Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript deals with the Cumulative Probability Models (CPMs) which, using nondecreasing transformations, nonparametrically estimate the results of continuous regression  classifying them by category. The asymptotic properties for the CPMs are examined and the uniform consistency of the  the nonparametric likelihood estimates of the regression coefficients and the estimated transformation function is proved. The authors are also described the joint asymptotic distribution of CPM and conducted some simulations and applications on a dataset of HIV-positive patients. According to them, CPM and original data seem to have similar output results.

In my opinion, the topic of the manuscript seems interesting, and it is written quite clearly and precisely. This is supported by the fact that the proofs of the two theorems, due to their enormous length and complexity, are placed in the "Supplementary material". In addition, the authors have, with considerable effort, conducted several simulations with different lengths , as well as real-world data application.  All that, in my opinion, justifies the application of the proposed techniques. My basic proposal is to highlight in some places what is new in the manuscript itself, as (some more) reasons for "replacing" continuous outputs of regression models with categorized data. Finally, there are some minor, mostly technical remarks and suggestions, of which I single out the following:

Lines 76-77: This sentence should be a little more detailed and precise, especially the term "uniformly consistent".

Line 96: Please explain for clarity that Φ(x) is the cumulative distribution function (CDF) of the standard Gaussian distribution.

Line 102: Please explain (in more detail and more formally) what it means: "increments of A(·) are concentrated at the observed Y_i". Also, I think here the authors could highlight what is new in their manuscript compared to Liu et al. (2017).

Line 141: Please provide some reference for the Helly theorem mentioned here. Also, rephrase this part of sentence "for any subsequence there exists a weakly convergent subsequence". It is neither clear nor quitely accurate.

Line 156: The term "modern empirical process" seems unusual and I think it should be changed (or explained in some detail).

Lines 186-189: The sentence here should be rearranged to be more detailed, more formal and more rigorous. In particular, explain what a "center of the distribution " is.

Lines 247-248: Finally, I think the authors here should explain in more (mathematical) detail how they used "restricted cubic splines".

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The authors investigate the asymptotic properties of the cumulative probability model proposed by Liu et al. (2017). The manuscript is clear, relevant for the field, and well-structured. The level of scientific rigor is apparent, and the attention to detail concerning every aspect of the approach is appreciated. I have some minor remarks that the authors might consider.

 

General comments

 

Title, abstract: The title is concise, specific, and relevant. The length of the abstract is appropriate. The abstract is coherent with the content of the paper. It sufficiently highlights the purpose of the study, main findings, results, and conclusions.

 

Introduction: The introduction can be expanded by providing a more comprehensive overview of asymptotic properties for CPMs. This will help readers understand the motivation behind the proposed theorems. It would be beneficial to show how the proposed approach differs from previous research and what unique contributions it offers. Currently, this differentiation is not sufficiently established.

 

Literature review: All the references are relevant, and the paper has a reasonable number of self-citations. However, the literature review is insufficient both in terms of depth and breadth. There is only one reference to the paper published in the last five years. It should be expanded to include a broader range of studies related to cumulative probability models, their applications, and the challenges associated with asymptotic properties. The authors should reference and discuss existing approaches, highlighting their strengths and weaknesses. This will provide a stronger foundation for the current study and emphasize its novelty in relation to prior research.

 

Methodology: The methodology is clear and adequately addresses the purpose of the study. However, the use of technical terms and symbols in (4) should be appropriately explained to ensure readers can follow the methodology.

 

Validation: The mathematical results are scientifically sound, and the experiments support them in an appropriate manner. The asymptotic properties established for CPMs are confirmed by providing a number of computational experiments using real-life and artificially generated data. The computational experiments are described in detail, so the results are reproducible given that the code and data are publicly available. The provided illustrations properly show the obtained results.

 

Conclusion: The conclusions are consistent with the arguments presented. The robustness of the proposed approach to the specific choices of L and U is shown. However, it is suggested to provide a recommendation for choosing L and U in practice.

Additionally, while the conclusion suggests future research directions, it would be helpful to have a more detailed discussion of the limitations of the proposed approach.

 

 

Specific comments:

1. L.101: The minus symbol should be used as a superscript.

2. L.134: A typo (2.4) -> (4).

3. Year in brackets should not be used for citations; reference numbers should be placed in square brackets [ ]; references must be numbered in order of appearance in the text.

4. Proofs of the theorems should be included in the paper (in the main text or in the appendix). 

5. The authors are encouraged to move additional simulation results to the appendix, not to the supplementary file.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

In this revised version, the authors have fulfilled almost all my remarks and suggestions. I think that in this form the manuscript can be considered for publication.

Author Response

Thank you!

Reviewer 2 Report

Comments and Suggestions for Authors

The authors sufficiently revised their paper and eliminated all the drawbacks I wrote about earlier. Only one remark should be noted about the structure of the manuscript. Remember the situation when Pierre de Fermat found a remarkable proof of the theorem but did not write it because there was not enough space in the margin of the book. This led to three centuries of attempts to prove this result. Therefore, I still recommend moving all proofs to the appendix of the paper since they are essential to understanding the paper. There should be no difficulties with this because Mathematics has no restrictions on the maximum length of manuscripts. Supplementary is for additional data and files from the research.

   

 

 

Author Response

Thank you! We have now moved the proofs to the main manuscript as appendices. As the proof of each theorem takes a few pages, for clarity, we put the proof of Theorem 1 as Appendix 1 (in pages 18-23) and that of Theorem 2 as Appendix 2 (in pages 24-28). We have also updated the text where the proofs are referred to.