A simple formula for enumerating comparisons in trials and network meta-analysis [version 1; referees: awaiting peer review] v1

Shokraneh, Farhad; Adams, Clive E

Abstract

We present use of a simple formula to calculate the number of pairwise comparisons of interventions within a single trial or network meta-analyses. We used the data from our previous network meta-analysis to build a study-based register and enumerated the direct pairwise comparisons from the trials therein. We then compared this with the number of comparisons predicted by use of the formula and finally with the reported number of comparisons (indirect or direct) within the network meta-analysis. A total of 133 trials included in the network generated 163 comparisons (16 unique direct comparisons for 8 interventions). The formula predicted an expected 28 indirect or direct comparisons and this is the number that were indeed reported. The formula produces an accurate enumeration of the potential comparisons within a single trial or network meta-analysis. Its use could help transparency of reporting should a shortfall occur between comparisons actually used and the potential total.

Citation

Shokraneh, F., & Adams, C. E. (2019). A simple formula for enumerating comparisons in trials and network meta-analysis [version 1; referees: awaiting peer review] v1. F1000Research, 8, Article 38. https://doi.org/10.12688/f1000research.17352.1

• Journal Article Type: Article
• Acceptance Date: Dec 20, 2018
• Online Publication Date: Jan 9, 2019
• Publication Date: Jan 9, 2019
• Deposit Date: Jan 9, 2019
• Publicly Available Date: Jan 10, 2019
• Journal: F1000Research
• Electronic ISSN: 2046-1402
• Publisher: F1000Research
• Peer Reviewed: Peer Reviewed
• Volume: 8
• Article Number: 38
• DOI: https://doi.org/10.12688/f1000research.17352.1
• Keywords: Pairwise Comparisons, Study-Based Registers, Clinical Trials, Randomised; Controlled Trials, Network Meta-Analysis, Systematic Reviews
• Public URL: https://nottingham-repository.worktribe.com/output/1459873
• Publisher URL: https://f1000research.com/articles/8-38/v1
• Additional Information: Referee status: Indexed; Referee Report: 10.5256/f1000research.18976.r43427, Julie Broderick, Department of Medicine, Division of Physiotherapy, Trinity Centre for Health Sciences, Trinity College Dublin, Dublin, Ireland, 13 Feb 2019, version 1, indexed; Referee Comment: Farhad Shokraneh; Posted: 27 Mar 2019; Dear Dr Julie Broderick, Thanks for spending your time for reviewing our work and commenting on it. In the following lines, we replied to your comments and made changes to the paper to cover your suggestions. COMMENT: In my opinion, this is a novel approach which allows the quantification of comparators in trials and network meta-analyses. Following on from this, the comparators which are possible versus how many were reported can be discussed in further depth. This formula has great applicability in terms of network meta-analysis methodology which is still very much a developing area. Some very minor points; please add a table which demonstrates the use of the formula to quantify how many comparators are possible for trials from 2 interventions up to 10.REPLY: Thanks for your positive comment. We added a single visual proof for this formula to show how it works. The networks of 2 to 10 interventions will create networks in shapes of line, triangle, rectangle, pentagon, hexagon, heptagon, octagon, and decagon.CHANGE: We added: “The networks of 2 to 10 interventions will create networks in shapes of line, triangle, rectangle, pentagon, hexagon, heptagon, octagon, and decagon”. COMMENT: Pg 3, paragraph with the heading 'Reported comparisons within the trials' - line 2 change 'there' to 'the'.REPLY: Thank you and sorry for this typo.CHANGE: We change ‘there’ to ‘the’. COMMENT: Explanation of Figure 4 in text is not clear. REPLY: We added a few sentences in the text to clarify it.CHANGE: “This diagram shows that only some of the comparisons from trials in study-based register could be included in pairwise meta-analysis. In addition, the number of comparisons in network meta-analysis (calculated by formula) is larger and inclusive of all the comparisons in the network of interventions and includes all the possible unique comparisons even if the comparisons are not in trials or in pairwise meta-analysis”. Thanks again for your valuable comments. Best Regards,Farhad Shokraneh; Referee Report: 10.5256/f1000research.18976.r44117, G. Mustafa Soomro, St. James Hospital, Portsmouth, UK, 04 Mar 2019, version 1, indexed; Referee Comment: Farhad Shokraneh; Posted: 27 Mar 2019; Dear Dr G. Mustafa Soomro, Thanks for spending your time for reviewing our work and commenting on it. In the following lines, we replied to your comments and made changes to the paper to cover your suggestions. COMMENT: This is a useful paper for demonstrating and discussing how an established formula could be used for calculating all possible pairs of comparisons for interventions in a network meta-analysis. Thus, reviewers would be able to find out how many potential comparisons have not been carried out. I have some minor suggestions: In the abstract the following sentence should be amended for clarity perhaps as follows: “A total of 133 trials included in the network generated 163 comparisons (16 unique direct comparisons for 8 interventions)”. Amendment: “A total of 133 trials of 8 interventions were selected which included 163 comparisons. The network of these showed 16 unique direct comparisons.”REPLY: Thank you. We revised the sentence as suggested.CHANGE: “A total of 133 trials of 8 interventions were selected which included 163 comparisons. The network of these showed 16 unique direct comparisons”. COMMENT: On page 2, it says that N is a triangular number. Either this point is not relevant, or why being a triangular number is important should be described.REPLY: We agree. We deleted this part.CHANGE: “N is a triangular number” was deleted. COMMENT: I think they should say that the formula they have used is an established formula from combinatorics for calculating number of pairs for a number of items in a set.REPLY: Thank you for adding this. We agree and we added your suggestion in the text right after formula.CHANGE: “This is an established formula from combinatorics for calculating number of pairs for a number of items in a set”. COMMENT: On page 3 under “Comparisons in network meta-analysis plots”, line 2 and line 11 “though” should be changed to "through".REPLY: Thank you for detecting these errors. We collected both.CHANGE: we replaced ‘though’ with ‘through’. Thanks again for your valuable comments. Best Regards,Farhad Shokraneh; Grant Information: The author(s) declared that no grants were involved in supporting this work; Copyright: This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.