Cited by 2,785Probability and Statistical ResearchStatistical Distribution Estimation and ApplicationsGame Theory and Voting Systems

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A Comparison of Alternative Tests of Significance for the Problem of $m$ Rankings

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Friedman, Milton·1940·The Annals of Mathematical Statistics

DOI 10.1214/aoms/1177731944

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The use of ranks to avoid the assumption of normality implicit in the analysis of variance
Friedman, M. (1937) · Journal of the American Statistical Association · doi:10.1080/01621459.1937.10503522 · Cited by 7,005
Self-citation
Statistical Tables for Biological, Agricultural and Medical Research.
Reed, L. J., Fisher, R. A., & Yates, F. (1944) · Journal of the American Statistical Association · doi:10.2307/2280633 · Cited by 3,012
The Problem of $m$ Rankings
Kendall, M. G., & Smith, B. B. (1939) · The Annals of Mathematical Statistics · doi:10.1214/aoms/1177732186 · Cited by 992
The Correlation Ratio for Ranked Data
Wallis, W. A. (1939) · Journal of the American Statistical Association · doi:10.1080/01621459.1939.10503552 · Cited by 58

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“Specifically, the Friedman Test (Friedman, 1937, 1940) was used to assess differences among graders, and Conover’s post-hoc tests (Conover, 1999; Conover, & Iman, 1979) were conducted for pairwise comparisons.”
Reputational interdependence: an examination of category reputation in higher education
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Semiochemicals from ovaries of gravid females attract ovipositing female houseflies, Musca domestica
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Significance of non-parametric statistical tests for comparison of classifiers over multiple datasets
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Multi-label feature selection based on rough granular-ball and label distribution
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“To compare and analyze the performance of the proposed multi-label feature selection methods, we have performed two statistical tests such as Friedman test [11] and Bonferroni–Dunn test [7].”
Method
BIRCHSCAN: A sampling method for applying DBSCAN to large datasets
Ventorim, I. D. M., Luchi, D., Rodrigues, A. L., & Varejão, F. M. (2021) · Expert Systems with Applications · doi:10.1016/j.eswa.2021.115518 · Cited by 37
A classification method in machine learning based on soft decision-making via fuzzy parameterized fuzzy soft matrices
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“66, respectively, with 4 ( k − 1 ) degrees of freedom and the critical value for the Friedman test (Friedman 1940) given for k = 5 and N = 15 is 9 .”
Method
How to Guide Task-oriented Chatbot Users, and When: A Mixed-methods Study of Combinations of Chatbot Guidance Types and Timings
Yeh, S.-F., Wu, M.-H., Chen, T.-Y., Lin, Y.-C., Chang, X.-J., Chiang, Y.-H., & Chang, Y.-J. (2022) · CHI Conference on Human Factors in Computing Systems · doi:10.1145/3491102.3501941 · Cited by 38
“A Friedman test [25] indicated that diferences among the rankings were statistically signifcant (�2(8)=320.”
BackgroundMethod
An empirical assessment of smote variants techniques and interpretation methods in improving the accuracy and the interpretability of student performance models
Sahlaoui, H., Alaoui, E. A. A., Agoujil, S., & Nayyar, A. (2023) · Education and Information Technologies · doi:10.1007/s10639-023-12007-w · Cited by 45
Design of reject rules for ECOC classification systems
Simeone, P., Marrocco, C., & Tortorella, F. (2011) · Pattern Recognition · doi:10.1016/j.patcog.2011.08.001 · Cited by 37
Two stage architecture for multi-label learning
Madjarov, G., Gjorgjevikj, D., & Džeroski, S. (2011) · Pattern Recognition · doi:10.1016/j.patcog.2011.08.011 · Cited by 37
