancestor-descendant fashion), Multidivtime (Bayesian modelling of ancestor-descendant autocorrelated rate evolution) and BEAST (Bayesian modelling of rate evolution without an autocorrelation assumption or fixed topology) instead infer a more gradual diversification of Neornithes. Second, PATHd8 alone supports a prominent radiation of avian families in the Cenozoic, a scenario statistically rejected in many instances by the remaining four analyses. Although published PATHd8 divergence time estimates for Neoaves using nuclear DNA produced similarly young estimates 57, a reanalysis of these same data using Multidivtime roundly refuted the findings 16, echoing the incongruence of PATHd8 reported here. While the better reconciliation between molecular and fossil age estimates that PATHd8 offers seems satisfying at first, the unique discrepancy of this method from the other much more rigorous and biologically realistic methods raises concern.

Given the apparent lack of autocorrelation of substitution rates, together with the intrinsic topological uncertainty in the Neornithes tree, the analyses in BEAST best reflect our current understanding of early avian evolution (Figure 4). Without the restrictive assumptions inherent in other 'relaxed clock' methods, these analyses unambiguously support a Cretaceous origin and diversification of basal avian lineages. Even when considering large inferred credibility intervals, 37 early avian divergences are restricted to the Cretaceous, similar to that found through the analysis of nuclear DNA sequences 16. It should be noted, however, that these nodes mostly represent order- and superfamily-level divergences; the majority of families sampled here (80 of 100 in BEAST) have their origins either overlapping or restricted to the Paleogene, consistent with interpretations from the fossil record 77. In this respect, our results are similar to those inferred from a comprehensive study of the tempo of early mammalian evolution 81. The results of both studies indicate that significant cladogenesis occurred in the Cretaceous, but that many crown groups diversified in the Cenozoic.

While there is a growing consensus for the Cretaceous origin of many Neornithes orders and families, the rate and timing of their diversification (and the influence of the K-Pg mass extinctions upon that diversification) is not yet resolved. MtDNA may have further utility in making progress on the problem, as our analysis of posterior credibility interval widths indicates that longer sequences would likely improve divergence time estimates (Figure 6). However, we recognize that mtDNA is limited in that all sites belong to a single 'super-locus', and so a significant reduction of uncertainty (e.g. the slope in Figure 6) will ultimately require the supplement of multiple independent nuclear loci. In addition, while the present study was focused at the family level, increased taxon sampling will break up long branches (benefiting phylogenetic reconstruction) and improve branch length estimates. Nevertheless, our results suggest fertile ground for future molecular research into this problem. For example, we find that: (i) variation in the number of substitutions across branches can be accommodated by a high variance stationary-rate model 82; and (ii) rates are not autocorrelated across the avian tree in an ancestor-descendent fashion. This suggests potential for development of a hybrid model that accommodates both pieces of information and that future studies test assumptions of rate autocorrelation before invoking them, at least for the tree depth that we consider here.

In regards to the 'rock-clock' debate 3349, we feel that much of the perceived dissonance between fossil- and molecular-based age estimates stems from a comparison of different phylogenetic entities: molecular phylogeneticists generally deal with stem-group origins, while paleontologists generally concentrate on crown-group diversification 4. Moreover, it is too rarely emphasized that when dating the same node a genuine discrepancy is expected, as coalescent times (T_gene; see Figure 1) will predate cladogenesis while morphological diversification (T_morphology; see Figure 1) will postdate cladogenesis. The reality is that in normal practice neither group directly addresses the main parameter of interest, the timing of speciation (T_species; see Figure 1), the pattern of which is essential to the understanding of the origins of biodiversity. However, tools do exist to better estimate speciation times from both fossils (e.g. accounting for 'ghost' lineages 31) and genetic data (e.g. explicitly modelling ancestral effective population sizes to account for coalescent times 83). Further, molecular methods can be augmented with greater information from the fossil record, possibly by incorporating models of preservation bias into temporal constraints 31. Newly developed methods exist where probability distributions can be applied to calibrated nodes in a Bayesian framework 478485. Although we recognize that this approach is superior in that it can lend more credence to the fossil record than standard minimum-age constraints, we refrained from using such methods here as there is currently insufficient information with which to specify meaningful prior distributions for most avian diversification times. Realization of such distributions will require more communication between paleontologists and molecular phylogeneticists 8687.

