Associate Editor:

I have some comments in addition to the comments by reviewers.

1. In section 2,2, Author mentioned intractability of OUOUCIR model but justify by OUBMBM model as an example. It would be better directly explain intractability of OUOUCIR model.

Thank you very much for your insightful suggestion, the editorial change has been made. The intractability of the OUOUCIR model is directly explained in section 2.2 in this revision.

2. There is no explanation/discussion based on results of Tables 4 and 5.

I am sorry. I rechecked the codes and reran simulations. After summarized 5 independent runs where each run using 50000 replicates with acceptance rate \(\delta=0.01\) resulting 500 samples, Tables 4 and 5 are updated based on 5*500 posterior samples in this revision. In addition, a big table (Table 6) is added to show the distributions (prior, posterior, posterior mean and true parameter values). Explanation as well as discussion based on these three tables are added in the manuscript.

3. At the end of section 4.1, authors mentioned the results from Figures 7-8 in supplementary material. Although Figures are in the supplementary material, it would be better provide a summary of the result here.

Thank you very much for suggestion. The Figures are updated in this revision where the Bayesian coverage probability for the new models (OUOUCIR and OUBMCIR) are reported. A summary that indicates the higher estimated coverage probability is added in the manuscript. The plots of the coverage probabilities of each parameter under the OUOUCIR model is provided in Figure 4 in this revision while for the plots of the coverage probability of each parameter under the OUBMCIR model is provided in Figure S1 in the online supplemental material.

4. In Table 5, Including the true value of the parameters makes easier to compare. Also, why \(b_1\) is not well estimated (true value is 1)? Any reason?

Thank you for pointing out this issue. The true values of the parameters are added in the header of the Table from the simulation study for easier comparison with the posterior mean in each column. And after redo the simulation with more rigorous setting (i.e. 5 independent runs are performed where each run uses 1% tolerance rate from 50000 samples results in 5*500=2500 posterior samples for each parameter), the results shows that we have more accurate estimate of \(b_1\) where the posterior mean can be closed to the true value. But to be honest, there still exist bias for some runs, it may be useful to develop better procedure/algorithm in the near future to get more accurate estimates.

5. In Figure 4, OUBMCIR has some issue to identify. What could be the reason? Can you see any evidence while applying ABC?

Thank you very much for your insightful comment. It is a very interesting question, and after exploring with several different setting of parameters values and taxa size, I found that the main reason for the identifiability issue of the OUBMCIR model is due to the taxa size where the OUBMCIR model requires more taxa size to identify the correct model. Results are updated in Figure 5 in this revision where the OUBMCIR model can be identified well for balanced tree with 256 taxa and higher (over 77% correctly identifies the model). And in the online supplemental material Figure S2, the result shows that the OUBMCIR model can identified well (over 75% correctness) for birth-death tree with larger taxa size (500 taxa). We report this update to the reviewer here.

6. p. 19, line 3: “All priors are set to uniform distribution” Any reason? Why different from the setting in the simulation study?

Thank you so much for your insightful comment. Below is an explanation from my understanding. For empirical study, one of concerns is that we do not have information of the internal nodes states (ancestral values), we only have the trait value at the tip of tree with known branch lengths and topology. When the procedure is performed for generating the posterior samples, there is higher uncertainty for the trait value at each internal nodes. Hence, I feel that it may be more appropriate to assume that all possible values of parameter are equally likely. Therefore the uniform priors are considered to used for all parameters. In this revision, the results using the uniform priors in reported in Tables 4, 5 and 6 for keeping the manuscript better organized with the empirical analysis and cross validation (so all simulations and empirical analyses use uniform priors in this revision.) On the other hand, for the use of informative prior the simulation results can be accessed in Tables S1, S2 and S3 in online supplemental material.

7. Some figures such as Figures 3 and 6 seems too big. Could the figures be scaled for readability?

Thank you very much for your suggestions. Those figures have been re-scaled in this revision for better readability.

