PONE-D-20-31434

Random intercept model including heteroscedasticity in a logarithmic scale: correction terms and prediction in the original scale

PLOS ONE

Dear Dr. Ramírez-Aldana,

Thank you for submitting your manuscript to PLOS ONE. One of the referees have recommended that you make amendments to your article. We therefore invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript by Dec 25 2020 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

• A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.
• A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.
• An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols

We look forward to receiving your revised manuscript.

Kind regards,

Ivan Kryven

Academic Editor

PLOS ONE

Journal Requirements:

When submitting your revision, we need you to address these additional requirements.

1. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at

https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and

https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf

2. We suggest you thoroughly copyedit your manuscript for language usage, spelling, and grammar. If you do not know anyone who can help you do this, you may wish to consider employing a professional scientific editing service.  

Whilst you may use any professional scientific editing service of your choice, PLOS has partnered with both American Journal Experts (AJE) and Editage to provide discounted services to PLOS authors. Both organizations have experience helping authors meet PLOS guidelines and can provide language editing, translation, manuscript formatting, and figure formatting to ensure your manuscript meets our submission guidelines. To take advantage of our partnership with AJE, visit the AJE website (http://learn.aje.com/plos/) for a 15% discount off AJE services. To take advantage of our partnership with Editage, visit the Editage website (www.editage.com) and enter referral code PLOSEDIT for a 15% discount off Editage services.  If the PLOS editorial team finds any language issues in text that either AJE or Editage has edited, the service provider will re-edit the text for free.

Upon resubmission, please provide the following:

• The name of the colleague or the details of the professional service that edited your manuscript
• A copy of your manuscript showing your changes by either highlighting them or using track changes (uploaded as a *supporting information* file)
• A clean copy of the edited manuscript (uploaded as the new *manuscript* file)

3. Please include captions for your Supporting Information files at the end of your manuscript, and update any in-text citations to match accordingly. Please see our Supporting Information guidelines for more information: http://journals.plos.org/plosone/s/supporting-information.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

**********

2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

**********

3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: No

Reviewer #2: Yes

**********

4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: This paper proposes and evaluates correction methods for analysis using log-transformed data to fit a linear mixed model and using the fit to obtain predictions on the original scale.

(1) The correction follows from the property of the log-normal distribution. If mu and var are the mean and variance on the log-scale than exp(mu + var/2) is the mean on the original scale. This is well-known, so the claim made in the paper that the correction is proposed "for the first time" is probably overstating the case.

(2) That being said, the authors have a particular application. My feeling is that the whole story line should be more focused on the application case. Most importantly, I am missing a clear statement what exactly is to be estimated in the context of the example. Without that context, it is cumbersome to follow all the equations. Not because of the equations, but because it is unclear what we are estimating!

(3) While I think it is important to focus more clearly on the example, I also think the scope needs to be broadened a bit for the benefit of readers using mixed models but not necessarily the one considered by the authors. The corrections are pretty straight-forward and applicable with other mixed models. All that's required is to work out the relevant marginal variance to use in the correction.

(4) Model (2) only has a single subscript "i", whereas equation (4) has two subscripts "ij". It is not clear what these subscripts refer to and why they are changing in this way. Again, providing the specific context of your example may help here.

(5) In line 169, I am not sure why there is a need to switch to matrix notation. What does this really help? I am also not sure expressions like in eq. (12) are correct, where you are mixing matrix and scalar expressions.

(6) Line 219: What is the rationale underlying the smearing estimator, and what is it's advantage over the plug-in estimator based on what I said in (1)? Some explanation in the paper would be useful.

(7) Line 267: It is important to tell readers how you simulated the data and why.

(8) Line 274: Why did you even consider fitting this GLMM? It's not the model used to simulate the data, is it? Conversely, if it is assumed the GLMM generated the data, why use any other approach for analysis? Clearly, a rationale for using these alternate models is missing.

(9) Line 281: On a similar note, it is not clear to me why data were generated from a GLMM in a second simulation. I think you need a clear justification and story line for this choice.

(10) The manuscript file did not contain and of the diagnostic residual plots, so I was not able to convince myself that the log-normality assumption was satisfied. How do you actually check this when the data is heteroscedastic, as is the case under your model?

(11) And now the obvious question: what to do if you are not so lucky that the log-transformation works well? You mention Box-Cox transformation in the discussion. What if I need to use x^0.3, say? How does your approach generalize? If you can cover that more general case, and also tell people what to do in case their LMM looks different than yours, you can vastly broaden the scope and impact of your paper. As it stands, it's a rather narrow case study.

Reviewer #2: Authors have proposed correction terms to predict the response in the original scale, which are easy to calculate and implement, for a random intercept model (RIM) with or without heteroscedasticity with response in a logarithmic scale. Different estimators of the predicted response are given (some of them already present in the literature)

In addition, simulations and a real dataset are presented in the paper to show the importance of using the correction terms to obtain more accurate predictions.

I have no comment whatsoever. Overall the manuscript is well written and organized. I enjoyed reading the paper and we can say that this paper contributes to the growing literature on statistical modeling.