The molecular data set comprises 135 avian species and 100 traditionally recognized families, as well as five outgroup taxa (turtles, n = 3; crocodilians, n = 2; see Additional file 2). The names for avian taxa shown in our figures and tables generally follow Gill and Wright 88. Turtles were used solely to root phylogenies and were not used for dating purposes. For each taxon a total of 5248 base pairs (bp) of mitochondrial DNA (mtDNA) was either sequenced directly using primers reported by or modified from 89 or downloaded from GenBank. mtDNAs from taxa newly collected by us and first reported here (GenBank accession numbers EU166921–EU167086, EU372666–EU372688, EU391159). Protein-coding genes were aligned at the amino acid level, while RNA genes were aligned as nucleotides using secondary structure models following 90. From the original matrix, 654 alignment positions could not be aligned unambiguously and so were excluded from subsequent analyses, yielding a final matrix of 4594 bp (Table 4).

We include as many fossil calibration points as possible in disparate parts of the avian tree to maximize information from the fossil record, and reduce dependency on any one calibration estimate. Given that serious bias can result if even a single fossil has been misdiagnosed in its taxonomic affinity, we limit our calibration points to those fossils that have been rigorously analyzed in a cladistic context (Table 5). Our fossil calibrations are nearly identical to those used by Brown et al 16, which is a modification of the fossil complement used by Ericson et al 57. These internal avian calibration points are all implemented as minimum age constraints in the various dating analyses. We also use two bounded external calibrations derived from the fossil record that date the caiman-alligator (71-66 MY) and crocodile-bird (251-243 MY) splits 91. This last calibration has recently been independently supported by molecular data using soft calibration bounds 92.

Inferring dates on an incorrect tree topology will lead to estimates that are suspect, if not wholly incorrect. We seek to accommodate the existing uncertainty about avian phylogenetic relationships by dating nodes on two alternative trees. The first topology considered is taken from Figure 27.10 of Cracraft et al 53, which is a consensus tree based on previous molecular- and morphology-based phylogenetic reconstructions. This tree is conservative in that all represented branching events are well supported by independent lines of evidence; uncertain relationships among lineages are presented as hard polytomies. This topology is referred to as T_Consensus. The second topology considered was derived from a partitioned-model maximum likelihood search using the program RAxML-VI-HPC 2.2.3 93. The data were divided into four partitions: first, second and third codon positions of mitochondrial cytochrome-b, ND1 and ND2 genes, and concatenated mitochondrial 12S rDNA and tRNA genes (L, I, Q, M, W, A, N, C, Y). A collapsed consensus tree derived from Cracraft et al (thick branches only of Figure 27.10 in 53) was used as a backbone constraint to make tree searches more efficient (T_Constraint; see Additional file 1). Several hundred heuristic searches were performed under the GTRMIX model assuming a range of values for both the initial rearrangement setting (i; range 5–20) and number of rate categories (c; range 5–25). The topology of the maximum likelihood estimate (MLE) is referred to as T_Optimal.

For the non-Bayesian dating methods, we accommodate uncertainty in branch length estimation through a non-parametric bootstrapping procedure. For each original data partition, 100 pseudoreplicate datasets were generated using the program SEQBOOT of the PHYLIP 3.6 package 94; these bootstrapped matrices were concatenated to produce 100 pseudomatrices with the same size and number of partition-specific characters as the original matrix. For T_Optimal, branch lengths and substitution model parameters were estimated using a partitioned GTR+G model in RAxML. For T_Consensus, branch lengths and all substitution model parameters were estimated from each bootstrap matrix on the fixed topology using the GTR+I+G substitution model in PAUP* 95 because RAxML cannot evaluate a tree containing mutlifurcations. Using these procedures we generated 100 trees with branch lengths (phylograms) for each topology.

