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Thank you for your revisions, which address all of my previous comments. The one remaining reviewer has raised only a possible update to a reference (their suggestion was only published online last month, although the preprint has been around for a little longer). On a related note, I noticed that you are using the 2008 edition of Klenke and not the 2013 second edition (with updates to the page numbers used here). Both of these are minor enough that you could, if you wished, make the updates at the proof stage and so I will accept your manuscript now and leave these details to your discretion.
There seems to be a typo on Line 172: “…if and only if the function symmetric.” where an “is” seems to be missing and should be corrected if so.
There is also an extremely minor inconsistency that you could easily ignore where on Lines 60, 94, 97, 153, and 311 the ordering is “Foote, 2007 [and] Foote et al., 2007”, but on Line 295 this is reversed to be “Foote et al. (2007) and Foote (2007)”.
Well done on writing an interesting paper that I suspect will generate quite a bit of discussion!
[# PeerJ Staff Note - this decision was reviewed and approved by David Roberts, a PeerJ Section Editor covering this Section #]
I thank the authors for revising their manuscript, which I am happy to recommend for publication. If you have the chance, I would suggest that you replace the PyRate reference with a newer article that shows more clearly the effects of assuming a hat-shaped model of preservation on the estimation of species longevity as well as the skyline alternative models: DOI: 10.1017/pab.2019.23
Congratulations on the very interesting contribution.
Thank you for the revised version of your manuscript. I accept your point about the supplementary material.
The remaining reviewer has made some useful comments, all of which warrant your response in a revised version of your manuscript. I have also made some minor comments below. Note that in your document, the line numbers sometimes encompass several lines and so some of the references to line numbers here refer to sections of multiple lines. If you have any problems finding the text I am referring to, please just let me know.
Line 15: “averaged trajectories” might be clearer here (especially given the earlier reference using the same phrase to refer to something else) as “trajectories averaged”.
Line 159: Missing “the” in “function is THE largest symmetric function”?
Line 159: “given” rather than “biven”.
Equation 2: Should the semi-colon be a comma here?
Line 169: Missing space before the reference.
Line 186: As equation 5 is quite a bit later than this reference to it, which could lead to a reader wondering if the reference is wrongly numbered or unable to find the equation, you could 1) insert the equation here instead, 2) remove the reference to the equation, or 3) make it clear that the equation is appreciably later in the manuscript. I’d favour options 1 and 2 as they preserve the order of reference to equations.
Line 203: Perhaps “By rescaling the trajectories of a group of taxa to start AT zero and to end AT one…”
Line 203: Perhaps “…this raises A question…”
Appendix: The opening quotation mark is reversed in footnote 1.
As I mentioned in my previous review there are lots of methods that estimate macroevolutionary parameters without averaging, which I think deserve being mentioned here. In particular, most of the methods to estimate origination and extinction rates are *not* based on the procedure described in the section of the paper titled: “The Way Data are Processed”. That procedure has indeed been used to describe rise and fall of species and occasionally the rise and fall of clades. However, most methods to infer diversification dynamics do not rely on such procedure: e.g. Alroy 2014 Paleobiology, Foote 2001 Paleobiology, Silvestro et al. 2014 Syst Biol, Sakamoto et al. 2016 PNAS, Liow and Finarelli 2014 Proc B. I think the author should better acknowledge this in the paper, and clarify that the procedure they describe has been mostly used to describe abundancy or occupancy of taxa throughout their lifespan.
On line 285: the authors mention that in paleobiology “origination and extinction times of taxa are known”. With few exceptions of extremely well-preserved taxa, origination and extinction times are typically not known and several methods exist to estimate them.
I find that the point of the paragraph “Predicting Extinction Risk” is unclear. I suggest clarify how this links to the previous and following paragraphs.
Lines 36-46: While the PyRate method includes a hat-shaped model, the model is used to estimate the origination and extinction times of each taxon, and not to estimate origination and extinction rates, which are instead estimated under skyline model (i.e. not based on waxing and waning).
Both reviewers are very positive about your manuscript and have made some constructive and helpful suggestions. These, along with my comments below, should be addressed in a point-by-point fashion in your rebuttal, showing for each what you have changed in the manuscript or explaining why no changes were made for that particular point.
As commented on by Reviewer #1, some of the material in the appendices could be incorporated into the manuscript proper, and Reviewer #2’s interest in your measure of symmetry might also encourage you to move more material into the manuscript.
Line 35: You might prefer to keep the order consistent here and on Line 24 above and Lines 56–57 below (plus Lines 131, and presumably elsewhere).
Line 39: Missing space before the reference. Also on Line 80.
Line 41: While stylistic, I suggest using “for example” instead of “e.g.” here. Also Line 120 and 314 (and perhaps also 32 and 91). I’d suggest expanding “i.e.” on Lines 115 and 201 also.
Line 43: Spurious “has”?
Line 54: “through”.
Line 78: Either some words are missing or the reference style is not what was intended here.
Line 106: “which” not “what”.
Line 117: Should these two references be in the same set of parentheses or is this style intentional with the “model A”? Similar questions about Lines 148, 166, and others.
Lines 140–145: Perhaps use a numerical list rather than bullets here?
Line 170: This is stylistic, but I suggest either deleting “the” before “Brownian” or adding “process” after “motion” here. Same point on Lines 172 and 173 (and elsewhere).
Line 193: Spurious space before period.
Line 150: The opening quotes are closing quotes “''” and should be “``” in LaTeX instead. Same point for Lines 153, 156, 200 (7th line here with heading the 1st line), 228, 247, 248, 251, 266, 271, 295, and 301 (and possibly elsewhere).