FIRE IMPACT ON A MAQUIS SOIL SEED BANK IN CABAÑEROS NATIONAL PARK (CENTRAL SPAIN)
Ferrandis, P., Herranz, J. M., & Martínez-Sánchez, J. J. (1999) · Israel Journal of Plant Sciences · doi:10.1080/07929978.1999.10676747 · Cited by 37
Adaptive degree penalization for link prediction
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Designing of classifiers based on immune principles and fuzzy rules
Lei, Z., & Li, R. (2007) · Information Sciences · doi:10.1016/j.ins.2007.11.019 · Cited by 37
Feature selection using binary monarch butterfly optimization
Sun, L., Si, S., Jing, Z., Xu, J., Lin, Y., & Lv, Z. (2022) · Applied Intelligence · doi:10.1007/s10489-022-03554-9 · Cited by 36
Parallel Global Best-Worst Particle Swarm Optimization Algorithm for solving optimization problems
Kumar, L., Pandey, M., & Ahirwal, M. K. (2023) · Applied Soft Computing · doi:10.1016/j.asoc.2023.110329 · Cited by 36
Semi-supervised attribute reduction based on label distribution and label irrelevance
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A COPRAS-based Approach to Multi-Label Feature Selection for Text Classification
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GLUE-X: Evaluating Natural Language Understanding Models from an Out-of-Distribution Generalization Perspective
Yang, L., Zhang, S., Qin, L., Li, Y., Wang, Y., Liu, H., Wang, J., Xie, X., & Zhang, Y. (2023) · Findings of the Association for Computational Linguistics: ACL (Findings of ACL) · doi:10.18653/v1/2023.findings-acl.806 · Cited by 81
Multi-label feature selection based on stable label relevance and label-specific features
Yang, Y., Chen, H., Mi, Y., Luo, C., Horng, S., & Li, T. (2023) · Information Sciences · doi:10.1016/j.ins.2023.119525 · Cited by 36
Outlier detection in a multiset-valued information system based on rough set theory and granular computing
Song, Y., Lin, H., & Li, Z. (2023) · Information Sciences · doi:10.1016/j.ins.2023.119950 · Cited by 38
Improving performance of extreme learning machine for classification challenges by modified firefly algorithm and validation on medical benchmark datasets
Bačanin, N., Stoean, C., Marković, D., Zivkovic, M. Z., Rashid, T. A., Chhabra, A., & Šarac, M. (2024) · Multimedia Tools and Applications · doi:10.1007/s11042-024-18295-9 · Cited by 38
“First, the Friedman test [102] along with two-way variance analysis by ranks is used and the results are depicted in Table 8.”
Concordance coefficients to measure the agreement among several sets of ranks
Teles, J. M. V. (2012) · Journal of Applied Statistics · doi:10.1080/02664763.2012.681460 · Cited by 36
Integrated Rough Fuzzy Clustering for Categorical data Analysis
Saha, I., Sarkar, J. P., & Maulik, U. (2018) · Fuzzy Sets and Systems · doi:10.1016/j.fss.2018.02.007 · Cited by 36
A multiobjective optimization-based sparse extreme learning machine algorithm
Wu, Y., Zhang, Y., Liu, X., Cai, Z., & Cai, Y. (2018) · Neurocomputing · doi:10.1016/j.neucom.2018.07.060 · Cited by 36
Exploring Joint AB-LSTM With Embedded Lemmas for Adverse Drug Reaction Discovery
Santiso, S., Pérez, A., & Casillas, A. (2018) · IEEE Journal of Biomedical and Health Informatics · doi:10.1109/jbhi.2018.2879744 · Cited by 36
“Finally, the Friedman test [62] was employed with 10-fold cross-validation to corroborate that the differences between the approaches explored were significant.”
Method
A parameter-free particle swarm optimization algorithm using performance classifiers
Harrison, K. R., Ombuki-Berman, B. M., & Engelbrecht, A. P. (2019) · Information Sciences · doi:10.1016/j.ins.2019.07.016 · Cited by 36
Maximizing three-hop influence spread in social networks using discrete comprehensive learning artificial bee colony optimizer
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The constraints of theory: What is the impact of the Theory of Constraints on Operations Strategy?