8. I assume that the supplementary material (section 7) will be a separate file.

Thank you very much for your points, the manuscript has been reorganized. The supplementary material containing 4 Tables and 2 Figures is now a separate file and is included in this submission.

9. Although editing English has been made, it still has several issues. I advise the author to get professional English editing service.

Thank you very much for suggestion. I am sorry that the English issues cannot meet the standard in previous revision. After seeking editorial help, the manuscript in the revision has been again carefully proofread (correct typos and grammar errors) and edited by a native speaker and professional expert in this research area. I hope the English in this version can meet the standard.

10. A few example of typos, minor issues, possible grammatical errors, etc.

• p.4, line 2: access -> assess

Thank you, the editorial change has been made.

• p.4, line 8: “the dynamic of the models of adaptive trait evolution” seems awkward

Thank you, the editorial change has been made.

• p.4, table 1 caption: “Property of model of adaptive trait evolution” seems awkward as well

Thank you, the editorial change has been made.

• p.5, line 2: “SimModelOnePathV2.R” does not fit in the text without any explanation

Thank you for your comment, the SimModelOnePathV2.R is the R file for generating figure 1. We have removed it in the text.

• p.5, eq (7): What is AZ? There was explanation in the first version but now it is removed.

Thank you for your question. The \(A\) is a 3 by 3 matrix containing the forces parameter \(\alpha_y,\alpha_\theta\) and \(\alpha_\tau\) while \(Z_t=(y_t,\theta_t^y,\tau_t^y)'\) is the random vector. \(AZ\) is the vector of the linear combination of the force parameter and the random vector. The explanation is added to this revision.

• p.7, line -5: Is ‘1’ in “OU1 process” typo?

Thank you for your question, OU1 process is the basic model in this area in Hansen 1997 paper. The editorial change is made to clarify this.

• p.12, The sentence starts with “Approximate Bayesian Computation..” seems awkward

Thank you, the editorial change has been made.

• p.15, line -5: What does it mean by “Student_t(1,0,2.5)”? The author needs to explain the notation.

Thank you for your suggestion, the editorial change has been made. It intends to say using a t distribution with 1 degree of freedom (a Cauchy distribution).

• p.16, line -2, -5: values for % are not correct. 0.56% -> 56%, 0.18% -> 18%

Thank you, the editorial change has been made.

Reviewer no. 2:

I only have two minor comments:

1. It would be very helpful to add one short paragraph at the beginning of the Results (section 4), to summarize which results will be presented. For instance, the section 4.1 “Simulation” starts very abruptly and the reader needs to go back into the text to verify what simulations the author is referring to, and how these simulations were generated. Section 4.2 has this introductory paragraph and it it, thus, very easy to follow the rest. I suggest something similar for section 4.1.

Thank you very much for your constructive suggestion. An introductory paragraph of which results will be presented in provided in section 4.1 where two simulations (parameter estimation for all models and Bayesian coverage probability for each parameter in the new models) are described. Because this section is reorganized after several extensive simulations (Table 6) are performed, the arrangement has some changes in this revision for presenting the results. Hope this improves the quality of this paper to meet the standard of CSDA.

2. In your new cross-validation results you find that it is hard to distinguish between OUBMCIR and OUOUCIR when the true model is OUBMCIR (for all taxa sizes). Would you discuss why this might be the case?

Many thanks for your useful concern. A paragraph for discussing this issue has been added in this revision. After redo the simulation by using larger taxa size (please see Figure 4: using balanced tree of of 128, 256 and 512 taxa and Figure S2 in online supplemental material using balanced tree 500 taxa), it is easier to distinguish between OUBMCIR and OUOUCIR by using larger taxa size.

Moreover, two set of simulations are performed using different type of trees (balanced tree and birth-death tree). After comparing the results using Figure 5 and Figure S2 in supplemental material, I found that models can be better identified under balanced tree case than under birth-death trees case. This may be due to fact that the death-birth tree is randomly simulated at each replicate while the balanced tree is fixed throughout simulation. So the tree type does affect the identifiability issue of model by cross validation analysis, in particular for the OUBMCIR model.

