IDEAS home Printed from https://ideas.repec.org/a/spr/compst/v38y2023i1d10.1007_s00180-022-01230-7.html
   My bibliography  Save this article

Adjusting the adjusted Rand Index

Author

Listed:
  • Martina Sundqvist

    (Université Paris-Saclay, AgroParisTech, INRAE, UMR MIA-Paris
    Institut Curie-PSL Research University, Translational Research Department, Breast Cancer Biology Group)

  • Julien Chiquet

    (Université Paris-Saclay, AgroParisTech, INRAE, UMR MIA-Paris)

  • Guillem Rigaill

    (Université Paris-Saclay, CNRS, INRAE, Univ Evry, Institute of Plant Sciences Paris-Saclay (IPS2)
    Université Paris-Saclay, CNRS, Univ Evry)

Abstract

The Adjusted Rand Index (ARI) is arguably one of the most popular measures for cluster comparison. The adjustment of the ARI is based on a hypergeometric distribution assumption which is not satisfactory from a modeling point of view because (i) it is not appropriate when the two clusterings are dependent, (ii) it forces the size of the clusters, and (iii) it ignores the randomness of the sampling. In this work, we present a new "modified" version of the Rand Index. First, as in Russell et al. (J Malar Inst India 3(1), 1940 ), we consider only the pairs consistent by similarity and ignore the pairs consistent by difference to define the MRI. Second, we base the adjusted version, called MARI, on a multinomial distribution instead of a hypergeometric distribution. The multinomial model is advantageous because it does not force the size of the clusters, correctly models randomness and is easily extended to the dependent case. We show that ARI is biased under the multinomial model and that the difference between ARI and MARI can be significant for small n but essentially vanishes for large n, where n is the number of individuals. Finally, we provide an efficient algorithm to compute all these quantities ((A)RI and M(A)RI) based on a sparse representation of the contingency table in our aricode package. The space and time complexity is linear with respect to the number of samples and, more importantly, does not depend on the number of clusters as we do not explicitly compute the contingency table.