Pacheco, D. A. D. J., Antunes, J. A. V., & de Matos, C. A. (2020) · International Journal of Production Economics · doi:10.1016/j.ijpe.2020.107955 · Cited by 36
A new density-based subspace selection method using mutual information for high dimensional outlier detection
Riahi-Madvar, M., Akbari, A., Nasersharif, B., & Raahemi, B. (2021) · Knowledge-Based Systems · doi:10.1016/j.knosys.2020.106733 · Cited by 36
Memory-based approaches for eliminating premature convergence in particle swarm optimization
Chaitanya, K. D. V., Somayajulu, D. V. L. N., & Krishna, P. R. (2021) · Applied Intelligence · doi:10.1007/s10489-020-02045-z · Cited by 36
Parameter estimation of q-Gaussian Radial Basis Functions Neural Networks with a Hybrid Algorithm for binary classification
Fernández‐Navarro, F., Hervás‐Martínez, C., Gutiérrez, P. A., Peña, J. M., & Granados, F. L. (2011) · Neurocomputing · doi:10.1016/j.neucom.2011.03.056 · Cited by 36
Profiling instances in noise reduction
Delany, S. J., Segata, N., & Namee, B. M. (2012) · Knowledge-Based Systems · doi:10.1016/j.knosys.2012.01.015 · Cited by 36
Greedy Randomized Adaptive Search and Variable Neighbourhood Search for the minimum labelling spanning tree problem
Consoli, S., Darby-Dowman, K., Mladenović, N., & Pérez, J. A. M. (2008) · European Journal of Operational Research · doi:10.1016/j.ejor.2008.03.014 · Cited by 36
Summarization of changes in dynamic text collections using Latent Dirichlet Allocation model
Kar, M., Nunes, S., & Ribeiro, C. (2015) · Information Processing & Management · doi:10.1016/j.ipm.2015.06.002 · Cited by 36
Pre-extinction in sensory perconditioning.
Coppock, W. (1958) · Journal of Experimental Psychology · doi:10.1037/h0046379 · Cited by 36
Class-specific feature selection via maximal dynamic correlation change and minimal redundancy
Xiao, M., Xu, H., & Ju, C. (2023) · Expert Systems with Applications · doi:10.1016/j.eswa.2023.120455 · Cited by 35
A Novel Flexible Inertia Weight Particle Swarm Optimization Algorithm
Amoshahy, M. J., Shamsi, M., & Sedaaghi, M. H. (2016) · PLoS ONE · doi:10.1371/journal.pone.0161558 · Cited by 35
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Domingo, D., Gracia, M. T. L., Gracia, A. L. M., & de la Riva, J. (2017) · European Journal of Remote Sensing · doi:10.1080/22797254.2017.1336067 · Cited by 35
“Furthermore, Friedman nonparametric test was applied in order to compare the performance of the different models (Friedman, 1940).”
Method
Automatic feature weighting for improving financial Decision Support Systems
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A novel many-objective evolutionary algorithm based on transfer matrix with Kriging model
Ma, L., Wang, R., Chen, S., Cheng, S., Wang, X., Lin, Z., Shi, Y., & Huang, M. (2019) · Information Sciences · doi:10.1016/j.ins.2019.01.030 · Cited by 35
Visual question answering via Attention-based syntactic structure tree-LSTM
Liu, Y., Zhang, X., Huang, F., Tang, X., & Li, Z. (2019) · Applied Soft Computing · doi:10.1016/j.asoc.2019.105584 · Cited by 35
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Multi-scale covering rough sets with applications to data classification
Huang, Z., & Li, J. (2021) · Applied Soft Computing · doi:10.1016/j.asoc.2021.107736 · Cited by 35
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Sun, L., Wang, T., Ding, W., & Xu, J. (2022) · IEEE Transactions on Fuzzy Systems · doi:10.1109/tfuzz.2022.3222941 · Cited by 35
“To demonstrate the statistical analysis of all of these comparative algorithms for each evaluation index, the Friedman test [42] and Nemenyi test [43] were employed.”
Method
Network-based statistical comparison of citation topology of bibliographic databases
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Negative Correlation Ensemble Learning for Ordinal Regression
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LIMFS: Label interaction-aware multi-label feature selection
Yu, Y., Li, B., Qian, J., Shu, W., & Miao, D. (2026) · Expert Systems with Applications · doi:10.1016/j.eswa.2026.131177 · Cited by 34
The responsibility weighted Mahalanobis kernel for semi-supervised training of support vector machines for classification
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Baltic Sea nutrient reductions – What should we aim for?
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Nonadditive similarity-based single-layer perceptron for multi-criteria collaborative filtering
Hu, Y. (2013) · Neurocomputing · doi:10.1016/j.neucom.2013.09.027 · Cited by 34
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Calma, A., Reitmaier, T., & Sick, B. (2018) · Information Sciences · doi:10.1016/j.ins.2018.04.063 · Cited by 34
Building selective ensembles of Randomization Based Neural Networks with the successive projections algorithm
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Optimal selection of molecular descriptors for antimicrobial peptides classification: an evolutionary feature weighting approach
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“The statistical analysis of the MLAs’ performance identified (by the Friedman test) a significant difference in the BalAcc ( χ 2 f ( 3 ) = 55.2, p -value = 6.224e-12 ) of the four MLAs on multiple datasets.”