Owing to the concern that assumptions of particular analytical methods may systematically bias divergence time estimates, we compare several methods for accommodating among-lineage rate heterogeneity for the purpose of estimating the divergence times of modern avian lineages. To facilitate direct comparison, all methods utilize the same fossil calibrations and tree topologies outlined above.

First, the program r8s 1.7 66 was used to estimate rates and dates for internal nodes in the phylogeny via penalized likelihood (PL). This semiparametric procedure combines a parametric model that allows for different rates on each branch in the tree 96 with a non-parametric penalty function that penalizes rates that change too quickly along the tree from ancestor to descendent branches 97; a smoothing parameter (λ) determines the relative contribution of the penalty function. The optimal smoothing value was determined individually for each topology through a sequence-based cross-validation procedure 96 using penalty = add and the normalized (χ²) errors. Confirmation of the optimal smoothing values was obtained via multiple optimizations with different initial conditions (by setting num_restarts = num_time_guesses = 3). Confidence intervals for node ages were determined using the distribution of estimated ages across bootstrapped trees assuming the optimal smoothing value from the original phylograms. Summary of the mean estimate and associated error for a given node was accomplished using the profile command in r8s.

Second, the program PATHd8 9899 also smoothes rates across the tree, but does so by calculating an average path length from an internal node to all its descending terminals; deviations from a molecular clock are corrected through reference to fossil calibrations. Important distinctions from r8s above are that PATHd8 smoothes rates between sister groups (rather than from ancestor to descendent nodes) and it does this locally rather than over the entire tree. The same 100 phylograms as analyzed with r8s above were used to generate confidence intervals on divergence times, although summary statistics were calculated manually.

As a third approach, the Bayesian MULTIDISTRIBUTE package 100 represents an attempt to explicitly model rate change across a tree 67101102. Here, the logarithm of the rate at the end of a branch is modelled with a normal distribution, the mean of which has an expected value equal to the rate at the beginning of the branch; thus, rates are implicitly assumed to be autocorrelated from ancestor to descendent nodes, although this autocorrelation may decay with increasing branch lengths. The posterior probability distribution of divergence times is approximated by samples from a Markov chain Monte Carlo (MCMC) chain. The data were partitioned as noted above. For each partition, estimates of the transition/transversion rate ratio and the gamma site class-specific rates under the F84+G model (the most complex model supported by the MULTIDISTRIBUTE package) were calculated in the baseml program of the PAML 3.15 package 103. The output from baseml was used as the input for the MULTIDISTRIBUTE program estbranches, which produced MLEs of branch lengths and their approximate variance-covariance matrix. Although differing in branching order, T_Consensus and T_Optimal had similar overall tree lengths; as a result, the same priors were applied to both topologies in the program Multidivtime: rtrate = rtratesd = 0.012, and brownmean = brownmeansd = 0.01. As one of our external calibration points bounds the root node (crocodile-bird split at 251-243 MY), date priors were less important and were defined narrowly (bigtime = 255 MY, rttm = 247 MY, rttmsd = 1.5 MY). The program was run without the assumption of correlated changes in rates across data partitions. Following a burnin of 10⁶ samples, 10⁴ samples were taken at a sampling interval of 10². All analyses were repeated with different inseed values to check for convergence of the MCMC chain.

Fourth, the overdispersed clock method of Cutler 82 represents a strong departure from other approaches to dating in the way it models branch length heterogeneity. Instead of treating a variable number of substitutions across lineages as indicative of changes in substitution rate across the tree, Cutler's method assumes that rate is stationary, but with high variance. Thus, rather than assuming that the number of substitutions along a lineage is Poisson distributed (where the variance is equal to the mean), the observed variation across branches is accommodated by the larger variance afforded through a Gaussian distribution. As a result, branches with either particularly high or low numbers of substitutions need not be clustered on the tree; in other words, heritability (phylogeny) plays no role in the observed numbers of substitutions. The program Dating5 104 calculates χ² statistics for both the constant-rate Poisson and stationary models and compares the overall fit of the models using a likelihood ratio test. In addition, the program calculates R, the index of dispersion (the ratio of the variance to the mean number of substitutions) under the stationary model. Dating was restricted to T_Optimal as the current version of Dating5 does not allow for polytomies. In addition, we could not achieve likelihood convergence with the partitioned data, and so reported results are from concatenated sequences. From asymptotic likelihood theory, 95% confidence intervals were calculated using a threshold of 2 log likelihood units around the MLE.

