PONE-D-22-21044Bayesian model selection for multilevel models using marginal likelihoodsPLOS ONE

Dear Dr. Edinburgh,

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Alessandro Barbiero, Ph.D. in Statistics

Academic Editor

PLOS ONE

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“We would like to thank Dr Torben Sell (University of Edinburgh) for his insight and advice about challenges in MCMC sampling. TE is funded by Engineering and Physical Sciences Research Council (EPSRC) National Productivity Investment Fund (NPIF) EP/S515334/1, reference 2089662. A CC BY or equivalent licence is applied to the AAM arising from this submission.”

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Additional Editor Comments:

I would suggest, as one of the referees does, that the authors include a simulation study to assess their proposition.

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Reviewers' comments:

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Reviewer #1: Partly

Reviewer #2: Yes

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

Reviewer #1: Yes

Reviewer #2: Yes

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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: Yes

Reviewer #2: Yes

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Reviewer #1: Yes

Reviewer #2: Yes

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5. Review Comments to the Author

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Reviewer #1: Review on

“Bayesian model selection for multilevel models using marginal likelihoods”

This article proposes the use of marginal likelihoods instead of full likelihood to perform model selection for multilevel models. The methods are illustrated using generalized linear models where the responses are normally distributed, and the identity link is used. In this setting, the authors demonstrate that marginal likelihoods can be obtained analytically by integrating out the fixed and random effects. This reduces the dimensionality when computing the model evidence using SMC as only the variance parameters remain to be integrated out.

The manuscript is generally very well written with clear explanations although there are some typographical errors to be corrected. Below are some comments and suggestions for the authors’ consideration in revising the manuscript. A point-by-point response to these comments is appreciated.

1.I think using the term “marginal likelihood” to represent what is obtained after integrating a subset of the parameters is rather confusing because this term is usually taken to mean what is obtained after integrating out all the parameters. Perhaps “partial marginal likelihood” would be more accurate? Or the authors can consider other terms that provide a better description?

2.Page 5 line 151: Should $\\nu$ be $\\eta$ in “integrated over $\\beta$ and $\\nu$”?

3.Page 5 line 170: Typo in “include the this using the notation above”

4.Page 5 line 172: Typo in “Rearranging the square brackets” because there are no square brackets in the expression above this line.

5.Page 6 line 176: Since the product is over i and j, I wonder if the power for $2 \\pi \\sigma_y$ should be $nJ$ instead of just $n$? Same problem in equation (6).

6.Page 6 line 177: A \\sigma_\\eta^2$ is missing in the numerator of the very last term.

7.Page 6 line 179: Typo in “rearranging for with $\\beta$ as in the”

8.Page 7 line 186: In line 4 of the equation, the power of $|\\Sigma_\\eta(\\nu)|$ should be J instead of m. Similarly I wonder if the power of (2\\pi\\sigma_y) should be nJ instead of n? Same problem in equation (7).

9.There is only one example given on Minnesota radon contamination. Will the authors consider including a simulation study? For this real dataset, it is unclear how the models should be ranked since there is no true model.

10.There is a lack of discussion on the comparison of SMC(marginal likelihood) and SMC(full likelihood). It is not clear what are the advantages of SMC(marginal likelihood)? Is the computation time reduced? How does this reduction depend on the number of observations and parameters? If this is the case, the authors should present some results on the timings? What other advantages are there? The authors need to present a clearer discussion of the advantages of their approach.

11.Is it possible to design an experiment with a gold standard where the value of the likelihood can be computed? Then both approaches can be compared against this gold standard.

Reviewer #2: This paper gives a derivation of the marginal likelihood for a general multilevel linear model using a specific prior setting. The derivations look accurate, and related literature is cited accordingly. I have the following comments:

1. Please fix "... the this ..." on line 170.

2. You can skip the equation before Eq. (5).

3. Lines 156 - 159, $\\Sigma_{\\eta}$ is set to $\\nu I$ which implies that the elements of $\\eta$ are independent. What is the implication of this on the multilevel linear model? Is it a restrictive approach? Is it possible to do the derivations if the elements of $\\eta$ are not assumed to be independent?

4. Please link the result in Eq. (7) back to the elements given between lines 120 - 128. Or it would be good to provide an algorithm to outline the implementation of the derivation there in practice. This is only a part of the whole implementation and should be understandably by the practitioners.

5. The example lacks an informative data definition. Some descriptive statistics would help.

6. Please give the data definition and the model's settings starting on line 203 in separate paragraphs for clarity. It is hard to follow in its current form.

7. Once we get Table 1, what will we do with it? The link between the application and model selection is missing. What is the implication of your model selection effort? What does Minnesota radon contamination look like from the perspective of the best model selected by your approach?

8. What if I use an RJMCMC approach? Would I get different results in terms of the estimates of radon contamination? In that sense, what is the contribution of the work to the existing literature?

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Reviewer #1: No

Reviewer #2: No

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PONE-D-22-21044R1Bayesian model selection for multilevel models using integrated likelihoodsPLOS ONE

Dear Dr. Edinburgh,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we 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 Jan 15 2023 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: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols.

We look forward to receiving your revised manuscript.

Kind regards,

Alessandro Barbiero, Ph.D. in Statistics

Academic Editor

PLOS ONE

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Reviewers' comments:

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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)

Reviewer #2: All comments have been addressed

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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: Yes

Reviewer #2: Yes

**********

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

Reviewer #1: Yes

Reviewer #2: Yes

**********

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: Yes

Reviewer #2: Yes

**********

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: Yes

Reviewer #2: Yes

**********

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: I thank the authors for addressing my earlier comments. I only one more comment. I like the newly added simulation study as it clearly shows the advantages of the integrated and full likelihood in terms of computation time, accuracy of model evidence and posterior estimates. However, I think the discussion on Table 1 is too brief. There is a paragraph under the section "Discussion" on page 21, lines 410-418 concerning the results on Table 1, and I suggest moving that to the section on "Example: Simulation study".

Reviewer #2: (No Response)

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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

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Bayesian model selection for multilevel models using integrated likelihoods

PONE-D-22-21044R2

Dear Dr. Edinburgh,

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Kind regards,

Alessandro Barbiero, Ph.D. in Statistics

Academic Editor

PLOS ONE

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