MethodInfluential
Machine learning: Modeling increment in diameter of individual trees on Atlantic Forest fragments
Júnior, I. D. S. T., Torres, C. M. M. E., Leite, H. G., de Castro, N. L. M., Soares, C. P. B., Castro, R. V. O., & Farias, A. A. (2020) · Ecological Indicators · doi:10.1016/j.ecolind.2020.106685 · Cited by 34
Static source code metrics and static analysis warnings for fine-grained just-in-time defect prediction
Trautsch, A., Herbold, S., & Grabowski, J. (2020) · IEEE International Conference on Software Maintenance (ICSM) · doi:10.1109/icsme46990.2020.00022 · Cited by 34
“data for normality and homoscedacity and then automatically selects suitable tests for the data: repeated measures ANOVA as omnibus tests with a post-hoc Tukey HSD test [35] in case the data is normal and homoscedastic and Friedman test [36] as omnibus test with a post-hoc Nemenyi test [37] otherwise.”
Method
KAT: A Knowledge Adversarial Training Method for Zero-Order Takagi–Sugeno–Kang Fuzzy Classifiers
Qin, B., Chung, F.-L., & Wang, S. (2020) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2020.3034792 · Cited by 34
“The statistical results of the Friedman ranking test [37]–[39] on the testing accuracies by all the adopted methods on all the adopted datasets are reported in Tables SV and SVI in the supplementary material.”
Method
Legal information retrieval for understanding statutory terms
Šavelka, J., & Ashley, K. D. (2021) · Artificial Intelligence and Law · doi:10.1007/s10506-021-09293-5 · Cited by 37
“In Demšar (2006) the Friedman test (Friedman 1937, 1940), a non-parametric equivalent of the repeated-measures ANOVA, is recommended.”
Method
Quasi-reflection learning arithmetic optimization algorithm firefly search for feature selection
Bačanin, N., Budimirović, N., Venkatachalam, K., Jassim, H. S., Zivkovic, M. Z., Askar, S. S., & Abouhawwash, M. (2023) · Heliyon · doi:10.1016/j.heliyon.2023.e15378 · Cited by 34
Involvement of 5-HT receptor subtypes in the discriminative stimulus properties of mescaline
Appel, J. B., & Callahan, P. M. (1989) · European Journal of Pharmacology · doi:10.1016/0014-2999(89)90041-1 · Cited by 34
IENet: a robust convolutional neural network for EEG based brain-computer interfaces
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Sentiment Classification Using Two Effective Optimization Methods Derived From The Artificial Bee Colony Optimization And Imperialist Competitive Algorithm
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Non-dominated sorting differential evolution algorithm for the minimization of route based fuel consumption multiobjective vehicle routing problems
Psychas, I.-D., Marinaki, M., Marinakis, Y., & Migdalas, A. (2016) · Energy Systems · doi:10.1007/s12667-016-0209-5 · Cited by 33
Combining Mask Estimates for Single Channel Audio Source Separation Using Deep Neural Networks
Grais, E. M., Roma, G., Simpson, A. J. R., & Plumbley, M. D. (2016) · Interspeech (USB) · doi:10.21437/interspeech.2016-216 · Cited by 33
An efficient character recognition method using enhanced HOG for spam image detection
Naiemi, F., Ghods, V., & Khalesi, H. (2019) · Soft Computing · doi:10.1007/s00500-018-03728-z · Cited by 33
A Novel Model Based on Nonlinear Manifold Detection for Software Defect Prediction
Ghosh, S., Rana, A., & Kansal, V. (2018) · International Conference on Intelligent Computing and Control Systems (ICICCS) · doi:10.1109/iccons.2018.8663026 · Cited by 33
“Friedman test belongs to the category of nonparametric statistical test [7] which is applied basically for finding out significant difference, if any among performances of various machine learning approaches associated with Nonlinear MDTs and assigning rank to them as per order.”
Method
Hybrid Brain Storm Optimization algorithm and Late Acceptance Hill Climbing to solve the Flexible Job-Shop Scheduling Problem
Alzaqebah, M., Jawarneh, S., Alwohaibi, M., Alsmadi, M. K., Almarashdeh, I., & Mohammad, R. M. A. (2020) · Journal of King Saud University - Computer and Information Sciences · doi:10.1016/j.jksuci.2020.09.004 · Cited by 33
ALSA: Adversarial Learning of Supervised Attentions for Visual Question Answering
Liu, Y., Zhang, X., Zhao, Z., Zhang, B., Cheng, L., & Li, Z. (2020) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2020.3029423 · Cited by 33
“We employ Friedman–Nemenyi test [64] with 95% con-ﬁdence level to analyze the difference between ALSA and the baselines.”
