(a) Patterns of speciation events throughout the Late Cretaceous
Interestingly, although NNodes were higher than expected in the last two stages when compared to the root state or earlier stages of the Late Cretaceous (as modelled as intercept offsets), when time-dependent processes are included, then the effects are slightly negative in both Time and √Time models (electronic supplementary material). This means that when a constant rate of diversification through time is assumed (in both the Time and √Time models), the observed NNodes is lower than expected towards the end of the Cretaceous. This lower-than-expected NNodes given constant-rate time-dependent processes, does not, however, imply that there is an under-sampling of nodes towards the end of the Cretaceous, but rather that constant-rate time-dependent diversification processes are poor descriptors of the distribution of NNodes through time. The Time2 process models this pattern better (through DIC comparisons contra ) and even preserves the slight over-sampling effects of the last two stages, more reflective of the underlying data structure.
(b) Implications for diversification declines in dinosaurs
Contrary to arguments in favour of an asymptotic model of diversification in dinosaurs [13, 14] and that observed declines in diversification rates are artefacts of edge effects or incomplete fossil sampling [5, 6], our results do not support either claim. As we demonstrate above, nodes are not under-sampled proximally to the K-Pg boundary [5, 6] but they are in fact, well sampled (Fig. 1). After accounting for this slight over-sampling, we detect significant signals of diversification decline towards the K-Pg boundary (Fig. 3).
It is worth noting that while the time-square root model is used as a benchmark for an asymptote model of diversification (where the rate of NNodes accumulation slows down towards an asymptote) [7, 13], in practice the predicted values do not converge onto a net-zero diversification rate but instead on a constant net-positive rate (Fig. S1b, e). This means that the time-square root model does not represent a diversification process towards an asymptote, but instead a slower yet steady increase through time (Fig. S1b), only at a lower rate than the time-linear model (Fig. S1d, e). This is the primary reason why this model is consistently rejected in favour of the time-quadratic model (Figs. 3, S1c, f) – it fails to sufficiently model the slowdown in diversification rates (Fig. S1b).
Whether or not such modelled diversification declines can be interpreted as evidence that the dinosaur subclades were undergoing natural deflation before their final extinction 66 Myr ago is debatable. Our preferred interpretation is that the three major subclades of dinosaurs were merely diversifying at an ever-decreasing rate, considerably lower than expected for clades of their sizes and long evolutionary histories. There is no doubt that dinosaurs dominated the terrestrial ecosystems globally until they catastrophically went extinct at the K-Pg boundary; dinosaurs were hugely abundant and globally ubiquitous. A diversification decline in this context then implies that dinosaurs were nowhere near as diverse as would be expected given their long-standing dominance. This translates ecologically into a reduction in resilience whereby less diverse groups of assemblages make up the predominant faunal ecosystems (regardless of their dominance)  which are dependent on long-lasting environmental stability. Such ecosystems are then vulnerable to catastrophic disasters such as an asteroid impact .
Our finding that dinosaurs were not under-sampled in the Maastrichtian in comparison to the other Cretaceous stages, is supported by empirical palaeontological-geological data  and modelled preservation rates  but has previously been rejected by other empirical  and modelling  studies. Part of the confusion has arisen perhaps because analysts switch between global-scale and regional-scale studies. In fact, the sampled fossil record of dinosaurs in North America through the Campanian and Maastrichtian provides a large part of the global signal, with some matching data from European sites, considerable data from Argentina and Mongolia, but with very uncertain dating, and virtually nothing from Africa, Australia and the rest of Asia [18, 20, 21]. Counts of dinosaur-bearing geological formations worldwide showed that numbers reached a peak in the Maastrichtian, implying excellent opportunities to sample [16, 22]. The high numbers of dinosaur localities, collections, formations, and skeletal quality in the Campanian and Maastrichtian were also noted as evidence for good sampling and the reality of the Late Cretaceous downturn in dinosaurian diversity dynamics . Similar results emerged from modelling of dinosaurian preservation rates as part of a Bayesian study of speciation dynamics of North American dinosaurs ; preservation rates remained low (< 2 occurrences per lineage per million years) through much of the earlier Cretaceous, and then climbed to 4 in the Campanian and 7.5 in the Maastrichtian, so improving by > 400% from Santonian to Maastrichtian .
