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Potential Impact of Updated Bayesian Deduction in Medicine: Application to Colonoscopy Prioritization

Cancers 2025, 17(23), 3845; https://doi.org/10.3390/cancers17233845

by Pierre Collet^1,2,3,*, Felipe Quezada-Diaz^2,3

and Carla Taramasco^1,2,3

Reviewer 1: Anonymous

Reviewer 2:

Sumedha Nitin Prabhu

Reviewer 3: Anonymous

Reviewer 4: Anonymous

Reviewer 5: Anonymous

Cancers 2025, 17(23), 3845; https://doi.org/10.3390/cancers17233845

Submission received: 16 September 2025 / Revised: 14 October 2025 / Accepted: 15 October 2025 / Published: 29 November 2025

(This article belongs to the Special Issue Recent Advances in Diagnosis and Management of Colorectal Cancer)

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This study is based on data from a symptomatic patient cohort in Chile [25]. The proposed prioritization strategy using CBI for multiple FIT tests is highly dependent on the epidemiological background and healthcare resource environment of that population. Therefore, the results obtained in this paper are valid only under specific conditions, and their generalizability is limited.

First, CRC incidence, age distribution, and lifestyle factors vary widely across countries and regions. This means that the thresholds for risk stratification and the optimal number of tests may differ. Second, the performance of FIT tests can vary depending on the specific product used and the sample processing conditions. Therefore, the results obtained in one country may not be directly comparable to those in other countries. Third, variations in healthcare systems and cost structures (e.g., colonoscopy availability and cost levels) may impede the reproducibility of the cost-saving and waiting list–reduction effects documented here. Furthermore, given that the current study focused on symptomatic patients, applying the same approach directly to asymptomatic screening populations would not be appropriate.

The proposal of this study is conceptually interesting. However, additional cohort studies in different populations—including asymptomatic groups—and validation in other countries are indispensable for confirming its effectiveness and broader applicability. Such validation would further strengthen the clinical significance of the study.

Suggestions for Figure 1.

Please clarify the x-axis label as "Number of FIT tests (times)" and the y-axis label as "Probability of CRC (%)."

Please provide a legend indicating the following: the red line indicates the cumulative CRC probability; the orange to green lines indicate the transitions of each sub-cohort.

Suggestions for Figure 2.

Please revise the x-axis label to "Number of FIT tests (times)" and the y-axis label to "Number of patients."

Please define abbreviations such as "BMI" (presumably Bayesian Multiple Inference) in a legend or footnote.

Please add a legend to explain the meaning of each line color (FIT+, FIT–, FIT++, etc.), as the current labeling is dispersed and not intuitive.

Regarding the description of data analysis methods

The current explanation remains largely conceptual and does not explicitly present the relationship between prior, likelihood, and posterior distributions in mathematical form. In order to ensure optimal comprehension of the sequential updating process of Bayesian inference, equations should be included to facilitate readers' understanding of the analytical steps. For instance, demonstrating how different FIT result combinations (e.g., FIT+++, FIT+++–, FIT–––) transform prior into posterior probabilities with a concrete calculation example would enhance clarity. A simplified algorithmic outline would also be beneficial. Additionally, incorporating posterior predictive checks or metrics, such as ROC/PR curves, to assess the predictive performance of the proposed model, would further enhance the reliability of the findings.

Author Response

Common paragraph for all reviewers:

Dear reviewers,

We really appreciate the quality and depth of your reviews and comments. They really allowed us to improve the paper, up to the point that a lot of it has been rewritten to take your remarks into account. Some remarks of one reviewer overlapped with other remarks of other reviewers. We tried to take all of them into account for this new version.

We hope that you will find that this new version is better than the previous one. We are also open to another round of improvement should you wish to make new comments on this new version.

Thanks again for your work. It was much appreciated.

Reviewer 1:

Comment 1: This study is based on data from a symptomatic patient cohort in Chile [25]. The proposed prioritization strategy using CBI for multiple FIT tests is highly dependent on the epidemiological background and healthcare resource environment of that population. Therefore, the results obtained in this paper are valid only under specific conditions, and their generalizability is limited.

Response 1: This is a strength of the Bayesian approach, not a flaw: the problem with frequentist approaches is that they need thousands of points to make a statement. This means that it is rare that a unique hospital has enough data to get interesting results from a frequentist point of view.

As a result, most frequentist studies use data from multiple hospitals or even multiple countries in order to get more cases and data points, but… living conditions in the US are different from living conditions in China, so results from a Chinese cohort cannot be used for US patients. Chile is a very diverse country, with the driest desert in the North (Atacama desert) down to Antarctica in the South. Therefore, data on CRC collected from the North will not apply for patients from the South of Chile. What is nice with this approach is that:

we can use the prevalence observed in a specific cohort of patients of a specific hospital as a prior prevalence.
For sensitivity (False Negatives), it is given by the threshold of the FIT or FOBT test that is used. This can be measured over thousands of patients over the world. But the way a local laboratory manipulates the samples may have an influence on False Negatives too. So in this case, we can start with validated international values, and as new cases are obtained in a hospital, we can adapt the value of specificity as the results come in, for a specific hospital.
For specificity (False Positives), as stated in the paper, it will vary depending on the cohort. It will be very high for asymptomatic patients and much lower for symptomatic patients where the symptom of CRC will be common as the symptom for haemorrhoids, for instance. Here again, we can use the prior observed specificity from a specific cohort from a specific hospital, and update it as values are coming in.