Reviewer no. 3:

1. The authors proposed OUOUCIR and OUBMCIR models. However, there are a lot of texts on describing OUOUBM and OUBMBM, which have been studied by others. I think those contents are redundant and should be more concise. The paper should focus on the new models.

Thank you very much for your constructive suggestions. The editorial change has been made and now the manuscript focuses more on the new models (the OUBMCIR model and the OUOUCIR model). In particular, the explanation of the mathematical property of SDEs in the OUOUCIR model is added in section 2.1 and the explanation of the intractability for the OUOUCIR model is provided in section 2.2. Simulation for computing the Bayesian coverage probability focuses on the new models and the results is reported in section 4.1.2 (Please see Figure 4 in the manuscript and Figure S1 in supplemental material). Furthermore, the interpretation of the result from simulation on cross validation in this revision focuses more on discussion of OUBMCIR model and OUOUCIR model than other models.

2. The application of ABC methods on trait evolution has been studied by referred papers [30] and [31], which somehow affect the novelty and contribution of this work, though it is applied to the proposed new models.

Thank you very much for your constructive criticism, I full agree that the ABC methods has been studied by referred papers. Let me try to explain more about the contribution in this work. The new model in this work mainly focuses on using CIR process to model the rate of adaptive trait evolution \(\tau_t^y\). And due the distribution for the trait variable \(y_t\) in the new models has intractable likelihood, ABC is applied here for model inference. However, the models in the work [30] (for branching OU process) and [31] (for univariate continuous trait only) have known distribution and model likelihood although the authors instead/emphasize the use ABC approach for inference. Hope this explanation can distinguish my work from the work [30,31] in literature.

3. The revised version still has quite many problems on the writing, such as typos, grammar issues.

Thank you very much for your comments here. The typos and grammar errors from last revision are corrected in this revision. This manuscript has been carefully edited by a professional researcher (also a native speaker) in this research area. Hope this can meet the high standard in CSDA.

In the manuscript

Figure/Table	link
Figure 1:	https://tonyjhwueng.info/ououcir/SimModelOnePath.html
Figure 2:	https://tonyjhwueng.info/ououcir/3taxaexample.html
Figure 3:	https://tonyjhwueng.info/ououcir/3taxaTraitdemoMod.pptx
Figure 4:	http://www.tonyjhwueng.info/ououcir/summaryplotHDI.html
Table 4 and 5	http://www.tonyjhwueng.info/ououcir/unifsimtable.html
Table 6:	http://www.tonyjhwueng.info/ououcir/unifdistplot
Figure 5:	http://www.tonyjhwueng.info/ououcir/cvbalancedtree.html
Figure 6:	http://www.tonyjhwueng.info/ououcir/sanchezLaskercoralPhyloPlot.html
Figure 7:	http://www.tonyjhwueng.info/ououcir/graphviz.html
Table 7 and 8:	http://www.tonyjhwueng.info/ououcir/paramstable.html

In the supplemental material.

Figure/Table	link
Tables S1 and S2:	http://www.tonyjhwueng.info/ououcir/nonunifsimtable.html
Table S3:	http://www.tonyjhwueng.info/ououcir/nonunifdistplot
Figures S1:	http://www.tonyjhwueng.info/ououcir/summaryplotHDI.html
Figure S2:	http://www.tonyjhwueng.info/ououcir/cvbirthdeathtree.html
Table S4:	http://www.tonyjhwueng.info/ououcir/EmpiricalMaincodeV2/treetraitV2/

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Thank you so much. An R package ouxy is developed and can be accessed at https://cran.r-project.org/web/packages/ouxy/index.html and a method article titled: “Building an adaptive trait simulator package to infer parametric diffusion model along phylogenetic tree” is included within this submission.

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