Line 218, 2nd line: Missing possessive apostrophe in “a taxons life”. Same on Line 218, 3rd line.
Lines 235–236: “…in the long run…” would be more usual phrasing. Same on Line 236.
Line 262: “…but rather the…” would be the usual phrasing here.
Line 264: Double “and” here.
Line 273: A word or words appear to be missing here. Also Line 321.
Line 309: Do you mean “The result above implies…” here.
Lines 317–325: Please write this as one or more paragraphs rather than a bulleted list.
Line 324: You could swap “used” and “model” here.
Line 330: Rather than “is attached”, you could describe how it can be accessed. Same point on Line 354.
see General comments for the author
see General comments for the author
see General comments for the author
I read with interest the manuscript by Hohmann and Jarochowska, who describe how the use of diversity (or abundance) trajectories from the fossil record is affected by intrinsic biases. Essentially their findings show that regardless of the generating process, the trajectories averaged across multiple taxa are expected to show a symmetric wax and wane trend, which is therefore not necessarily informative of an underlying biological process.
I think the finding of this paper are interesting and as far as I can tell well justified mathematically. It is certainly useful to point out the limitations of using diversity trajectories, which is similar to what phylogenetists have come to realize in recent years for lineages-through-time plots. At the same time, I think the authors should do a better job at acknowledging that there are many alternative methods to make inferences about the processes underlying diversity trajectories, which are commonly used in (paleo)biology studies.
I think many studies (e.g. Quental and Marshall 2013 Science) do not actually average multiple trajectories but look at individual clades, do the concerns raised by the authors affect these studies too?
Also, while it is true that several studies used trajectories to make ecological/evolutionary inferences, many others attempt to infer the the underlying processes without looking at the trajectory per se. For example, Liow and Finarelli 2014 (Proc B) used presence/absence data to infer diversification dynamics; Silvestro et al. 2015 (PNAS) used birth-death models to estimate the processes underlying the wax and wane of different carnivore clades; Hagen et al. 2017 (Syst Biol) developed a model to estimate age dependent extinction. I think the Discussion should mention that these (and other) methods are alternative ways to test hypotheses from fossil data, going beyond the use of trajectories, and ideally discuss whether these methods solve the problems of “necessary symmetry”.
As part of better placing this manuscript in the context previous literature, I think the authors should refer to the paper by Budd and Mann 2018 (Evolution), which is discussing the “push of the past” bias derived from conditioning on clades to be large. This seems quite tightly related with the “Symmetry by conditioning” bias shown on p. 5 of this manuscript.
It is worth noting that there are many clades that do not show symmetric wax and wane, for example ammonites, mammalian carnivores, most plant clades (e.g. angiosperms families) and all clades that survived (and recovered from) a mass extinction.
I think the description of the non-parametric approach developed by the authors should be given in the main text rather than in the appendix. Right now the method is only mentioned twice (Introduction and Discussion) in the main text, and without reading the appendix it is hard to tell what it is really about.
The data displayed in Fig. 1 should be explained: what are the “trajectories of individual taxa” referring to? Are taxa species, genera, families? Do their trajectories quantify diversity or abundance through time? Additional information is available in the Appendix, but I think the reader should be given the information to understand it based on the caption and the main text.
A figure schematically showing the procedure described on p. 3 (way the data are processed) would be useful.
The word “data” is plural so I would replace “The Way Data is Processed” with “The Way Data are Processed”
Line 176 “it *is* time reversible”
Line 180: “…same, independent whether they are plotted” rephrase, e.g. “independently of whether”
Fig. 1 I think adding a plot of the actual trajectories would be very useful here. Also, I suggest replacing the large red and blue circles with dashed vertical lines.
See comments to authors.
See comments to authors.
See comments to authors.
In 2007, a few papers were published showing that various measures of the "footprint" of species and genera rise and fall symmetrically, on average, over the lifespan of taxa. The important caveat is "on average." Hohmann and Jarochowska show that several mathematical factors lead to exaggerated symmetry even if individual taxon patterns are not symmetric. It's an important result, because a number of studies following the original 2007 papers modeled individual species histories as if they were symmetric. For example, the assumption is part of the original implementation of the PyRate method for estimating evolutionary rates (Silvestro et al. 2014, Systematic Biology 63:349). But, beyond showing that symmetry may be an artifact, the authors contribute a new measure of symmetry for individual taxa. I have not tested this measure, but it seems reasonable. My reading of the manuscript is that the authors are not questioning the tendency for individual taxa to expand and contract but are arguing that symmetry may be an artifact and that processes producing the rise and fall pattern are difficult if not impossible to infer from the average trajectories. (For example, I was surprised to find in my 2007 Paleobiology paper that the average trajectory of species richness, conditioned on extinction of the higher taxon, is expected to be symmetrical even if speciation and extinction rates are not equal.)
As one of the people guilty of promoting the idea of symmetry, I'm delighted to see this paper.
As far as I can tell, what the authors present is all sound. My main concern is that it is not always clear which elements of what they present are new and which are inherent in earlier work. For example, they cite my 2014 paper for the idea that a bounded random walk on average will yield a symmetric rise and fall (and I'm sure that idea was not original with me). But isn't that exactly what their Figure 2 is showing? More broadly, I think it would put the new work in a clearer context if the authors distinguished previous claims that average trajectories rise and fall symmetrically from claims that individual taxa do so. I also think that symmetry of individual taxa may not always be a bad approximation. For example, Tietje and Kiessling (2013) showed that the time of extinction of a genus could be reasonably well predicted based on the time it reached its maximal geographic range.
But these are really small suggestions that I think will increase the impact of the paper; they do not reflect the analytical quality of the work. I think minor changes to the text are all that are needed, and I look forward to seeing this paper published.
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