Suggested Citation

  • Martina Sundqvist & Julien Chiquet & Guillem Rigaill, 2023. "Adjusting the adjusted Rand Index," Computational Statistics, Springer, vol. 38(1), pages 327-347, March.
    • Handle: RePEc:spr:compst:v:38:y:2023:i:1:d:10.1007_s00180-022-01230-7
    DOI: 10.1007/s00180-022-01230-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00180-022-01230-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00180-022-01230-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Eddelbuettel, Dirk & Francois, Romain, 2011. "Rcpp: Seamless R and C++ Integration," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 40(i08).
    2. Lawrence Hubert & Phipps Arabie, 1985. "Comparing partitions," Journal of Classification, Springer;The Classification Society, vol. 2(1), pages 193-218, December.
    3. Ahmed N. Albatineh & Magdalena Niewiadomska-Bugaj & Daniel Mihalko, 2006. "On Similarity Indices and Correction for Chance Agreement," Journal of Classification, Springer;The Classification Society, vol. 23(2), pages 301-313, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. José E. Chacón, 2021. "Explicit Agreement Extremes for a 2 × 2 Table with Given Marginals," Journal of Classification, Springer;The Classification Society, vol. 38(2), pages 257-263, July.
    2. Stefano Tonellato & Andrea Pastore, 2013. "On the comparison of model-based clustering solutions," Working Papers 2013:05, Department of Economics, University of Venice "Ca' Foscari".
    3. Theresa Ullmann & Anna Beer & Maximilian Hünemörder & Thomas Seidl & Anne-Laure Boulesteix, 2023. "Over-optimistic evaluation and reporting of novel cluster algorithms: an illustrative study," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 17(1), pages 211-238, March.
    4. José E. Chacón & Ana I. Rastrojo, 2023. "Minimum adjusted Rand index for two clusterings of a given size," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 17(1), pages 125-133, March.
    5. Antonio D’Ambrosio & Sonia Amodio & Carmela Iorio & Giuseppe Pandolfo & Roberta Siciliano, 2021. "Adjusted Concordance Index: an Extensionl of the Adjusted Rand Index to Fuzzy Partitions," Journal of Classification, Springer;The Classification Society, vol. 38(1), pages 112-128, April.
    6. Matthijs Warrens, 2008. "On Similarity Coefficients for 2×2 Tables and Correction for Chance," Psychometrika, Springer;The Psychometric Society, vol. 73(3), pages 487-502, September.
    7. Carter Allen & Yuzhou Chang & Brian Neelon & Won Chang & Hang J. Kim & Zihai Li & Qin Ma & Dongjun Chung, 2023. "A Bayesian multivariate mixture model for high throughput spatial transcriptomics," Biometrics, The International Biometric Society, vol. 79(3), pages 1775-1787, September.
    8. Valerie Robert & Yann Vasseur & Vincent Brault, 2021. "Comparing High-Dimensional Partitions with the Co-clustering Adjusted Rand Index," Journal of Classification, Springer;The Classification Society, vol. 38(1), pages 158-186, April.
    9. Johann Kraus & Christoph Müssel & Günther Palm & Hans Kestler, 2011. "Multi-objective selection for collecting cluster alternatives," Computational Statistics, Springer, vol. 26(2), pages 341-353, June.
    10. Isabella Morlini & Sergio Zani, 2012. "Dissimilarity and similarity measures for comparing dendrograms and their applications," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 6(2), pages 85-105, July.
    11. Ahmed Albatineh & Magdalena Niewiadomska-Bugaj, 2011. "Correcting Jaccard and other similarity indices for chance agreement in cluster analysis," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 5(3), pages 179-200, October.
    12. Matthijs J. Warrens & Hanneke Hoef, 2022. "Understanding the Adjusted Rand Index and Other Partition Comparison Indices Based on Counting Object Pairs," Journal of Classification, Springer;The Classification Society, vol. 39(3), pages 487-509, November.
    13. Jeffrey L. Andrews & Ryan Browne & Chelsey D. Hvingelby, 2022. "On Assessments of Agreement Between Fuzzy Partitions," Journal of Classification, Springer;The Classification Society, vol. 39(2), pages 326-342, July.
    14. Ekaterina Kovaleva & Boris Mirkin, 2015. "Bisecting K-Means and 1D Projection Divisive Clustering: A Unified Framework and Experimental Comparison," Journal of Classification, Springer;The Classification Society, vol. 32(3), pages 414-442, October.
    15. Jonathon J. O’Brien & Michael T. Lawson & Devin K. Schweppe & Bahjat F. Qaqish, 2020. "Suboptimal Comparison of Partitions," Journal of Classification, Springer;The Classification Society, vol. 37(2), pages 435-461, July.
    16. Andrzej Młodak, 2021. "k-Means, Ward and Probabilistic Distance-Based Clustering Methods with Contiguity Constraint," Journal of Classification, Springer;The Classification Society, vol. 38(2), pages 313-352, July.
    17. Isabella Morlini & Sergio Zani, 2012. "A New Class of Weighted Similarity Indices Using Polytomous Variables," Journal of Classification, Springer;The Classification Society, vol. 29(2), pages 199-226, July.
    18. Alicja Grześkowiak, 2016. "Assessment of Participation in Cultural Activities in Poland by Selected Multivariate Methods," European Journal of Social Sciences Education and Research Articles, Revistia Research and Publishing, vol. 3, January -.
    19. Yunpeng Zhao & Qing Pan & Chengan Du, 2019. "Logistic regression augmented community detection for network data with application in identifying autism‐related gene pathways," Biometrics, The International Biometric Society, vol. 75(1), pages 222-234, March.
    20. Wilson J. Wright & Peter N. Neitlich & Alyssa E. Shiel & Mevin B. Hooten, 2022. "Mechanistic spatial models for heavy metal pollution," Environmetrics, John Wiley & Sons, Ltd., vol. 33(8), December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:compst:v:38:y:2023:i:1:d:10.1007_s00180-022-01230-7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.