Method
RNN-VirSeeker: A Deep Learning Method for Identification of Short Viral Sequences From Metagenomes
Liu, F., Miao, Y., Liu, Y., & Hou, T. (2020) · IEEE Transactions on Computational Biology and Bioinformatics · doi:10.1109/tcbb.2020.3044575 · Cited by 35
“In order to prove the significant improvement of RNNVirSeeker over the other three methods more comprehensively, Friedman test [37], [38] and Nemenyi [39] test were utilized as statistical measures, which compared the performance of multiple algorithms on multiple datasets.”
Method
Intelligent Scheduling with Reinforcement Learning
Cunha, B., Madureira, A. M., Fonseca, B., & Matos, J. E. (2021) · Applied Sciences · doi:10.3390/app11083710 · Cited by 35
Hierarchical feature selection with multi-granularity clustering structure
Guo, S., Zhao, H., & Yang, W. (2021) · Information Sciences · doi:10.1016/j.ins.2021.04.046 · Cited by 33
An ensemble-based convolutional neural network model powered by a genetic algorithm for melanoma diagnosis
Pérez, E., & Ventura, S. (2021) · Neural Computing and Applications · doi:10.1007/s00521-021-06655-7 · Cited by 34
“Friedman’s test [79] was conducted in cases where a multiple comparison was carried out, Hommel’s post hoc test [80] was employed to perform a multiple comparison with a control method, Shaffer post hoc test [81] was employed to perform pairwise comparisons, and ﬁnally Wilcoxon Signed-Rank test…”
Method
Modified artificial bee colony algorithm with differential evolution to enhance precision and convergence performance
Üstün, D., Toktaş, A., Erkan, U., & Akdağlı, A. (2022) · Expert Systems with Applications · doi:10.1016/j.eswa.2022.116930 · Cited by 33
Improvement and application of hybrid real-coded genetic algorithm
Song, H., Wang, J., Song, L., Zhang, H., Bei, J., Ni, J., & Ye, B. (2022) · Applied Intelligence · doi:10.1007/s10489-021-03048-0 · Cited by 33
A modified weighted mean of vectors optimizer for Chronic Kidney disease classification
Houssein, E. H., & Sayed, A. A. (2023) · Computers in Biology and Medicine · doi:10.1016/j.compbiomed.2023.106691 · Cited by 33
Ensemble of kernel extreme learning machine based elimination optimization for multi-label classification
Zhang, Q., Tsang, E. C. C., He, Q., & Guo, Y. (2023) · Knowledge-Based Systems · doi:10.1016/j.knosys.2023.110817 · Cited by 33
A lazy bagging approach to classification
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“The statistical results as well as the Wilcoxon rank sum test (Wilcoxon 1945) and Friedman (Friedman 1940) also were reflected in Table 2.”
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“The Friedman test [43], considered as the nonparametric equivalent of the repeated measures ANOVA test, relies on a multiple comparisons for computing the groups differences using a con-trol method.”
Method

Authors: Friedman, M.
Corresponding author: Friedman, Milton
Venue: The Annals of Mathematical Statistics
Publisher: Institute of Mathematical Statistics
Published: 1940-03
Volume: 11
Issue: 1
Pages: 86-92
ISSN: 0003-4851 (ISSN-L)
DOI: 10.1214/aoms/1177731944
OpenAlex: W2016944307
Cited by: 2,785
References: 4
Citations by year: 2026: 191 · 2025: 256 · 2024: 244 · 2023: 253 · 2022: 264 · 2021: 231 · 2020: 209 · 2019: 173 · 2018: 133 · 2017: 106 · 2016: 132 · 2015: 93 · 2014: 82 · 2013: 72 · 2012: 67
FWCI: 5.07
Type: journal article
Language: en
Open access: bronze
doi.org
Locations: The Annals of Mathematical Statistics (publishedVersion)
Project Euclid (Cornell University) (submittedVersion)
Topics: Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Game Theory and Voting Systems
Keywords: tests of significance, rank tests, comparison of statistical tests

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A Comparison of Alternative Tests of Significance for the Problem of $m$ Rankings

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