**********

6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

PONE-D-20-31434R1

Random intercept and linear mixed models including heteroscedasticity in a logarithmic scale: correction terms and prediction in the original scale

PLOS ONE

Dear Dr. Ramírez-Aldana,

Thank you for revising your manuscript. As you can see Reviewer 1 still has several comments and we therefore invite you to prepare a new version of the manuscript addressing Reviewer's feedback. Following on Reviewer's advise, please condense the description of the key steps of the approach and include more elaborated discussion of  examples and extensions.

Please submit your revised manuscript by Feb 06 2021 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

• A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.
• A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.
• An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols

We look forward to receiving your revised manuscript.

Kind regards,

Ivan Kryven

Academic Editor

PLOS ONE

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: (No Response)

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: (No Response)

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: (No Response)

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: (No Response)

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: (No Response)

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The revision is a substantial improvement because the example now plays a more central role throughout, making it easier to assess the relevance of the proposed method.

(1) The proposed method itself is rather straightforward. If we log-transform data and fit a linear mixed model assuming normality, then the analysis implies that the untransformed data have a log-normal distribution. The naive back-transformation of means from the log-scale to the original scale does now provide unbiased estimators of the expected value on the original scale. This is all well known, and it is also known that the mean on the original scale is simple exp(mu)*(exp(sigma^2/2). All the authors are, therefore, proposing is that we should multiply the naive estimator exp(mu_hat) with the improved estimator exp(mu_hat)*(exp(sigma^2_hat/2). This works straightforwardly for a linear mixed model, we just need to work out what the marginal variance is in each case.

It is not clear why the authors are using so much matrix algebra to make this very simple point. The matrix algebra is clearly unnecessary, even if generalizing this to other models. Thus, I see a lot of scope form simplification. Certainly, the matrix algebra does not make things easier in any way as asserted in L202. How could things be possibly any easier than in the scalar multiplication shown in eq. (9)? And that's all it takes!

(2) In the same vein, the authors' proposal to use simulation in case other transformations than the logarithm are used is fine, though not new either. Moreover, this can be said in a single sentence. Then three pages the authors are devoting to this can be shortened substantially, if not removed altogether. Instead, they could focus on analytical results that are available for the power-normal distribution:

Freeman, J., and R. Modarres. 2006. Inverse Box–Cox: The powernormal distribution. Stat. Probab. Lett 76:764–772. doi:10.1016/j.spl.2005.10.036

(3) The authors make a bit point about heteroscedasticity, but the form of heteroscedasticity is very restrictive (L145): they assume that the variance is sigma12/w, where w are known weights. This is, of course, the most favourable case, but most of the time such weights are unknown. This begs two questions: (i) what to do if heteroscadasticity is of a different form and (ii) what are the weights in the central real example? They mention in L32 what the weights could be, but what about their example?

(4) As regards the simulation, it would be very useful to explain how the parameter setting is modelled on the real example. As it stands, it seems like the settings studied fall from the skies, which is not very convincing. In particular, coming back to the weights, what is the rationale for the simulation of the weights in L283-284? What do the weights represent in the real-life application the simulation is hopefully representing here?

Further remarks:

L141: Something wrong here with the representation of the normal as N(sigma^2, sigma^2)

L154: y => and

**********

7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

PONE-D-20-31434R2

Random intercept and linear mixed models including heteroscedasticity in a logarithmic scale: correction terms and prediction in the original scale

PLOS ONE

Dear Dr. Ramírez-Aldana,

Thank you for resubmitting your manuscript to PLOS ONE. Reviewer 1 has noticed that one of their comments was not understood as intended, please see below. We therefore invite you to respond to this comment by incorporating the necessary changes to the manuscript.

Please submit your revised manuscript by Mar 26 2021 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

• A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.
• A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.
• An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols

We look forward to receiving your revised manuscript.

Kind regards,

Ivan Kryven

Academic Editor

PLOS ONE

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: (No Response)

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: (No Response)

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: (No Response)

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: (No Response)

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: (No Response)

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The authors acknowledge that in case of heteroscedasticity the weights are typically unknown and hence need to be estimated. They further mention two options in R to estimate such functions and note that these involve parameters. It may be added that common forms of heteroscedasticity may also involve correlation, as is the case, e.g., with repeated measures data. The authors further response is based on this assertion: "As far as we know, once these parameters are estimated, the weights could be assumed as known, by simply calculating these values for each individual using the estimated parameters." This is true as far as the point estimates for fixed and random effects are concerned, and only when residual errors are independently distributed, but it is incorrect as far as the inference is concerned. Adjustments are needed both regarding the standard errors and the degrees of freedom. See the papers by Kenward and Roger (1997 and 2009 in Biometrics and CSDA).

I would like the authors to re-consider this point, and I am happy to take another look once that's done.

**********

7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

Random intercept and linear mixed models including heteroscedasticity in a logarithmic scale: correction terms and prediction in the original scale

PONE-D-20-31434R3

Dear Dr. Ramírez-Aldana,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

An invoice for payment will follow shortly after the formal acceptance. To ensure an efficient process, please log into Editorial Manager at http://www.editorialmanager.com/pone/, click the 'Update My Information' link at the top of the page, and double check that your user information is up-to-date. If you have any billing related questions, please contact our Author Billing department directly at authorbilling@plos.org.

If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

Kind regards,

Ivan Kryven

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments: