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A Comparison of Alternative Tests of Significance for the Problem of $m$ Rankings
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“As for comparing the classifiers over multiple datasets, we followed the recommendations in Demˇ sar (2006) and used the Friedman test (Friedman, 1940) to reject the null hypothesis.”
From Deep learning for time series classification: a review (2019) · cites this paper as Friedman, 1940
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Cites
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-citationStatistical Tables for Biological, Agricultural and Medical Research.
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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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On comparing, classifying and clustering several dependent regression models
Mahmoudi, M. R., Mahmoudi, M., & Pak, A. (2019) · Journal of Statistical Computation and Simulation · doi:10.1080/00949655.2019.1615489 · Cited by 27
“To compare the regression models, the nonparametric tests such as Friedman test (see Friedman [26–28]) and (Conover [29]) can be applied by using the n couples ( ˆ y 11 , . . . , ˆ y m 1 ) , . . . , ( ˆ y 1 n , . . . , ˆ y mn ) .”
MethodZhang, Y., Hou, Y., Zhou, S., & Ouyang, K. (2020) · Sensors · doi:10.3390/s20143818 · Cited by 28
Semisupervised Learning via Axiomatic Fuzzy Set Theory and SVM
Jia, W., Liu, X., Wang, Y., Witold Pedrycz, C., & Zhou, J. (2020) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2020.3032707 · Cited by 29
“Next, the Friedman test can be utilized to estimate the performance of all algorithms [45].”
MethodEmploying machine learning techniques to assess requirement change volatility
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“We ranked alternative strategies, for each group, following the practice of Reference [21].”
MethodLiang, P., Lei, D., Chin, K., & Hu, J. (2022) · Knowledge-Based Systems · doi:10.1016/j.knosys.2022.109795 · Cited by 27
Mahapatra, A. K., Panda, N., & Pattanayak, B. K. (2022) · Arabian Journal for Science and Engineering · doi:10.1007/s13369-022-07408-x · Cited by 27
“Friedman and Holms tests are carried out to establish the worth of the suggested approach [49].”
MethodAn Analysis of the FURIA Algorithm for Fuzzy Rule Induction
Hühn, J., & Hüllermeier, E. (2009) · Studies in computational intelligence · doi:10.1007/978-3-642-05177-7_16 · Cited by 27
Constricted Particle Swarm Optimization based Algorithm for Global Optimization
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An experimental analysis of evolutionary heuristics for the biobjective traveling purchaser problem
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HFSTE: Hybrid Feature Selections and Tree-Based Classifiers Ensemble for Intrusion Detection System
Tama, B. A., & Rhee, K.-H. (2017) · IEICE Transactions on Information and Systems · doi:10.1587/transinf.2016icp0018 · Cited by 27
“The significant results of each classifier are then assessed using Friedman test [23] and Nemenyi post hoc test [24].”
MethodVariable neighbourhood search for the minimum labelling Steiner tree problem
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Multi-Label Feature Selection via Positive or Negative Correlation
Lin, Y., He, Z., Guo, L., & Ding, W. (2023) · IEEE Transactions on Emerging Topics in Computational Intelligence · doi:10.1109/tetci.2023.3302653 · Cited by 26
“Tofurther explore whether there are significant differences among the nine feature selection algorithms, the Friedman test [53] and the Nemenyi [54] test are utilized.”
MethodSong, H., Wang, J., Bei, J., & Wang, M. D. (2024) · Expert Systems with Applications · doi:10.1016/j.eswa.2024.124624 · Cited by 26
Chen, Y., Huang, Z., & Li, J. (2025) · Fuzzy Sets and Systems · doi:10.1016/j.fss.2025.109382 · Cited by 26
Analysis of randomisation methods in swarm intelligence
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Ordinal regression neural networks based on concentric hyperspheres
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A novel three-step procedure to forecast the inspection volume
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Multi-layer graph constraints for interactive image segmentation via game theory
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Perception of phasic pain is modulated by smell and taste
Riello, M., Cecchini, M. P., Zanini, A., Chiappari, M. D., Tinazzi, M., & Fiorio, M. (2019) · European Journal of Pain · doi:10.1002/ejp.1453 · Cited by 30
“More precisely, Friedman test (Friedman, 1940) was used to analyse factor valence (sweet, bitter and neutral) separately for smell and taste with regard to tactile threshold, pain threshold, pain tolerance and the corresponding pain ratings (intensity and unpleasantness).”
MethodBlaga, A. D., Frutos-Pascual, M., Creed, C., & Williams, I. (2020) · CHI Conference on Human Factors in Computing Systems · doi:10.1145/3313831.3376554 · Cited by 26
“We test for significance between the conditions and the metrics described using a non parametric Friedman test [31].”
MethodDai, J., Liu, Y., & Chen, J. (2021) · Information Sciences · doi:10.1016/j.ins.2021.03.049 · Cited by 26
Fuzzy rough discrimination and label weighting for multi-label feature selection
Tan, A., Liang, J., Wu, W.-Z., Zhang, J., Sun, L., & Chen, C. (2021) · Neurocomputing · doi:10.1016/j.neucom.2021.09.007 · Cited by 26
Feature selection using self-information uncertainty measures in neighborhood information systems
Xu, J., Qu, K., Sun, Y., & Yang, J. C. (2022) · Applied Intelligence · doi:10.1007/s10489-022-03760-5 · Cited by 26
“In this subsection, we adopt the statistical methods Fried-man test [52] and Nemenyi test [53] to discuss the overall classification performance of FSASI algorithm compared with other algorithms.”
MethodMorales-García, J., Bueno-Crespo, A., Martínez‐España, R., & Cecilia, J. M. (2023) · Journal of Ambient Intelligence and Smart Environments · doi:10.3233/ais-220441 · Cited by 26
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“II and III give the performance comparison [8], [29] results between M1 and its peers on D1-3 in estimation error and time cost; Tables IV Authorized licensed use limited to the terms of the applicable license agreement with IEEE.”
A music information system automatically generated via Web content mining techniques
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Sinopsis de pruebas estadísticas no paramétricas. Cuándo usarlas
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A new evolutionary algorithm for mining top-k discriminative patterns in high dimensional data
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Weighting scheme for a pairwise multi-label classifier based on the fuzzy confusion matrix
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“Additionally, the Friedman [39] test was also applied, followed by the Nemenyi post-hoc procedure [36].”
MethodLikelihood-fuzzy analysis: From data, through statistics, to interpretable fuzzy classifiers
Pota, M., Esposito, M., & Pietro, G. D. (2017) · International Journal of Approximate Reasoning · doi:10.1016/j.ijar.2017.10.022 · Cited by 26
Juhász, M. L. W., & Marmur, E. S. (2018) · Dermatologic Surgery · doi:10.1097/dss.0000000000001528 · Cited by 26
Cost-sensitive feature selection via the ℓ2,1-norm
Zhao, H., & Yu, S. (2018) · International Journal of Approximate Reasoning · doi:10.1016/j.ijar.2018.10.017 · Cited by 26
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A fuzzy β-covering rough set model for attribute reduction by composite measure
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Categorization and sensorimotor interaction with objects
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DC programming and DCA for parametric-margin ν-support vector machine
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“The procedure followed for statistical comparison is similar to Demsar [11], where we have obtain the ranking of the GRS models using the Friedman Statistics [18] test.”
MethodDeep Semisupervised Multitask Learning Model and Its Interpretability for Survival Analysis
Chi, S., Tian, Y., Wang, F., Wang, Y., Chen, M., & Li, J. (2021) · IEEE Journal of Biomedical and Health Informatics · doi:10.1109/jbhi.2021.3064696 · Cited by 27
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MPdist-based missing data imputation for supporting big data analyses in IoT-based applications
Lee, G. H., Han, J., & Choi, J. K. (2021) · Future Generation Computer Systems · doi:10.1016/j.future.2021.06.042 · Cited by 25
Tefek, M. F. (2021) · Journal of Computational Electronics · doi:10.1007/s10825-021-01796-3 · Cited by 25
“The Friedman Mean Rank test is a nonparametric statistical analysis technique for repeated measures developed by Milton Friedman [53].”
MethodRevisiting the Impact of Dependency Network Metrics on Software Defect Prediction
Gong, L., Rajbahadur, G. K., Hassan, A. E., & Jiang, S. (2021) · IEEE Transactions on Software Engineering · doi:10.1109/tse.2021.3131950 · Cited by 25
“The Friedman test [22] checks if there are statistically significant differences between performance scores generated by the models trained on the studied metric families.”
Incomplete label distribution feature selection based on neighborhood-tolerance discrimination index
Qian, W., Dong, P., Dai, S., Huang, J., & Wang, Y. (2022) · Applied Soft Computing · doi:10.1016/j.asoc.2022.109693 · Cited by 25
Manita, G., Chhabra, A., & Korbaa, O. (2023) · Applied Soft Computing · doi:10.1016/j.asoc.2023.110478 · Cited by 29
Increasing Retrieval Quality in Conversational Recommenders
Llorente, & Escalera, S. (2011) · IEEE Transactions on Knowledge and Data Engineering · doi:10.1109/tkde.2011.116 · Cited by 25
Using extension sets to aggregate partial rankings in a flexible setting
Aledo, J. A., Gámez, J. A., & Molina, D. (2016) · Applied Mathematics and Computation · doi:10.1016/j.amc.2016.06.005 · Cited by 25
Schulz, A., Christian Guckelsberger, B., & Janssen, F. W. (2016) · Semantic Web · doi:10.3233/sw-150188 · Cited by 25
“We therefore relied exclusively on the non-parametric tests suggested in literature: Friedman’s test [12,18] was used as a non-parametric alternative to a repeated-measures one-way ANOVA, and Nemenyi’s test [27,18] was used post-hoc as a replacement for Tukey’s test.”
MethodSampling algorithms for stochastic graphs: A learning automata approach
Rezvanian, A., & Meybodi, M. R. (2017) · Knowledge-Based Systems · doi:10.1016/j.knosys.2017.04.012 · Cited by 25
An Empirical Study on Online Price Differentiation
Hupperich, T., Tatang, D., Wilkop, N., & Holz, T. (2018) · ACM Conference on Data and Application Security and Privacy (CODASPY) · doi:10.1145/3176258.3176338 · Cited by 25
“For this purpose, we conduct the Friedman test [6, 7].”
MethodPerformance Comparison of Machine Learning Platforms
Roy, A., Qureshi, S., Pande, K., Nair, D., Gairola, K., Jain, P., Singh, S., Sharma, K., Jagadale, A., Lin, Y.-Y., et al. (2019) · INFORMS journal on computing · doi:10.1287/ijoc.2018.0825 · Cited by 27
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Ružičić, M. M., Miletić, M., & Dobrota, M. (2021) · Sustainability · doi:10.3390/su13073591 · Cited by 25
“Considering that the Friedman test is a non-parametric test, besides the mean value of co-operation level, we also present the Friedman mean rank to testify the differences.”
MethodInfluentialOrdinal and nominal classification of wind speed from synoptic pressurepatterns
Gutiérrez, P. A., Salcedo‐Sanz, S., Hervás‐Martínez, C., Carro‐Calvo, L., Sánchez‐Monedero, J., & Prieto, L. (2012) · Engineering Applications of Artificial Intelligence · doi:10.1016/j.engappai.2012.10.018 · Cited by 26
Thermal parameter inversion of low-heat cement concrete for Baihetan arch dam
Wang, F., Song, R., Yu, H., Zhang, A., Wang, L., & Chen, X. (2024) · Engineering Applications of Artificial Intelligence · doi:10.1016/j.engappai.2023.107823 · Cited by 24
Li, T., Qian, Y., Li, F., Liang, X., & Zhan, Z. (2024) · IEEE Transactions on Big Data · doi:10.1109/tbdata.2024.3378090 · Cited by 24
A robust multi-label feature selection based on label significance and fuzzy entropy
Yang, T., Wang, C., Chen, Y., & Deng, T. (2024) · International Journal of Approximate Reasoning · doi:10.1016/j.ijar.2024.109310 · Cited by 26
Towards the Novel Reasoning among Particles in PSO by the Use of RDF and SPARQL
Fister, I., Yang, X., Ljubič, K., Fister, D., Brest, J., & Fister, I. (2014) · The Scientific World JOURNAL · doi:10.1155/2014/121782 · Cited by 24
Banknote Counterfeit Detection through Background Texture Printing Analysis
Berenguel, A., Terrades, O. R., Lladós, J., & Cañero, C. (2016) · IAPR International Workshop on Document Analysis Systems (DAS) · doi:10.1109/das.2016.34 · Cited by 24
Diffusive likelihood for interactive image segmentation
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Multi-Label Attribute Reduction Based on Variable Precision Fuzzy Neighborhood Rough Set
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Wang, Y., Li, X., Wong, K., Chang, Y., & Yang, S. (2021) · IEEE Transactions on Cybernetics · doi:10.1109/tcyb.2021.3069434 · Cited by 24
Liu, X., Dai, J., Chen, J., & Zhang, C. (2021) · Applied Soft Computing · doi:10.1016/j.asoc.2021.107593 · Cited by 24
Extreme learning machine versus classical feedforward network
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“To identify hyperparameter values that allow reaching high performance on multiple various datasets we utilised a pair of statistical tests used to compare classifiers [7]—Friedman’s test [10] and Nemenyi’s post-hoc test [31].”
MethodAfkhami, M., Zahraie, B., & Ghorbani, M. (2022) · Journal of Arid Environments · doi:10.1016/j.jaridenv.2022.104715 · Cited by 25
Chen, H., Jing, X., Zhou, Y., Li, B., & Xu, B. (2022) · Information and Software Technology · doi:10.1016/j.infsof.2022.106892 · Cited by 24
COVID-19’s Impact on the Restaurant Industry
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Jia, L., Wang, Z., Lv, S., & Xu, Z. (2022) · IEEE Access · doi:10.1109/access.2022.3212067 · Cited by 31
Qu, K., Xu, J., Han, Z., & Xu, S. (2023) · Applied Intelligence · doi:10.1007/s10489-022-04398-z · Cited by 24
“The Friedman test [49] based on algorithm ranking is employed to judge whether the assumption that all algorithms have the same classification ability is valid.”
MethodBayesian network based probabilistic weighted high-order fuzzy time series forecasting
Wang, B., Liu, X., Chi, M., & Li, Y. (2023) · Expert Systems with Applications · doi:10.1016/j.eswa.2023.121430 · Cited by 24
Modeling Facial Soft Tissue Thickness for Automatic Skull-Face Overlay
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Stream mining: a novel architecture for ensemble-based classification
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Generalised Gaussian radial basis function neural networks
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Time series forecasting by recurrent product unit neural networks
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Coussement, K., & Buckinx, W. (2011) · European Journal of Operational Research · doi:10.1016/j.ejor.2011.05.027 · Cited by 24
NEARNESS AS A FUNCTION OF LATERAL ORIENTATION IN PICTURES
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Data-driven blood glucose level prediction in type 1 diabetes: a comprehensive comparative analysis
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A Fully Complex-valued Fast Learning Classifier (FC-FLC) for real-valued classification problems
Sivachitra, M., Savitha, R., Suresh, S., & Vijayachitra, S. (2014) · Neurocomputing · doi:10.1016/j.neucom.2014.04.075 · Cited by 23
Multi-label classification methods for improving comorbidities identification
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Mold Level Predict of Continuous Casting Using Hybrid EMD-SVR-GA Algorithm
Lei, Z., & Su, W. (2019) · Processes · doi:10.3390/pr7030177 · Cited by 23
“First, the original mold level signal is decomposed by EMD into several intrinsic mode functions (IMFs).”
MethodInfluentialWeak multi-label learning with missing labels via instance granular discrimination
Tan, A., Ji, X., Liang, J., Tao, Y., Wu, W.-Z., & Pedrycz, W. (2022) · Information Sciences · doi:10.1016/j.ins.2022.02.011 · Cited by 23
“To further reflect the superiority of the proposed method, Friedman test [11] and Bonferroni-Dunn test [9] are utilized to evaluate the statistical significance of the compared algorithms.”
MethodSparse multi-label feature selection via dynamic graph manifold regularization
Zhang, Y., & Ma, Y. (2022) · International Journal of Machine Learning and Cybernetics · doi:10.1007/s13042-022-01679-4 · Cited by 23
“Bonferroni-Dunn test [42] and Friedman test [43] were used to analyze the statistical significance and differences among experimental results of all indicators of each algorithm on eight data sets.”
MethodShu, W., Yu, J., Yan, Z., & Qian, W. (2022) · International Journal of Approximate Reasoning · doi:10.1016/j.ijar.2022.11.020 · Cited by 23
AMFSA: Adaptive fuzzy neighborhood-based multilabel feature selection with ant colony optimization
Sun, L., Chen, Y., Ding, W., Xu, J., & Ma, Y. (2023) · Applied Soft Computing · doi:10.1016/j.asoc.2023.110211 · Cited by 23
Moosaei, H., & Hladík, M. (2023) · Neural Networks · doi:10.1016/j.neunet.2023.07.039 · Cited by 23
Online hierarchical streaming feature selection based on adaptive neighborhood rough set
Shu, T., Lin, Y., & Guo, L. (2024) · Applied Soft Computing · doi:10.1016/j.asoc.2024.111276 · Cited by 23
Metadata
- 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)
- 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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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