This study proposes a novel method to characterize trait distributions in ecological communities, and uses simulations to test the suitability of the method for distinguishing different community assembly scenarios. Their method, based on a mathematical relationship between the skewness and the kurtosis of distributions, aims to characterize a common shape of trait distributions across a set of communities. They compare the simulation outputs among themselves, and find that the method allows for a clearer distinction between scenarios than other known methods, in particular when accounting for stochastic processes. They also compare the simulation results to previous empirical work on global dryland systems (published in 2017 by the lead author), and conclude that a common generalizable assembly rule akin to the “disruptive selection” scenario can be found in these systems.

This study is well-presented and clearly written, and I enjoyed reading it. The case made by the authors for the need to account for the mathematical relationship between the 3rd and 4th moments of distributions is very compelling, and the possibility to classify and locate different types of distribution in a 2 dimensional space (cf. Box 1) is an exciting prospect. It is quite possible that there is indeed something very interesting to explore here, and I would be happy to be convinced.
However, as it stands, there are number of points, both methodological and related to the interpretation of results, which leave me puzzled as to the usefulness and applicability of the method. The discussion, and perhaps also the simulations, still lack some important points to provide a clearly applicable framework for the method. I detail my main concerns below.

1. Species pool effects are not taken into account or even discussed. Given the importance of species pool definitions in determining the result of simulations or other null expectations, I expected to find a discussion (or even a test !) of the influence of the species pool on the outcome of simulations. In particular, I had to wonder if particularities of the empirical species pool may be one of the explanations behind the “mismatch” in coefficients with the empirical model on drylands. The R code (greatly appreciated!) provides different options for the trait distribution in the species pool, but only results for uniform distributions are presented. This is too bad since uniform trait distribution across species pools are not common (or even likely?). How would a gaussian, log normal trait or even multimodal distribution affect the results? Moreover, how does the definition of the species pool (geograhic range vs. metacommunity) affect the expectations?

2. The comparison with empirical data is unconvincing, as it relies solely on an informal comparison of numbers in the text (l.383-384). How can we judge the extent to which the two equations, (K = 1.3*S^2 +1.3) vs (K = 1.S^2 +2), are different? Given the extensive simulation framework provided, I would expect some statistical test of the empirical data against the different simulation scenarios (e.g. using the original species trait pool and community richness). Moreover, the empirical equation discussed in the text is given with too little context, omitting the fact that it was fitted for one trait (SLA), but that another equation was found for another trait (height, with a higher intercept value). A large part of the discussion, and the entire conclusion, rely on this comparison with the empirical dataset. Such a heavy emphasis on these empirical results would need to be supported by a more in depth comparison with the simulations and discussion of the possible differences.

3. Related to point #2, more information and guidelines are needed to make this method applicable for use with empirical data:
   o How can SKR be used for future empirical research ?
   o How realistic is it to expect to have the same SKR maintained across communities along an environmental gradient? The empirical example provided concerns a case when all communities are apparently structured by the same type of process (ie apparent disruptive selection on SLA). Is that a reasonable assumption when communities are distributed along a gradient? If the shape of the trait distribution shifts along a gradient, e.g. with disruptive selection at one end vs. directional at the other (a possible expectation along stress-resource gradients), how is the SKR approach to be used? Gradient analyses are based on characterizing patterns at the community level, and as far as I understand it, the SKR can only be calculated across communities. How many communities are necessary for a robust assessment of the parameters of the regression?
   o How to use the parameters of the SKR in empirical data? How to interpret them? For instance, can the ecolottery simulations presented here be used as a null model, or should permutational models (as in Gross et al. 2017) be preferred?
   o What is the minimum number of species per community for a correct estimation of the 3rd and 4th moments (cf. discussion by Loranger et al 2018)?

4. Assumptions underlying the filtering functions for the four scenarios need to be justified and tested for sensitivity to edge effects. I suspect that the simulations are prone to artefacts and edge effects when an environmental gradient is simulated, to the point where the simulations may no longer represent the intuitive idea of the four assembly rule scenarios. I have used the equations for the “filtering functions” provided in the Rcode to take a quick look at the behaviour of probability distributions under the 4 scenarios (hopefully interpreting the method properly – if not, sorry!).
   The main issue I found concerned what can be called “edge effects”, when the optimal trait values (equivalent to the “Env” parameter in the simulations) reach the edge of the species pool. From running a few simple simulations based on the authors’ code for the filter functions (cf. attached Rmd document), it is apparent that when the optimal trait values reach the extreme of the species trait pool (e.g. Env = -1.8), stabilizing selection turns into directional selection, while disruptive selection also looks likes directional selection but in the other direction as the optimal trait value (ie skewed in the opposite direction!). This may sound a bit abstract so I provided some figures and code to illustrate what I mean (attached .Rmd and pdf).

