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Multiple Criteria Optimization (MCO): A gene selection deterministic tool in RStudio

Identifying genes with the largest expression changes (gene selection) to characterize a given condition is a popular first step to drive exploration into molecular mechanisms and is, therefore, paramount for therapeutic development. Reproducibility in the sciences makes it necessary to emphasize objectivity and systematic repeatability in biological and informatics analyses, including gene selection. With these two characteristics in mind, in previous works our research team has proposed using multiple criteria optimization (MCO) in gene selection to analyze microarray datasets. The result of this effort is the MCO algorithm, which selects genes with the largest expression changes without user manipulation of neither informatics nor statistical parameters. Furthermore, the user is not required to choose either a preference structure among multiple measures or a predetermined quantity of genes to be deemed significant a priori. This implies that using the same datasets and performance measures (PMs), the method will converge to the same set of selected differentially expressed genes (repeatability) despite who carries out the analysis (objectivity). The present work describes the development of an open-source tool in RStudio to enable both: (1) individual analysis of single datasets with two or three PMs and (2) meta-analysis with up to five microarray datasets, using one PM from each dataset. The capabilities afforded by the code include license-free portability and the possibility to carry out analyses via modest computer hardware, such as personal laptops. The code provides affordable, repeatable, and objective detection of differentially expressed genes from microarrays. It can be used to analyze other experiments with similar experimental comparative layouts, such as microRNA arrays and protein arrays, among others. As a demonstration of the capabilities of the code, the analysis of four publicly-available microarray datasets related to Parkinson´s Disease (PD) is presented here, treating each dataset individually or as a four-way meta-analysis. These MCO-supported analyses made it possible to identify MMP9 and TUBB2A as potential PD genetic biomarkers based on their persistent appearance across each of the case studies. A literature search confirmed the importance of these genes in PD and indeed as PD biomarkers, which evidences the code´s potential.

The focus of the manuscript is the presentation of the automated R code for MCO, which can support meta-analysis of up to five databases simultaneously. So, even if MCO is mathematically unbound in the number of datasets that it can accommodate, the R code has been designed for up to five datasets.
Finally, in the cases outlined for the illustrative example, four databases are available as described in Table  1, selected as explained in the Results section. These four datasets are used to support the analysis and meta-analysis cases detailed on Tables 2 and 3.
To make sure that the distinction is clear, the material was organized in subsections: MCO algorithm, MCO tool, and MCO tool -Application, where the particular bounds are discussed.

Reviewer 1, Issue 3
It is not clear how we get F? Also, rather than use 1, 2, 3, 4, and 5, it would be better use K to denote the number of datasets. Do rewrite Algorithm 1 to better illustrate it.

Follow up to Reviewer 1, Issue 3
Thanks for the advice. Besides the disambiguation provided previously, we have adopted the letter K to denote the number of databases of interest to the user to be analyzed through MCO (See Algorithm 1). For the R code, K falls between 1 and 5, and for the particular cases outlined in Tables 2 and 3, K is 1, 3, or 4 depending on the case.
On the other hand, F is the only algorithm-related parameter that is user-defined. If F=1, the Pareto Frontier is identified with all data points involved in the analysis. If F=2, the first frontier is calculated with all data points involved and then the solutions to this first frontier are removed to proceed to a second run of the algorithm. This procedure is necessary to accommodate experimental error in the datasets. Obtaining more than 1 efficient frontier results in a hierarchy of solutions.
In our experience, MCO converges to tens of solutions when analyzing tens of thousands initial genes. The number of frontiers (F) used on each analysis case in our manuscript is duly specified for single datasets (10 frontiers) and for meta-analysis (1 frontier).

Reviewer 1, Issue 4
The pictures are of poor quality.

Follow up to Reviewer 1, Issue 4
We have tested our figures using the software provided by PLOS. All of them passed the quality inspection. In attention to the reviewer's comment, we have increased the quality of our pictures even beyond these tests. The improved compliant set is now included in our resubmission.
In particular, we adjusted figures 1,3, 4,5 and 6. We also improved the following: the content in Fig 1 has been reorganized for clarity, the letter size was made larger for all graphs in the manuscript, and the size and letter type of the table in Fig 6 was modified to make it more readable. All figures passed the testing provided by the PACE program, as requested by the Journal instructions.

