Bayesian Mediation Analysis with an Application to Explore Racial Disparities in the Diagnostic Age of Breast Cancer

Round 1

Reviewer 1 Report

Bayesian Mediation Analysis—With an Application to Explore

Racial Disparities in the Diagnostic Age of Breast Cancer"

This paper proposes three methods for Bayesian mediation analysis to reach conclusions about mediation effects.   The authors applied these three methods to explore the racial disparity in the diagnostic age of breast cancer patients in Louisiana.  However, there are some Major points that must be answered:

1:  The introduction must be rewritten in a broader way than it is, so that it clarifies the objectives, novelty, and method used, as well as comparison with other works in the literature. It also clarifies the obstacles that may appear when implementing the proposed strategy.

2:  Page 2 , Lines 79-89, need to be rephrased again clearly.

3. I would like to inquire about the causes of death in the selected sample. Is it possible to classify causes of death according to the number of risks mentioned and classified in the paper?

4. Page 5, Authors should talk more about their choice of prior distribution.

5- The algorithms for the three proposed methods must be added.

6:  The conclusion must be written well, and I also need to review the code for the number part

Author Response

We appreciate the thorough review and constructive feedback from the reviewer. Below, we have addressed the following points:

1:  The introduction must be rewritten in a broader way than it is, so that it clarifies the objectives, novelty, and method used, as well as comparison with other works in the literature. It also clarifies the obstacles that may appear when implementing the proposed strategy.

Authors: We appreciate the comment from reviewer 1 and have rewritten most of the introduction. We added a literature review of the previous mediation analysis methods, their weaknesses, and the novelty and contributions of our proposed methods. The background and analysis purposes of the application example were added to section 2.

2:  Page 2 , Lines 79-89, need to be rephrased again clearly.

Authors: We rephrased the lines and found the expression is clearer. Thanks for pointing this out.

3. I would like to inquire about the causes of death in the selected sample. Is it possible to classify causes of death according to the number of risks mentioned and classified in the paper?

Authors: I am not very sure what is asked in the comment. However, it is possible to identify breast cancer patients by the causes of death. Cases identified from death certificates only mean that the patients have not been diagnosed with breast cancer when alive. We exclude those cases when analyzing the cancer diagnosis age.

4. 4. Page 5, Authors should talk more about their choice of prior distribution.

Authors: Good point. We added a discussion on the choice of prior distributions.

5- The algorithms for the three proposed methods must be added.

Authors: The references for the three proposed methods were added.

6:  The conclusion must be written well, and I also need to review the code for the number part

Authors: Thanks for pointing this out. The codes were added to the supplementary material. We also rewrote the conclusion section to make it clearer.

Reviewer 2 Report

Please see the attached. 

Comments for author File: Comments.pdf

Author Response

We appreciate the thorough review and constructive feedback from the reviewers. Below, we have addressed the following points:

1. In line 125, it mentions that 1579 non-Hispanic white and 696 black female patients, respectively. The data is unbalanced. Did you consider the effect of unbalance in the data analysis you used and ANOVA (which you mentioned in line 139)? Is there any collinearity among the considered covariates?

Authors: This is a great question. The data are observed. We included all breast cancer patients diagnosed in 2011 in Louisiana.  The unbalanced data may reduce power, but we still found significant differences in the age of cancer diagnosis. We also checked that the two populations have no significant difference in the variance of the diagnosis age. There could be collinearity among considered covariates. Collinearity may result in higher estimated variances but is allowed in the analysis.

2. In line 223, a gamma prior with shape 1 and rate 0.1 was assigned to all variance terms? The shape of gamma distribution is sensitive to shape and rate parameters. Authors should mention the reasoning to choose such settings or include some references.

Authors: The Gamma distribution is a conjugate distribution for the variances in Bayesian linear regressions. The Gamma distribution is chosen for computational convenience. We have added a reference for the choice.

3. In MCMC, did the posterior converge well? Authors may provide trace and ACF plots.

Authors: Many thanks to the reviewer for the comment. The posterior distributions converge well. We have checked the trace and ACF plots to confirm this. We also added a discussion on the convergence in the results.

Reviewer 3 Report

P. 3, line 114  Was there any clustering in the data?  Given that it is from the CDC and is spatially indexed, I wondered if the data were clustered by neighborhood, city, county, state, etc?  If so, the authors should consider using a multilevel modeling approach, or a fixed effects model.

P. 4, line 136  Please provide references for the conditions that a variable must satisfy to be a mediator.

P. 6, line 224  I wondered why the authors used only a single chain?  Typically, Bayesian modeling is done using 2-4 chains and then the results from the different chains are used to assess convergence.

Author Response

We appreciate the thorough review and constructive feedback from the reviewer. Below, we have addressed the following points:

P. 3, line 114  Was there any clustering in the data?  Given that it is from the CDC and is spatially indexed, I wondered if the data were clustered by neighborhood, city, county, state, etc?  If so, the authors should consider using a multilevel modeling approach, or a fixed effects model.

Authors: This is a great point! The spatial variables were compiled at the census tract level. There are 1148 census tracts in Louisiana. The 2100 patients were scattered around Louisiana, with a tract of 0 to 5 patients. For this situation, it is best to use a one-level model to ensure the convergence of models. For bigger area units such as the county, it is more beneficial to use multilevel models. We added a discussion in Section 5.

P. 4, line 136  Please provide references for the conditions that a variable must satisfy to be a mediator.

Authors: Thanks for pointing this out. The reference has been added.

P. 6, line 224  I wondered why the authors used only a single chain?  Typically, Bayesian modeling is done using 2-4 chains and then the results from the different chains are used to assess convergence.

Authors: This is a great point. We recommended one chain for simplicity and efficiency. We also used the trace and ACF plots to check the convergence of the posterior distributions. If there are no computational constraints, multiple chains traversing the same parameter space better justify the convergence and are recommended. We added the discussion in Section 4.

Round 2

Reviewer 1 Report

No other comments

Reviewer 2 Report

I recommend the paper to be accepted.