The authors have mostly avoided these edge effects by limiting the variations of “Env” between [-1,1] while the trait values are by design spread between [-2, 2]. This is a convenient solution, as it means the simulations do not become absurd (e.g. the shape of distributions remains overall bimodal: l 342-346), but it feels not quite satisfying. What if the optimal trait value in a community is really actually at the edge of the trait range of the species pool? How would simulations then differentiate between different assembly scenarios and distribution types? These edge effects, inherent to having a fixed species pool, need to be discussed as a caveat of the method. An additional related caveat is also the possible (even likely) case that along a gradient, trait ranges are reduced within communities compared to the species pool (ie range filtering). In this case, the filtering functions would be a bad simulation of expected trait distributions. Perhaps a solution would be to devise better filtering functions to avoid these artificial edge effects. In particular, the disruptive selection filter appears necessarily skewed in an opposite direction as the rest as soon as Env =! 0, which does not seem quite right to me.

Additional comments :

l. 49: “scaling trait distributions across communities” The meaning of “scaling” here is difficult to grasp, especially as a “naïve” reader facing the abstract.

l. 53-54 (and elsewhere): The mention of plant communities here comes as a bit of a surprise, as the method appeared so far to be for all types of communities. Perhaps it should be mentioned that plant communities were used as a case study here, but that the method could be applied to other taxons (as long as communities include single tropic levels ?).

l. 107: cite Kraft et al. 2008 as a precursor study using kurtosis in community ecology.

l.168: “stochastic processes” Which type of stochastic processes are included? These should be made a bit more explicit here. For instance, these could be parameters for dispersal, immigration, death, etc.

l. 180-183: The relationship between values of “Env” and changes in trait filtering is not explicit anywhere. I had to look at the R code to figure out that “Env” is used as an equivalent to a trait parameters in the filtering functions. It should be made explicit at least in the table of parameters in the supplementary material. Moreover, the interpretation of what Env does to trait distribution is a bit strange. On the one hand, Env can be interpreted as the ‘optimum trait value’ in the “stabilizing selection” scenario (ie Env = the mean trait value of the gaussian distribution). However in the other two deterministic scenarios, it is very unclear what this “Env” value represents. If I understand correctly (cf. my little simulations), for the disruptive and the directional scenarios, Env is equal to the trait value with the minimum probability (the bottom of the trough in the bimodal distribution). This makes it all a bit confusing.

l. 192: “Richness was fixed”. Please make it clear that you are referring to species richness in the species pool, not in single communities.

l. 193: as mentioned in my main comments, the choice of a uniform trait distribution is unusual (in my experience) and should be justified.

l. 225: “A low overlap indicates…” What constitutes a “low” overlap, and when is it sufficiently “low” to appropriately discriminate scenarios? Currently the use of overlap is purely illustrative and a bit hand-wavy. I don’t think that additional statistical tests are really necessary, but a clear formulation of what constitutes low vs. average or high overlap should be provided.

l. 228: This title is a bit obscure and could be simplified.

l.239-240: Given that variance is one of the most commonly used metric of trait diversity in trait-based community ecology studies, it is interesting that it should have a good performance here. Perhaps this deserves to be discussed a bit more. I would be curious to know how well does it fair compared to using the SKR parameters, or to the “distance to minimum kurtosis” values in discriminating scenarios. Also, how do the SKR parameter vary with increasing variance, if at all ?

l. 244-245: This is an important information, and part of the necessary discussion I was referring to above. I always suspected this was one major flaw of variance, but it is nice to see it plainly shown.

l. 278-279: This is a bit of a concern, since it means that in a perfect scenario with low stochasticity (eg. No dispersal limitations), where all communities have well defined trait distribution, but not enough variations in skewness or “diversity” (ie kurtosis given skewness), then the SKR parameters cannot accurately be fitted. This is also visually apparent from the scatter of empirical data points in the figures from Gross. et al. 2017, as the linear regression is visibly “pulled” by only a subset of the communities with higher diversity. I wonder therefore if fitting a linear model is the best option given this tendency for many low kurtosis vs. few high kurtosis communities along the same SKR?

l. 327 : Here again the meaning of “scaling” is not self-evident. Later it is made clearer in lines 354-355, but it should be explained earlier, or reformulated altogether.

l. 339- 342. OK, but this is unlikely to be the case along some gradients, where it is precisely a shift in the shape of the distribution which can be interesting/expected.
Is there a way to use the SKR along a gradient where trait distributions are shifting from unimodal to bimodal distribution ?