Reviewer 1, Issue 5
What are the differences between the MCO method with the commonly used filter, wrapper, embedded, and hybrid gene selection methods?

Follow up to Reviewer 1, Issue 5
MCO is based on mathematical optimization. Mathematical optimization is applied when searching for the demonstrably best possible solution(s) to a particular problem.
The demonstrably best possible solution to an optimization problem is called a global optimum, and it might or might not be accessible or recognizable in general.
In our paper and our previous work, gene selection is casted as a multiple criteria optimization problem. In addition, our algorithm guarantees the global optimality of the solutions (that is, the resulting selected set of genes) thanks to the Pareto efficiency conditions. So, MCO guarantees global optimality, which -to the best extent of our knowledge-is a characteristic not afforded by the other methods.
In addition, MCO arrives to the globally optimal solutions to the gene selection problem without asking the user to define a priori a final number of solutions, a preference structure between the different performance measures, the value of a threshold, or the use of a particular normalization procedure. The vast majority of gene selection methods would have at least one of these requirements. The fact that MCO does not require any of them, makes our method objective and repeatable.
Finally, as demonstrated by the cases outlined in Table 3, MCO can select important genes based on the simultaneous consideration of multiple datasets, in spite of the underlying experiments being incommensurable (i.e. measured in dissimilar units). This is, indeed, a capability unique to MCO.

Reviewer 1, Issue 6
The discussion is kind of superficial, which needs writing Follow up to Reviewer 1, Issue 6 We are deepening biological interpretation based on Gene Ontology analysis in the Discussion Section to further support the biological significance of the results that are possible through the capabilities afforded by our method.

Reviewer 2, Issue 1
In classification research, now I consider this contribution as pretty good to make a paper valuable for the field of interest. I am a convincing advocate of introducing the rigor of gene classification research in high dimensional datasets. However, I read this manuscript and I suggest to author please apply to show the competent with others recent statistical comparison method.

Follow up to Reviewer 2, Issue 1
When it comes to approaching a general class of optimization problems, such as those falling within Gene Selection, it is necessary to keep in mind the no-free-lunch theorem (NFLT) which states "that a generalpurpose, universal optimization strategy is impossible. The only way one strategy can outperform another is if it is specialized to the structure of the specific problem under consideration". In the development of MCO as presented in our manuscript, two important characteristics were considered: (1) the strategy should not require the manipulation of parameters for it to result in a competitive solution, and (2) the resulting solution should be globally optimal in presence of conflicting performance measures. The first characteristic would bring repeatability and the second one, objectivity.
In Reviewer 1's issues 1 and 5, we outlined and explained the distinctive advantages of our Gene Selection method, namely certainty (optimality), objectivity, consistent convergence, non-parametric nature, and the possibility of truly carrying out simultaneous meta-analysis in spite of incommensurability. These characteristics have been also discussed in our preceding publications.
A comparative study is now available as supplementary material in attention to Reviewer 2's Issue 1 and issue 3. This comparison includes four methods of gene selection: MCO, CFS, IG, and eBayes. The four methods were applied to the four datasets utilized in this manuscript. MCO was applied standardly with 10 frontiers as advised here, while the other methods were varied in their adjustment parameters. In addition, four types of classifiers were used to assess their classification accuracy: support vector machine, KNN, Treebag, Linear Discriminant Analysis and RF.
The comparison involves classification accuracy, classification accuracy variance, and number of selected genes for the four gene selection methods across the datasets, the variation created by their parameters, and the classifiers. The results are as follows: In terms of classification accuracy ( Figure A1) , an ANOVA deemed that at least one of the methods had a different classification accuracy mean (p-value = 0.003). Furthermore, using Tukey´s multiple comparison scheme, it was determined that the CFS method provided the largest classification accuracy, while the other three (MCO, IG and eBayes) showed no significant difference between them at an alpha value of 0.05. This provides evidence of the competitiveness of MCO with the distinctive characteristic that no parameters need to be adjusted by the user.
Regarding classification accuracy variance (dispersion), there was evidence that at least one of the methods had a different variance than the rest (p-values of 0.012 and 0.090 for Multiple Comparison´s and Levene´s methods respectively). Furthermore, through the analysis of the confidence intervals, it was determined that MCO´s interval did not overlap with the other three method's intervals and that MCO´s interval contained lower values than the rest of the methods. This evidences MCO´s robust classification performance across different classifiers and datasets.
Finally, regarding the number of selected genes (parsimony), an ANOVA showed that at least one of the methods had a different mean number of selected genes (p-value = 0.011). Furthermore, using Tukey´s multiple comparison scheme, two methods were deemed statistically equivalent in terms of their number of selected genes: MCO, CFS, both of which showed a significantly lower mean number of selected genes than eBayes. The IG method did not show significant difference with CFS and eBayes due to its large dispersion. This result evidences the parsimonious gene selection of MCO.
So, in conclusion, this comparison shows the robustness of the solutions of MCO, the results of which ten d to be parsimonious, providing a competitive classification accuracy performance without requiring the adjustment of statistical or computational parameters and with the unique capability to support simultaneous meta-analysis of multiple datasets.