Therefore, this approach allows doctors to have very precise results that will apply to THEIR specific cohort, but possibly not to another cohort from another hospital with a different population.

Finally, the paper is about presenting a new method, using the results on a specific colorectal cancer cohort showing what can be achieved by this method. It is probable that the results for a Chinese cohort could be different.

Comment 2: First, CRC incidence, age distribution, and lifestyle factors vary widely across countries and regions. This means that the thresholds for risk stratification and the optimal number of tests may differ. Second, the performance of FIT tests can vary depending on the specific product used and the sample processing conditions. Therefore, the results obtained in one country may not be directly comparable to those in other countries. Third, variations in healthcare systems and cost structures (e.g., colonoscopy availability and cost levels) may impede the reproducibility of the cost-saving and waiting list–reduction effects documented here. Furthermore, given that the current study focused on symptomatic patients, applying the same approach directly to asymptomatic screening populations would not be appropriate.

Response 2: Totally, cf. response 1. Specific previous prevalence must be computed (using data history of the hospital) for a symptomatic cohort as well as specific information on previous FIT tests for computing local FP and FN. Without previous FIT test information, a starting point could be the generally observed values for FIT tests and then, updated as results come in. For population screening without previous FIT specificity / sensitivity information, previous prevalence (such as obtained from statistical studies like the one of Susana Mondschein in Chile is the starting point, with generally observed FP and FN rates for FIT tests.

Comment 3: The proposal of this study is conceptually interesting. However, additional cohort studies in different populations—including asymptomatic groups—and validation in other countries are indispensable for confirming its effectiveness and broader applicability. Such validation would further strengthen the clinical significance of the study.

Response 3: The approach is currently totally mathematical. It will apply flawlessly to any existing cohort provided we already have the prevalence and specificity of the specific cohort (sensitivity comes from the type of test that is used) because it is descriptive of the available information.

For new cohorts from a new hospital starting a new department of oncology, their doctors could look for similar cohorts from similar hospitals to start with their observed prevalence and specificity and see how the results from the new patients will update the prior knowledge taken from another cohort.

However, FIT++++ patients will always have a higher probability to have CRC than FIT+++– or FIT–––– patients. There is very low probability that a FIT–––– patient could have a higher probability to have CRC than a FIT++++ patient but of course, nothing should be excluded. It is possible that a FIT–––– has CRC and a FIT++++ patient does not. It is just not probable. It is important to remind the doctors that this method is descriptive of the prior prevalence, specificity and sensitivity of their cohort. When a new patient comes in, it will always be the doctor’s responsibility to prioritize the patient for a colonoscopy over another patient. This mathematical combined Bayesian approach that we propose is here to help the doctor take a decision on new patients coming in. We updated the paper to try to make this point more salient.

The way a hospital service can use this method is the following:

During an initial period where good prevalence, specificity and sensitivity values must be found for the specific cohort (before using it for removing patients from colonoscopies):

Perform colonoscopies on all patients.
While the patients are waiting for their colonoscopy, perform 4 FIT tests on them.
Use the result of FIT tests to prioritize patients (those who are FIT++++ and FIT+++– should be prioritized).

Once good prevalence, specificity and sensitivity values have been obtained (this can be determined when, observed results are consistent with those obtained from previous data:

It can be proposed to FIT–––– (or even FIT+–––) patients (in accordance with the ethics committee of the hospital) to skip the colonoscopy (with the great advantage that they are much more numerous than FIT++++ or FIT+++– patients).
For FIT++–– patients, they could be given a lower priority or prescribed more FIT tests.

Suggestions for Figure 1.

Please clarify the x-axis label as "Number of FIT tests (times)" and the y-axis label as "Probability of CRC (%)."

Please provide a legend indicating the following: the red line indicates the cumulative CRC probability; the orange to green lines indicate the transitions of each sub-cohort.

Response to suggestions for Figure 1:

The new caption of Fig1 is: Evolution of the probability of having CRC depending on the positivity or negativity of FIT tests. FIT+ goes up, FIT– goes down. The red line indicates the cumulative CRC probability for people who continually have positive FIT results. Having a single negative FIT test brings to the 2nd orange sigmoid. 2 negative FITs bring down the probability of CRC to the 3rd yellow sigmoid. 3 negative FIT tests bring down the probability of having CRC to the 4th green sigmoid.

The axis labels have been renamed “Number of FIT tests (times)” and “Probability of CRC (%)”.

Suggestions for Figure 2.

Please revise the x-axis label to "Number of FIT tests (times)" and the y-axis label to "Number of patients."

Please define abbreviations such as "BMI" (presumably Bayesian Multiple Inference) in a legend or footnote.

Please add a legend to explain the meaning of each line color (FIT+, FIT–, FIT++, etc.), as the current labeling is dispersed and not intuitive.

Response to suggestions for Figure 2:
The legend to Fig 2 is now: Evolution of the number of positive or negative patients for 1 to 5 FIT CBI tests on the L/MR cohort. The red dot on top shows all 808 patients that did one FIT test. The orange line shows the number of patients who, on a single FIT test, were positive (282 FIT+) or negative (526 FIT–). The yellow line shows how many would have been FIT++, FIT+– or FIT++ had they been proposed to do 2 FIT tests. The dark blue line shows how many would have been FIT+++, FIT++–, FIT+––, FIT––– had they been proposed to do 3 FIT tests. The green line shows how many would have been FIT++++, FIT+++–, FIT++––, FIT+–––, FIT–––– had they been proposed to do 4 FIT tests and finally, the turquoise line shows how many would have been FIT+++++, FIT++++–, FIT+++––, FIT++–––, FIT+––––, FIT––––– had they been proposed to do 5 FIT tests.

BMI has been changed to CBI (thanks for spotting it).

Axes have been renamed “Number of FIT tests (times)” and “Number of patients”.

Regarding the description of data analysis methods

The current explanation remains largely conceptual and does not explicitly present the relationship between prior, likelihood, and posterior distributions in mathematical form.
In order to ensure optimal comprehension of the sequential updating process of Bayesian inference, equations should be included to facilitate readers' understanding of the analytical steps. For instance, demonstrating how different FIT result combinations (e.g., FIT+++, FIT+++–, FIT–––) transform prior into posterior probabilities with a concrete calculation example would enhance clarity.
A simplified algorithmic outline would also be beneficial.
Additionally, incorporating posterior predictive checks or metrics, such as ROC/PR curves, to assess the predictive performance of the proposed model, would further enhance the reliability of the findings.

Response to “Regarding the description of data analysis methods”:

The contents of this paper is currently purely conceptual and purely descriptive of the analysis of the cohorts that we proposed to analyse. We are currently submitting a project to validate this approach with a clinical trial. So we modified the paper to highlight that this work is purely descriptive and not predictive. But as we indicated in the “Discussion” section of the paper, even confidence intervals are descriptive and not predictive.
We added Bayes’ equation and tried to be more clear on how everything was computed. We used a hypothetical numerical example (in the “Discussion” section) to explain how an indicator such as “age” could be added.
Combined Bayesian Inference can be computed thanks to a recursive function (here is the C function that we use for computing our results):

void mbi(int nBinary, char *sPos, double dCS, double dNS, double dFP, double dFN, int nNbIt, int nTestNb){

if (nNbIt==0) return;

int nIndex=nBinary;

double dPositive=(dCS-dNS)*dFP/100+(dNS-(dNS*dFN/100));

double dPosSick=dNS-(dNS*dFN/100);

double dPosHealthy=(dCS-dNS)*dFP/100;

double dPosSickProb=dPosSick/dPositive*100;

double dPosHealthyProb=dPosHealthy/dPositive*100;

dPROB[nTestNb-1][nIndex*2+1]=dPosSickProb;

double dNegative=(dCS-dNS)-((dCS-dNS)*dFP/100)+(dNS*dFN/100);

double dNegHealthy=(dCS-dNS)-((dCS-dNS)*dFP/100);

double dNegSick=dNS*dFN/100;

double dNegSickProb=dNegSick/dNegative*100;

double dNegHealthyProb=dNegHealthy/dNegative*100;

dPROB[nTestNb-1][nIndex*2]=dNegSickProb;

char *sNewPos=(char *)malloc(nTestNb+1);

strcpy(sNewPos, sPos);

strcat(sNewPos,"+");

mbi(nIndex*2+1, sNewPos ,dPositive, dPosSick, dFP, dFN, nNbIt-1, nTestNb+1);

strcpy(sNewPos, sPos);

strcat(sNewPos,"-");

mbi(nIndex*2, sNewPos, dNegative, dNegSick, dFP, dFN, nNbIt-1, nTestNb+1);

}

Do you think a cleaned up algorithmic version of this c code would be helpful ?

No predictive analysis has been performed, because right now, in absence of a real clinical test, we only propose a descriptive analysis of what was observed on the cohort of [25] based on the published data. However, all the values are mathematically exact, because they are based on the evidence found in the [25] paper.

Thanks a lot for your thorough review. It was really appreciated.

Reviewer 2 Report

Comments and Suggestions for Authors

The reviewer understands that Collet et al. have presented a manuscript entitled "Potential Impact of Compound Bayesian Inference in Medicine: Application to Colonoscopy Prioritization". The reviewer has a few suggestions and they would like to request authors to kindly answer all the questions by updating their manuscript.

1) The sensitivity (73%) and specificity (94%) values used for FIT are based on published literature. Given that test performance may vary across cohorts, laboratories, and healthcare systems, how sensitive are your conclusions to deviations from these parameters?
2) The work is primarily mathematical. Do the authors plan to validate the 4-FIT CBI protocol with real-world prospective clinical trials, and what statistical power would be needed to confirm the predicted >85–98% colonoscopy reduction?
3) The CBI framework assumes independence of consecutive FIT tests due to intermittent bleeding. How robust is this assumption in real-world settings, and how might correlated test results affect the validity of the model?
4) While CBI minimizes needless colonoscopies, it still has the possibility of missing CRC patients due to FIT false negatives. How do the authors recommend reconciling resource optimization with patient safety in clinical decision-making?
5) The study provides a basic cost comparison. Have the authors completed (or intend to conduct) a formal cost-effectiveness or cost-utility study (e.g., QALYs, ICER) to support the health economics case for policymakers?
6) How does CBI stack up against alternative multi-test or machine learning-based risk stratification approaches (e.g., incorporating blood biomarkers, genomic data, or AI classifiers) in terms of predicted accuracy and scalability?
7) The suggested CBI could reprioritize patients in places such as Chile, where waiting lists are longer than one year. How do the authors plan to incorporate such a strategy into existing clinical workflows while addressing ethical concerns about deprioritizing symptomatic but FIT- patients?
8) By mistake, section 2.3 appears twice in your manuscript. (2.3. Bibliographic analysis) and (2.3. Proposed Compound Bayesian Inference (CBI) and application to colonoscopy priorization). Kindly rectify that.
9) You have used different colors to illustrate different data values in almost all tables. Please provide a key for each color and its meaning. It is very difficult for the reader to understand. your color-coding language. Provide a color code-specific key below each table to present a reader-friendly data.
10) Provide units for X and Y axes in all figures.
11) Rename section 4. Discussion as 4. Discussions.
12) Provide additional attractive figures to support the results in from your manuscript.
13) Rename section 5. Conclusion as 5. Conclusion and Future Perspectives and update the section information regarding how you plan to use your work for serving the betterment of humanity.

Author Response

Common paragraph for all reviewers:

Dear reviewers,

We hope that you will find that this new version is better than the previous one. We are also open to another round of improvement should you wish to make new comments on this new version.

Thanks again for your work. It was much appreciated.

Reviewer 2:

Comment 1) The sensitivity (73%) and specificity (94%) values used for FIT are based on published literature. Given that test performance may vary across cohorts, laboratories, and healthcare systems, how sensitive are your conclusions to deviations from these parameters?

Response 1: The contents of this paper is purely conceptual and purely descriptive of the analysis of the cohorts that we proposed to analyse. We are currently submitting a project to validate this approach with a clinical trial, to see if the descriptive values have a predictive value. Until we do this, we do not have any data for predictivity validation, so we modified the paper to highlight that this work is purely descriptive and not predictive. But as we indicated in the paper, even confidence intervals are descriptive and not predictive.

Therefore, the ideas presented here are exclusively descriptive of what can be mathematically inferred (with Bayes’ theorem) from the result of a hypothetical FIT test:

on an asymptomatic population using the prevalence of CRC found in a statistical study from [18] and
on a real FIT + colonoscopy clinical test published in (now accepted) [25].

For the clinical test of [25], we could compute (as was done in the paper) the real specificity and sensitivity of the FIT test. They were very different from the published literature.

So for clinical studies or hospitals who would like to try this method, they should first compute their own values before using the results of this approach.

Comment 2) The work is primarily mathematical. Do the authors plan to validate the 4-FIT CBI protocol with real-world prospective clinical trials, and what statistical power would be needed to confirm the predicted >85–98% colonoscopy reduction?

Response 2: Yes. We are currently submitting a clinical trial project to confirm the reduction.

Based on the following papers:

Katsoula A, Paschos P, Haidich AB, Tsapas A, Giouleme O. Diagnostic Accuracy of Fecal Immunochemical Test in Patients at Increased Risk for Colorectal Cancer: A Meta-analysis. JAMA Intern Med. 2017 Aug 1;177(8):1110-1118. doi: 10.1001/jamainternmed.2017.2309. PMID: 28628706; PMCID: PMC5710432.
Young GP, Woodman RJ, Symonds E. Detection of advanced colorectal neoplasia and relative colonoscopy workloads using quantitative faecal immunochemical tests: an observational study exploring the effects of simultaneous adjustment of both sample number and test positivity threshold. BMJ Open Gastroenterol. 2020 Sep;7(1):e000517. doi: 10.1136/bmjgast-2020-000517. PMID: 32994195; PMCID: PMC7526287.

on the diagnostic accuracy of FIT tests in colorectal cancer (CRC) detection, we estimate that a total of 1500 participants would be necessary for a precision margin within a 95% confidence interval. However, all the literature shows that all patients do not follow the protocol, so in order to have 1500 exploitable data points, it will be necessary to recruit more patients. We are therefore planning on 2000 patients in total.

Comment 3) The CBI framework assumes independence of consecutive FIT tests due to intermittent bleeding. How robust is this assumption in real-world settings, and how might correlated test results affect the validity of the model?

Response 3: We don’t know for real, but interestingly the simulations show (cf. Fig. 2) that false negatives start to appear for 5 FIT tests. We are applying for funds to perform a real-world clinical trial to validate this study.

Comment 4) While CBI minimizes needless colonoscopies, it still has the possibility of missing CRC patients due to FIT false negatives. How do the authors recommend reconciling resource optimization with patient safety in clinical decision-making?

Response 4: A-posteriori analysis of the published data shows that FIT–––– people have near-zero risk of having CRC. Better, we show that they have a lower risk to have CRC than the prevalence of CRC in asymptomatic patients. This could be the basis for recommending that a colonoscopy could be cancelled, or delayed. But remember that this is only descriptive of what was observed in the past (it is true “evidence-based medicine”) and they show probabilities. The probability of a winning ticket is very low, but over a large number of participants, there are winners. Here as well, having a 0.0001% of having CRC cannot unfortunately discard the possibility that a FIT–––– person could have CRC. It is just very improbable.

Comment 5) The study provides a basic cost comparison. Have the authors completed (or intend to conduct) a formal cost-effectiveness or cost-utility study (e.g., QALYs, ICER) to support the health economics case for policymakers?

Response 5: This is an interesting idea that we have not explored yet. Thanks for suggesting it. We will conduct it (but did not have time to do it in this revision).

Comment 6) How does CBI stack up against alternative multi-test or machine learning-based risk stratification approaches (e.g., incorporating blood biomarkers, genomic data, or AI classifiers) in terms of predicted accuracy and scalability?

Response 6: Combined Bayesian Inference is not incompatible with other source of probabilities, such as blood biomarkers or others. In fact, they would probably provide a more “independent” source of information than several identical FIT tests. If a blood biomarker information is available, it can be inserted as a test with its own specific FP and FN rate. We also show in the new “Discussion” section how these sources could be added to the multiple FIT tests.

The problem with AI classifiers is that they are not explainable. The advantage of CBI is that it is 100% explainable with the previously available data because it is 100% mathematical.

Comment 7) The suggested CBI could reprioritize patients in places such as Chile, where waiting lists are longer than one year. How do the authors plan to incorporate such a strategy into existing clinical workflows while addressing ethical concerns about deprioritizing symptomatic but FIT- patients?

Response 7: Because the number of patients who really have CRC is very small, even in symptomatic cohorts, the number of FIT++++ patients is very small too, compared with the size of the whole cohort. Therefore, getting FIT++++ patients to have a prioritized colonoscopy will not massively impact the waiting of other patients. Then, it is already done. In the [25] study, the initial cohort has been divided into a “High risk” cohort of patients who directly had a colonoscopy and a low-medium risk cohort who was proposed a FIT test before the colonoscopy.

This means that such prioritization is already done. The advantage of our method is that we would have numerical values to justify the prioritization in front of an ethics committee.

Comment 8) By mistake, section 2.3 appears twice in your manuscript. (2.3. Bibliographic analysis) and (2.3. Proposed Compound Bayesian Inference (CBI) and application to colonoscopy priorization). Kindly rectify that.

Response 8: Thanks for spotting this. We worked a lot on the paper and in this revised version we made sure the the sections numbering was sequential.

Comment 9) You have used different colors to illustrate different data values in almost all tables. Please provide a key for each color and its meaning. It is very difficult for the reader to understand. your color-coding language. Provide a color code-specific key below each table to present a reader-friendly data.

Response 9: We added colour in an attempt to improve readability. The current “colour-coding language” is the following;

Green from 0% CRC probability to dark pink for 4% probability to black for >50% CRC probability.
We currently added the colour code to the description of the table. However, we can easily remove colours if it reduces readability rather than improving it.

Comment 10) Provide units for X and Y axes in all figures.

Response 10: Done.

Comment 11) Rename section 4. Discussion as 4. Discussions.

Response 11: Done.

Comment 12) Provide additional attractive figures to support the results from your manuscript.

Response 12: We tried to improve on the quality of the figures in this revised version.

Comment 13) Rename section 5. Conclusion as 5. Conclusion and Future Perspectives and update the section information regarding how you plan to use your work for serving the betterment of humanity.

Response 13: We reformulated the Conclusion and future perspectives and ended it with the following sentence:

“We hope that the new approach presented here (Combined Bayesian Inference) could contribute to improving not only CRC detection and treatment, but also general health care around the world.”

Thanks a lot for your thorough review. It was really appreciated.

Reviewer 3 Report

Comments and Suggestions for Authors

The article discusses the applicability of the Compound Bayesian Inference (CBI) method in colon cancer screening and demonstrates the potential benefits of using multiple FIT (Fecal Immunochemical Test) tests in both asymptomatic and symptomatic cohorts using a mathematical model. This topic is particularly relevant and of high clinical relevance, especially considering the long colonoscopy waiting times in Chile. However, some corrections should be made to the article:

The CBI approach assumes that consecutive FIT tests are independent of each other. However, in practice, biological processes (e.g., bleeding patterns of polyps) within the same individual may not demonstrate complete independence. The limitations of this assumption should be discussed more clearly.

The tables are very complex, with a coloring design that is far from scientific.

The decimal places of the numbers should be given equally.

The discussion section is inadequate and needs to be improved.

Some sections of the article contain irregularities in English and long sentences; simplifying the language would increase its impact.

The authors mathematically demonstrate that four FIT tests are optimal. However, in practice, collecting four separate samples may present challenges in terms of patient compliance and logistics. This aspect also needs to be discussed for clinical applicability.

Author Response

Common paragraph for all reviewers:

Dear reviewers,

We hope that you will find that this new version is better than the previous one. We are also open to another round of improvement should you wish to make new comments on this new version.

Thanks again for your work. It was much appreciated.

Reviewer 3:

Comment 1) The CBI approach assumes that consecutive FIT tests are independent of each other. However, in practice, biological processes (e.g., bleeding patterns of polyps) within the same individual may not demonstrate complete independence. The limitations of this assumption should be discussed more clearly.

Response 1: This can only be verified by a real-world clinical trial. We are currently filing a clinical trial project to validate the findings.

Comment 2) The tables are very complex, with a coloring design that is far from scientific.

Response 2: We added colour in an attempt to improve readability. The current colour-coding is the following;

Green from 0% CRC probability to dark pink for 4% probability to black for >50% CRC probability.
We have added a colour code description to the tables description. However, we can easily remove the colours if they reduce readability rather than improve it.

Comment 3) The decimal places of the numbers should be given equally.

Response 3: The number of decimal places is very important for prevalence, because prevalence will compound with each new FIT test. Therefore (and for reproducibility of the results) we felt it was necessary to include many decimals in the probability of being sick or healthy resulting from the positivity or negativity of a FIT test.

Conversely, when displaying the number of affected people, the number of decimal places is much less important, which is why we reduced it to 2.

But we can use an identical number of decimal places everywhere if you think it would improve readability.

Comment 4) The discussion section is inadequate and needs to be improved.

Response 4: We totally agree. The Discussion section has been completely rewritten, notably by thanks to your remarks in comment 6.

Comment 5) Some sections of the article contain irregularities in English and long sentences; simplifying the language would increase its impact.

Response 5: We worked a lot on the text in order to it. If you still find too long sentences, please point them to us so that we could reword them.

Comment 6) The authors mathematically demonstrate that four FIT tests are optimal. However, in practice, collecting four separate samples may present challenges in terms of patient compliance and logistics. This aspect also needs to be discussed for clinical applicability.

Response 6: This question has been exposed in section 3.1 and in the discussion.

Thanks a lot for your thorough review. It was really appreciated.

Reviewer 4 Report

Comments and Suggestions for Authors

This manuscript reports the findings from a mathematical framework for prioritizing colonoscopies based on a sequence of four faecal immunochemical tests (FIT). The methodology is reasonable and well thought out and the approach has utilized an appropriate dataset. The results on optimization of the number of FIT are of particular interest. The manuscript is clearly written and the results well summarised in the accompanying figures and tables. Whilst the approach has promise as a more robust means of prioritizing colonoscopies, the exact % probabilities computed are based on just a couple of references and would benefit from further validation with more datasets. Similarly, any implementation of such a prioritization framework would require future empirical data validating the approach. There are some points for the authors to consider and these are detailed below.

Page 1 lines 36-37 “. . . shows that over . . . (resp. asymptomatic) patients . . .” The meaning of the resp. details given in brackets is unclear. Either remove these bracketed details or change the language to make the significance clear. This is particularly important in the abstract that needs to be rapidly comprehensible.
Page 2 line 85 “. . . studied a lot [7, 8, 9, 10, 11, 12, …].” Don’t use an ellipsis (…) in the reference list as the reference list should be complete and not ill-defined. Also, consecutive references can be cited using a range, e.g. [7-12].
Page 3 lines 128-129 “. . . in Region XI . . . the capital).” You have given the same region number (XI) to two different geographic regions. I assume this is incorrect, so please correct as appropriate.
Page 4 lines 137-142 “It mathematically follows . . . = TN).” I am a bit concerned that the language here is too certain about the outcomes, e.g. “19.6 are sick”. As you are dealing with probabilities, then if the current model holds, then on average 19.6 persons would be sick, but of course there is error/uncertainty associated with the model and so real numbers from an appropriate sample population may differ slightly.
Page 7 line 272 “Out of the . . . sensitivity of 96.3%.” The patient numbers here are quite low and so this is likely to introduce some significant uncertainties into the computation of the sensitivity.
I am concerned that the errors and uncertainties in the analysis have not been adequately considered. Some discussion of this is needed, particularly in relation to the % probability figures presented throughout the manuscript. As a related issue, in many places the % figures have been presented to too many decimal places, e.g. percentages in the sick and healthy columns in Table 2. The number of decimal places should be consistent with the precision of the model.
Page 8 line 305 “Table 77.” There is no Table 77, nor even a Table 7. Correct the numbering here.
Page 9 lines 318-319 “Fig. 1 clearly shows . . . for this cohort.” I don’t quite agree with the wording of this statement. The probability % does increase somewhat, particularly between the fourth and fifth tests, but it could be argued that the gain is not worth the extra expense or, as your later analysis points out, the greater risk of false negatives.

Comments on the Quality of English Language

The quality of the English is good but not perfect. There are some instances of grammatical or typographical errors that should be corrected in an editorial check.

Author Response

Common paragraph for all reviewers:

Dear reviewers,

We hope that you will find that this new version is better than the previous one. We are also open to another round of improvement should you wish to make new comments on this new version.

Thanks again for your work. It was much appreciated.

Reviewer 4:

Comment 1: Page 1 lines 36-37 “. . . shows that over . . . (resp. asymptomatic) patients . . .” The meaning of the resp. details given in brackets is unclear. Either remove these bracketed details or change the language to make the significance clear. This is particularly important in the abstract that needs to be rapidly comprehensible.

Response 1: Thanks for your remark. We reformulated the paragraph Results in order for it to be more understandable.

Comment 2: Page 2 line 85 “. . . studied a lot [7, 8, 9, 10, 11, 12, …].” Don’t use an ellipsis (…) in the reference list as the reference list should be complete and not ill-defined. Also, consecutive references can be cited using a range, e.g. [7-12].

Response 2: The reference list cannot be complete, as several hundred papers have been written on FIT or FOBT tests. We removed the dots and used a range.

Comment 3: Page 3 lines 128-129 “. . . in Region XI . . . the capital).” You have given the same region number (XI) to two different geographic regions. I assume this is incorrect, so please correct as appropriate.

Response 3: Thanks for spotting this. We gave the regions their correct number.

Comment 4: Page 4 lines 137-142 “It mathematically follows . . . = TN).” I am a bit concerned that the language here is too certain about the outcomes, e.g. “19.6 are sick”. As you are dealing with probabilities, then if the current model holds, then on average 19.6 persons would be sick, but of course there is error/uncertainty associated with the model and so real numbers from an appropriate sample population may differ slightly.

Response 4: As explained in the new “Discussion” section, statistics is descriptive by essence. Confidence intervals can be computed but they are descriptive too! So in order to make this paper more simple to read, we decided not to complexify it with confidence intervals.

The 19.6/100000 value comes from the study of S. Mondschein who explained in her paper how she obtained this number (from “registries of mortality and hospital discharges, making follow-up of the individuals possible”). Of course, there are always errors and no data set is perfect, but 19.6/100000 is the observed prevalence of CRC in Chile’s metropolitan region.

Then, as explained in the new discussion session, Combined Bayesian Inference is exploiting the existing data to show what was the probability of FIT++++, FIT+++–, etc. people to have CRC using existing data.

So the computed numbers are exact. Their inaccuracy does not come from the method. It could however come from errors in the prevalence, specificity and sensitivity of FIT tests for the general screening section.

For the priorization of symptomatic patients, the accuracy of the obtained numbers is also related to the accuracy of prevalence, specificity and sensitivity. In this cohort of exactly 808 people, 27 were diagnosed with CRC by colonoscopy (the gold standard) so the prevalence is 27/808. The inaccuracy in prevalence could come from a wrong diagnosis of CRC by colonoscopy (which can happen, because, like everything, colonoscopy also has its false positives and false negatives). So we could have used a confidence interval for 27, that would have translated with a confidence interval for the prevalence and FP and FN values of the FIT test,

This would have made the description of the approach overly complicated for a paper. But because all this is computed, we can add for each prior value a confidence interval linked to the diagnosis method. A computer will deal with it easily and provide results with their confidence intervals. We retain the idea and will use it in the computer implementation of the method.

Comment 5: Page 7 line 272 “Out of the . . . sensitivity of 96.3%.” The patient numbers here are quite low and so this is likely to introduce some significant uncertainties into the computation of the sensitivity.

I am concerned that the errors and uncertainties in the analysis have not been adequately considered. Some discussion of this is needed, particularly in relation to the % probability figures presented throughout the manuscript. As a related issue, in many places the % figures have been presented to too many decimal places, e.g. percentages in the sick and healthy columns in Table 2. The number of decimal places should be consistent with the precision of the model.

Response 5: The model is mathematically exact. The computed sensitivity of the test comes from the observed fact that out of the 27 patients diagnosed with CRC, 1 was FIT negative. So the FIT test here delivered a false negative.
100/27 is 3.703703704… 100 - 3703703704 = 96,2962962963 (rounded to 96.3). No statistics were involved in finding this number. Only a division and a subtraction.

However, when we are computing the prevalence of a subcohort out of the prevalence of the previous subcohort, we are compounding the prevalences. And here, decimal places are important. This is why we keep a large number of decimal places where they are needed (and for reproducibility), and we rounded to 2 decimal places when the rounding would not impact the calculation. But we can round to 2 decimals everywhere if you think it would improve the readability of the paper.

We explained this in the paragraph before Table 2.

Comment 6: Page 8 line 305 “Table 77.” There is no Table 77, nor even a Table 7. Correct the numbering here.

Response 6: Thanks for spotting this. There is no more table 77 now.

Page 9 lines 318-319 “Fig. 1 clearly shows . . . for this cohort.” I don’t quite agree with the wording of this statement. The probability % does increase somewhat, particularly between the fourth and fifth tests, but it could be argued that the gain is not worth the extra expense or, as your later analysis points out, the greater risk of false negatives.

Response 6: This has been reworded into:
“Fig. 1. shows that performing a fifth or sixth test would not yield much additional significant information, showing a limit of the multiple FIT tests for this cohort. The probability % increases somewhat, but the gain is not worth the extra expense.“

Thanks a lot for your thorough review. It was really appreciated.

Reviewer 5 Report

Comments and Suggestions for Authors

In this manuscript, Pierre Collet and coworkers address an important global health challenge: optimizing colorectal cancer detection and prioritization in settings with limited colonoscopy availability. By applying Compound Bayesian Inference (CBI) to four sequential FIT tests, the authors demonstrate mathematically that unnecessary colonoscopies could be reduced by over 85% in symptomatic patients and nearly 99% in asymptomatic patients, while dynamically stratifying patient risk. The approach is innovative, cost-effective, and highly relevant for healthcare systems such as Chile, where colonoscopy waiting times can exceed one year and survival outcomes are compromised. Importantly, the method is transparent, explainable, and adaptable, with potential applications not only in colorectal cancer screening but also in integrating diagnostic, genomic, and clinical markers across other diseases. Although the current study is descriptive and theoretical, it provides a strong rationale for real-world validation, which, if successful, could save lives and reduce healthcare costs. Overall, the manuscript is suitable for publication in Cancers following minor revisions.

Comment for Revision:

References: Ensure that all references in the manuscript are rewritten according to the specific citation format required by the Cancers journal for clarity and consistency.

Author Response

Common paragraph for all reviewers:

Dear reviewers,

We hope that you will find that this new version is better than the previous one. We are also open to another round of improvement should you wish to make new comments on this new version.

Thanks again for your work. It was much appreciated.

Reviewer 5:

Comment for Revision:

Comment 1) References: Ensure that all references in the manuscript are rewritten according to the specific citation format required by the Cancers journal for clarity and consistency.

Response 1: Thanks for the nice analysis of our work. We really think Combined Bayesian Inference could help not only for CRC but for other diseases as well.

As for the bibliography format, the submission site states that:

Your references may be in any style, provided that you use the consistent formatting throughout. It is essential to include author(s) name(s), journal or book title, article or chapter title (where required), year of publication, volume and issue (where appropriate) and pagination. DOI numbers (Digital Object Identifier) are not mandatory but highly encouraged. The bibliography software package EndNote, Zotero, Mendeley, Reference Manager are recommended.

When your manuscript reaches the revision stage, you will be requested to format the manuscript according to the journal guidelines.

We believe all our citations contain the essential information citede above. We did not yet get precise guidelines for the bibliography section. We will go through the section when we get them from the journal (probably after the paper is accepted by the reviewers, when elaborating the “final proof” version.

Thanks again for your nice review.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have provided detailed and thoughtful responses to the reviewers' comments. The revised version demonstrates significant enhancement in clarity and presentation, particularly with regard to the figures, explanatory captions, and the elucidation of the descriptive nature of the proposed Bayesian approach. The effort to add Bayes' equation, a numerical example, and an algorithmic implementation (C code) is appreciated. These additions significantly enhance the transparency of the analytical process.

The responses also provide a reasonable rationale for the local specificity of the model and its adaptability to different hospital-based cohorts, which partially addresses concerns about generalizability. The revised manuscript now clearly positions the method as a conceptual and descriptive framework rather than a predictive model, which is appropriate given the current stage of development.

However, to further strengthen the manuscript before acceptance, I recommend the following minor revisions:

While the additional algorithmic description is valuable, please include a short paragraph in the Methods or Appendix describing the recursive updating process in plain language so that readers without programming experience can understand it.

Author Response

Thank you for your constructive second review. Thanks to your feedback, we have created a flowchart to describe the algorithm and greatly improved section 2.3, which we hope now properly describes our algorithm and approach.

We think this has greatly improved the paper.

Reviewer 4 Report

Comments and Suggestions for Authors

The authors have made significant changes to the manuscript that in my view have improved the clarity and presentation of the analysis. Their responses to the review comments are also adequate.

Author Response

Dear reviewer,

We did not find any requests for improvements in your second review. We think that you uploaded your comments on our first set of improvements as a second request for improvements by mistake.

Thank you for your positive review of our improved paper.

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Potential Impact of Updated Bayesian Deduction in Medicine: Application to Colonoscopy Prioritization

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