L. 366-367: I agree with the mathematical bias, but wouldn’t a change in skewness also potentially reflect environmental change ?

l. 369-373: These are all great options, and it would be interesting to see them applied to the empirical case study here. In particular, I wonder whether the ecolottery simulations (parametrized to mimic the empirical dataset) would be a better choice than the randomization procedures in the Gross and al. 2017 paper ? Moreover, how does the ‘distance to the lower boundary’ perform in comparison to the information provided by the SKR parameters?

l. 379-381: “the disruptive scenario reproduced the change in cover between contrasted functional groups in response to environmental changes (Fig S6)”: I do not see how the results support this claim. Please justify this statement. Fig S6 is a seemingly arbitrary subset of the full empirical dataset (only 75 communities), and I do not see tu relationship with the simulated results.

l. 383-384: cf. main comment above. I am unconvinced that these equations are either similar or different. How are these SKR parameters compared between empirical and simulated data? Some type of statistic is necessary to convincingly discuss how much they are similar, or different. For instance, one immediate solution would seem to be to compare the empirical parameters to the quantile distribution of simulated parameters (as with a null model).

l. 389-397: some other possible alternative reasons for a “mismatch” of global empirical data with a simple simulation:
more complex/multiple processes acting on the same trait, creating a not-quite-bimodal pattern ?
co-variation with other traits which do not follow a bimodal pattern ?
differences in species pool trait distributions ?
un-detected environmental gradients modifying the SKR slightly in some communities ?

l. 398-407: I think this discussion is not appropriate for this paper. A short reference to the Gross et al. 2017 discussion may be warranted, but here this is reaching beyond the scope of what the present results are supporting. It is a suprisingly applied and specific perspective for this paper, which is more a conceptual and methodological paper than an empirical one.

l. 411: “powerful mean to disentangle the effect of multiple stochastic and deterministic processes”. I do not think this conclusion is well supported by the study so far. How were multiple stochastic effects disentangled in the simulations, or in the empirical case study? Only one process was simulated and detected at a time, not multiple ones. Moreover, the approach does not allow to disentangle different processes across communities, since it relies on pooling all communities into one single regression. On the contrary, I understand the SKR as a method which allows to look for commonalities in trait distributions across species, not differences.
With this method, differences can only be found between groups of communities (but can we ever speak of “disentangled” if they are not “entangled” in the first place?). However, even then, I would urge caution when claiming to interpret trait patterns in terms of “deterministic processes”.

I am overall convinced with this revised version of the manuscript, which I found to be improved in terms of clarity and impact. The authors have carefully addressed most of the issues previously raised, and I appreciate the additional efforts provided in the form of new analyses.

I am generally satisfied with the additional analyses, although the species pool sensitivity analyses would deserve a few sentences to summarize the overall results, perhaps in the Appendix. As it is, the reader is left to make their own mind up. Perhaps it could also be mentioned earlier in the methods/results, rather than left as an afterthought in the discussion.

In this new version, the manuscript becomes clearly about presenting the SKR as a concept, and I agree with the choice to leave out references to empirical data. The conceptual contribution of the SKR to trait-based community ecology is now well presented in the text, the box and figures. On the other hand, the methodological aspect is left unexplored. This is ok, but it could be made clearer that the authors are not providing a systematic way to apply this approach to new empirical data.

Without going into methodology, it would be interesting to have a more detailed perspective concerning the type of ecological questions and datasets the approach could be applied to. In particular, the authors could expand a bit more on the contribution of this SKR approach as opposed to previous trait-based approach. Given that the SKR looks for commonality across communities, while other methods often look for differences, how could these approaches be combined conceptually? The new text in the discussion already does a good job at discussing this, but I find a conclusive summary about the original contribution of the approach to be lacking.

The text uses assembly “processes”, “rules” and “scenarios” seemingly interchangeably, but I think a distinction is needed between “assembly processes” and “assembly rules” (or “scenarios”). I understand assembly processes to be actual ecological processes, ie mechanisms, (e.g. limiting similarity and competition; facilitation; exclusion of stress intolerant species, etc.) influencing species coexistence and population dynamics. On the other hand, assembly rules (or “scenarios”) still refer to patterns, i.e. expected trait probability distributions (e.g. disruptive or stabilizing), which could be caused by a variety of processes (as also mentioned by the authors). The ecolottery model is defined by such statistical probability distributions, not by actual mechanistic processes of growth/reproduction/resource acquisition etc.
This focus on patterns is not a problem. Not only is it a necessary simplification for simulations, but also as a conceptually desirable feature in order to 1) detect overall rules (ie general emerging common patterns) and 2) test the discriminating ability of the SKR vs. other indices. But I think the text should avoid emphasizing the comparison between “processes” vs. “patterns” (e.g. l. 118-120; l. 319-320), as we are more in a situation of comparing simulated “Patterns” vs. “Metrics”.

Some more detailed comments in the text:

Title: I find it quite vague, could it be made more specific? For instance highlighting the search for commonality in assembly rules?

L.118: “Yet, we do not know which ecological processes underpin such empirical assembly rules,…”. Cf. main comment above. The simulations do not actually answer the question of underlying processes.

L. 147-149 & 154-157: These lines are repeated exactly in the introduction. Is it a mistake ?

L. 160: “either…” : Incomplete sentence? I would expect “either” to be answered by “or”.

L. 162-163” Disruptive filtering typically produces uniform or bimodal distributions (Fig. 1b).” This is one clear example of how the simulations are really simulating patterns (ie shapes of distributions) and not processes (cf. main comment above). Since the simulation algorithms are designed explicitly to create certain trait patterns/distributions, is it really surprising that we are obtaining such simulated trait patterns (cf. results l..268-278) ? There is a danger of being a bit circular in the interpretation here if those scenarios are presented as “processes”. Deterministic scenarios here are pattern driven, not process driven.

L. 198: “We randomly assigned trait values to each species”. Isn’t this just another way of repeating the previous sentence?

L. 185-187: I understand that this a convenient solution, and I am happy with the explanation provided by the authors regarding their method. However, here we are still lacking the conceptual link/rationale between the trait values and the environment parameter. Please add a sentence explaining (conceptually) HOW the Env parameter changes community assembly in the simulations, and why it makes sense ecologically (e.g. shift in optimal trait values for the stabilizing scenario; I am still not so clear on the ecological logic underpinning the other scenarios…)

L. 214 “the 100 simulated distributions”. I would rather call them “simulated communities” to be consistent with the previous sentence.

L. 255: “weaker R2”: this sounds negative, as if the commonality in SKR was less clear or robust for these scenarios. However, if I understand correctly, the low R2 here only come from the fact that simulated distributions were clustered, and not spread out. Doesn’t this mean that all simulated distributions in this scenario were actually VERY similar to each other in their K and S? That would suggest to me a very robust and stable pattern of SKR… Overall I still find the discussion of R2 values confusing in this paper, since high R2 are related to how stochastically variable (in Skewness and Kurtosis) the distributions could be in a given scenario, while low R2 might indicate how predictable (ie not variable) the shape was. I would suggest insisting less on these R2 values.

l. 268-278: It needs to be clearly mentioned here that these results are entirely expected given that the scenarios are specifically designed to create these trait distribution shapes. Otherwise this appears all a bit circular. What is more interesting is the degree of overlap and how well the metrics discriminate between scenarios.

l. 286 “the songest SKR”: That is in terms of R2? I am repeating myself here, but what does a strong SKR (ie high R2) tell us ecologically?
It mainly relies on the scatter of skweness (or kurtosis) under a scenario, but not necessarily on the importance of the SKR. Scenarios creating very similar trait distributions (ie clustered points in SKR space) create a “weak” SKR, but actually represent a very constant trait shape = high commonality of trait distributions.

L. 296-297: Thankfully! Since this shift in mean trait values is introduced by design in the simulation algorithm…

L. 321 replace by “discriminate well”

L. 324: I agree that the SKR detects well a certain aspect of assembly rules, i.e. the shape of the trait probability distribution, but not others, e.g. shift in trait means (ie trait-environment relationships) , or shifts in variance (trait range filtering). Maybe this could be discussed as complementary approaches? (cf main comment above).

L. 328-344: That is a great paragraph, and really conveys the advantage of the approach.
L. 372-373: I do not see how these simulations are telling us anything new about “real-world communities”. Maybe reformulation is necessary here?

L. 373-375: cf. my other comments about the meaning of the R2 here. Is this reliance on the R2 and the “spread” of communities in SKR space not a concern for applicability of the SKR approach? What if a set of communities are shaped following the “directional” scenario, for instance, and result in clustered S and K. Does it mean that the SKR approach will not be a powerful tool then?

L. 431-434: These conclusions are a bit vague and somewhat obvious (the fact that stochastic and deterministic process influence diversity patterns at different scales is not exactly news). Perhaps provide a more novel perspective for the application of the approach?

Figure 1C: Given the equation, shouldn't it be "S2=0, K = 1.33" ?