Reviewer 2, Issue 2
Quite good concept of paper and require complete language check should be performed to handle all typos and language issues.

Follow up to Reviewer 2, Issue 2
We have thoroughly revised our use of language. We used the online tool "Grammarly.com" to improve our exposition in this new version.

Reviewer 2, Issue 3
There are many similar papers published on this topic, how your paper is different from existing ones? Explain.

Follow up to Reviewer 2, Issue 3
The reviewer is kindly referred to our follow-up to Reviewer 1´s Issues 1 and 5, as well as Reviewer 2´s Issue 1.

Reviewer 2, Issue 4
Include some of the latest and relevant references for the benefit of the readers/authors of Cancer classification/ Parkinson disease based journal. The following citations will be very useful for the current, future and young research scholars in this research field from all over the globe. Thank you for your suggestions. We have added the most relevant ones in our literature review and references.

Reviewer 2, Issue 5
The evaluation section is also not clear. Did authors use cross-validation or an independent test set? Did they train their model for one benchmark and used the trained model on the rest of the benchmarks?

Follow up to Reviewer 2, Issue 5
The idea for this work is to result in biological relevance, thus our validation method goes along the lines of the biological evidence that emerges with our selection of genes. To this end, the discussion section was enhanced in this revised version. Although the classification problem was not the goal of this work, a complete comparison of classification performance is now added as an appendix following Reviewer 2´s Issue 1. This includes a comparison of four gene selection methods across multiple datasets, classifiers, and parameter variation. The cross-validation specifics have been added to this section, as these were varied for all methods for comparison purposes.

Reviewer 3, Issue 1
In this paper and their previous works, they only used median or mean value to obtain differential genes. However, in many similar works about the selection of differential genes in two-class problems, researchers usually used more complicated rank-based measures, such as t-test, wilcoxon rank sum test, signal-to-noise ratio, etc. Authors should further demonstrate their MCO efficacy using such measures.

Follow up to Reviewer 3, Issue 1
The existence of several types of PMs measures and using a single one of them after a normalization procedure to select genes has contributed to the problem of reproducibility in this task. This was one of the reasons that motivated the generation of MCO. The rationale to use the (absolute) difference of medians or the difference of means has had to do more with transparency and with keeping assumptions to a minimum in our analyses.
In the R code, there is a possibility of using the (absolute) difference of means, medians, third quartiles, or the kth-percentiles to give analysts further possibilities.
A revision of our code will include adding a column to the dataset with values from performance measures calculated by the analyst with the additional information if these values are to be either maximized or minimized. This will swiftly accommodate the instances described by the reviewer.
As long as a performance measure´s values are numerical and subject to either maximization or minimization, it can be perfectly analyzed using MCO, regardless of its measuring units and its complexity. Also, the possibility of carrying out meta-analysis (to up to five datasets simultaneously) remains unchanged in our code.