On the other hand, other authors argue that the latest Cretaceous was a time of poor sampling and that the apparent downturn in dinosaur diversity is an artefact [18, 19]. All the ‘sampling metrics’ used by Upchurch et al. (2011)  are closely correlated with each other and with dinosaurian diversity, with highs in all metrics in the latest Cretaceous; residuals of dinosaur diversity minus the geological metrics tend to flatten the face-value diversity curve because in fact the ‘sampling metrics’ are redundant with the diversity measures, and the meaning of the result is unclear . Further, Chiarenza et al.  showed that the area of exposed rocks in the western United States representing habitats suitable for dinosaurs to occupy declined through the Campanian to Maastrichtian. However, using ecological niche modelling for the whole of North America, they showed that habitats suitable for dinosaurs remained constant or increased slightly through the Campanian and Maastrichtian, and so they argued that sampling was becoming worse through those two stages and that the decline in dinosaurian diversity  was an artefact of poor sampling. The ecological niche modelling approach  makes many assumptions, not least that it can define areas where dinosaurs are absent but ought to be found, and this could be disputed in view of the limited input variables used to define ‘suitable’ habitats. Further, it is not explained  how discovering the ‘unsampled’ dinosaurs would affect the continental-scale taxon counts. Endemicity was not great in the Late Cretaceous, so the unsampled, or under sampled, dinosaurs of eastern North America would probably only add a few previously unknown species, but no new genera or new families. If the study was to estimate dinosaurian abundance or biomass, then missing fossils from such a wide area would substantially underestimate values, but if the target is biodiversity at species or genus level, it would likely make little difference, and certainly not to our phylogeny-based finding of a substantial decline in dinosaurian diversification rates over tens of millions of years. Phylogenetically, the addition of species from previously known Late Cretaceous families will not increase effective sample sizes. Adding another tyrannosaur or ankylosaur or ornithomimosaur, etc, will not drastically (or fundamentally) alter the picture of Late Cretaceous dinosaur diversity, but adding a hitherto unknown radiation of an entire clade or a hidden diversity of an ancient clade – e.g., basal theropods – might (both of which are highly unlikely for eastern North America). The majority view [7, 16, 17, 21, 22, 24, 25] is that dinosaurs, in North America at least, were remarkably well sampled through the Campanian and Maastrichtian, and showed a decline in diversity and in speciation dynamics through this time.
Finally, as SLE has historically been discussed at the level of specimens within individual rock sections, it is predicated on the fact that the stratigraphic sequence is more or less complete. That is, whether a fossil occurrence is observed at a particular horizon depends on the probability of observing a fossil occurrence given the presence of rock. Then, this sampling probability (or sampling rate) often can be over longer time intervals (e.g., over one million years) than the depositional rate of the stratigraphic sequence. In such cases, sampling will decrease towards the edge as the probability of finding a fossil will diminish with less rock to search in [1, 4].
Dinosaur discoveries over their entire evolutionary history, on the other hand, are strongly correlated with rock availability [16, 18] – dinosaur discoveries even drive the discovery of new fossiliferous formations. Thus, on a macro-evolutionary time scale over 150 + Myrs, sampling rate and depositional rate are expected to be closely linked with each other, making the SLE unlikely to be a major influencer of diversification estimates. That is not to say however, that there are no sampling effects, as we do demonstrate that NOcc is significant in our models and as dinosaur diversity is tightly correlated with the number of dinosaur-bearing formations [16, 18].
While there are many uncertainties surrounding diversification dynamics of extinct clades, what is clear is that our model of diversification does not appear to suffer from an artefact owing to the under-sampling of nodes close to the edge (i.e., SLE), but rather that there is sufficient sampling (even slight over-sampling) of nodes. Dinosaurs, except for hadrosauriforms and ceratopsids, were not diversifying as expected given their evolutionary history and sizes of their clades, with their diversification trajectories on a downward trend towards the K-Pg boundary.