Finally, the program BEAST 1.4.6 105 differs in two important ways from the dating methods listed above. First, it does not require a fixed topology; rather, branch lengths, topology, substitution model parameters and dates can be estimated simultaneously. This incorporation of uncertainty in topology may be particularly important for the present data set, where relationships amongst many clades are uncertain. Second, BEAST does not assume that substitution rates are necessarily autocorrelated across the tree. Although intuitively satisfying, autocorrelation of rates has not been demonstrated in the literature; in fact, Drummond et al 47 report that many empirical data sets do not exhibit significant autocorrelation of rates.

BEAST 1.4.6 offers two statistical distributions for describing the change in rate across a branch 47: rates can be drawn independently from either an exponential distribution (UCED) or a lognormal distribution (UCLN). To find which distribution fit the present data best, we initially fixed the tree topology to T_Consensus. The data were partitioned as above, with each partition assigned a GTR+I+G substitution model. BEAST MCMC runs of 25 × 10⁶ generations following a burnin of 10⁵ generations were performed for UCED, UCLN and CLOCK models. To arbitrate between models, we calculated Bayes factors by comparing harmonic mean model likelihoods 106. For both non-autocorrelated models, we also calculated the covariance among branch rates, which indicates the degree of ancestor-descendant autocorrelation of rates across the tree. Using the optimal model, we then accommodated topological uncertainty by removing the restriction of a fixed tree. However, we did constrain certain clades (the backbone constraints described above) to be monophyletic to facilitate the placement of calibration points and increase the efficiency of the MCMC search. Three replicate runs of 25 × 10⁶ generations were performed to check for convergence of the MCMC chain. Mean parameter estimates and 95% highest posterior densities (HPDs) were determined through analyzing the combined BEAST tree files in TreeAnnotator 1.4.6 107. We refer to these results with the topology T_Flexible.

For each of the five dating methods above we calculated the average discrepancy between molecular (MA) and fossil (FA) estimated ages for those nodes that had fossil calibrations. The MA used in these calculations was the mean estimate from MCMC samples (Multidivtime, BEAST), or the optimal estimate from the empirical data matrix (r8s, PATHd8, Dating5). The value MA-FA is equivalent to δ_{Realized MA-FA} described above, and gives an indication of the degree of agreement between the molecular data and the fossil record.

If present, episodic (or punctuated) molecular evolution could seriously bias molecular genetic estimates of divergence time by systematically overestimating the ages of all nodes that preceded the punctuation. We therefore tested for the presence of episodic evolution in two ways. First, we used the method of Pagel et al 52108 which quantifies the proportional contribution of punctuated (β) and gradual (g) evolution to path lengths in a phylogeny, based on extent of association between sequence change and cladogenesis events. This method requires a fully bifurcating tree, and so analyses were limited to our optimal reconstruction T_Optimal. To test for this signature we analyzed maximum likelihood trees from RAxML bootstrap analyses (n = 100). Second, to test whether substitution rate accelerated coincident with or following the K-Pg boundary we plotted standardized inferred substitution rate (per data partition) versus inferred divergence time from the Multidivtime analyses above. If the K-Pg boundary extinctions facilitated diversification of avian higher-level taxa, it could produce a spike in this plot 76 detected as a departure from the mean standardized rate. These two tests are complementary in that the first focuses specifically on the effects of speciation, whereas the second focuses on absolute time effects, which may or may not be related to speciation.

As in the case of episodic evolution, a simple lack of historical signal present in molecular sequences could generate erroneous divergence time estimates. We therefore assessed the 'information content' present in our mtDNA matrix by regressing posterior 95% credibility width against posterior mean divergence time. When the when the amount of data is saturated then this regression will produce a linear relationship (i.e. R² → 1), the slope of which indicates the amount of uncertainty that cannot be reduced through adding longer sequences 785, but can be reduced through adding independent loci.