While methods for comparing two learning algorithms on a single data set have been scrutinized for quite some time already, the issue of statistical tests for comparisons of more algorithms on multiple data sets, which is even more essential to typical machine learning studies, has been all but ignored. This article reviews the current practice and then theoretically and empirically examines several suitable tests. Based on that, we recommend a set of simple, yet safe and robust non-parametric tests for statistical comparisons of classifiers: the Wilcoxon signed ranks test for comparison of two classifiers and the Friedman test with the corresponding post-hoc tests for comparison of more classifiers over multiple data sets. Results of the latter can also be neatly presented with the newly introduced CD (critical difference) diagrams.

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Statistical Comparisons of Classifiers over Multiple Data Sets

a b s t r a c t The interest in nonparametric statistical analysis has grown recently in the field of computational intelligence. In many experimental studies, the lack of the required properties for a proper application of parametric procedures - independence, normality, and homoscedasticity - yields to nonparametric ones the task of performing a rigorous comparison among algorithms. In this paper, we will discuss the basics and give a survey of a complete set of nonparametric procedures developed to perform both pairwise and multiple comparisons, for multi-problem analysis. The test problems of the CEC'2005 special session on real parameter optimization will help to illustrate the use of the tests throughout this tutorial, analyzing the results of a set of well-known evolutionary and swarm intelligence algorithms. This tutorial is concluded with a compilation of considerations and recommendations, which will guide practitioners when using these tests to contrast their experimental results.

https://sci2s.ugr.es/sites/default/files/ficherosPublicaciones/1374_2011-Derrac-SWEVO.pdf

A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms

We propose a natural scene statistic-based distortion-generic blind/no-reference (NR) image quality assessment (IQA) model that operates in the spatial domain. The new model, dubbed blind/referenceless image spatial quality evaluator (BRISQUE) does not compute distortion-specific features, such as ringing, blur, or blocking, but instead uses scene statistics of locally normalized luminance coefficients to quantify possible losses of “naturalness” in the image due to the presence of distortions, thereby leading to a holistic measure of quality. The underlying features used derive from the empirical distribution of locally normalized luminances and products of locally normalized luminances under a spatial natural scene statistic model. No transformation to another coordinate frame (DCT, wavelet, etc.) is required, distinguishing it from prior NR IQA approaches. Despite its simplicity, we are able to show that BRISQUE is statistically better than the full-reference peak signal-to-noise ratio and the structural similarity index, and is highly competitive with respect to all present-day distortion-generic NR IQA algorithms. BRISQUE has very low computational complexity, making it well suited for real time applications. BRISQUE features may be used for distortion-identification as well. To illustrate a new practical application of BRISQUE, we describe how a nonblind image denoising algorithm can be augmented with BRISQUE in order to perform blind image denoising. Results show that BRISQUE augmentation leads to performance improvements over state-of-the-art methods. A software release of BRISQUE is available online: http://live.ece.utexas.edu/research/quality/BRISQUE_release.zip for public use and evaluation.

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No-Reference Image Quality Assessment in the Spatial Domain

We evaluate 179 classifiers arising from 17 families (discriminant analysis, Bayesian, neural networks, support vector machines, decision trees, rule-based classifiers, boosting, bagging, stacking, random forests and other ensembles, generalized linear models, nearest-neighbors, partial least squares and principal component regression, logistic and multinomial regression, multiple adaptive regression splines and other methods), implemented in Weka, R (with and without the caret package), C and Matlab, including all the relevant classifiers available today. We use 121 data sets, which represent the whole UCI data base (excluding the large-scale problems) and other own real problems, in order to achieve significant conclusions about the classifier behavior, not dependent on the data set collection. The classifiers most likely to be the bests are the random forest (RF) versions, the best of which (implemented in R and accessed via caret) achieves 94.1% of the maximum accuracy overcoming 90% in the 84.3% of the data sets. However, the difference is not statistically significant with the second best, the SVM with Gaussian kernel implemented in C using LibSVM, which achieves 92.3% of the maximum accuracy. A few models are clearly better than the remaining ones: random forest, SVM with Gaussian and polynomial kernels, extreme learning machine with Gaussian kernel, C5.0 and avNNet (a committee of multi-layer perceptrons implemented in R with the caret package). The random forest is clearly the best family of classifiers (3 out of 5 bests classifiers are RF), followed by SVM (4 classifiers in the top-10), neural networks and boosting ensembles (5 and 3 members in the top-20, respectively).

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Do we need hundreds of classifiers to solve real world classification problems

The propensity score is defined as a subject's probability of treatment selection, conditional on observed baseline covariates. Weighting subjects by the inverse probability of treatment received creates a synthetic sample in which treatment assignment is independent of measured baseline covariates. Inverse probability of treatment weighting (IPTW) using the propensity score allows one to obtain unbiased estimates of average treatment effects. However, these estimates are only valid if there are no residual systematic differences in observed baseline characteristics between treated and control subjects in the sample weighted by the estimated inverse probability of treatment. We report on a systematic literature review, in which we found that the use of IPTW has increased rapidly in recent years, but that in the most recent year, a majority of studies did not formally examine whether weighting balanced measured covariates between treatment groups. We then proceed to describe a suite of quantitative and qualitative methods that allow one to assess whether measured baseline covariates are balanced between treatment groups in the weighted sample. The quantitative methods use the weighted standardized difference to compare means, prevalences, higher-order moments, and interactions. The qualitative methods employ graphical methods to compare the distribution of continuous baseline covariates between treated and control subjects in the weighted sample. Finally, we illustrate the application of these methods in an empirical case study. We propose a formal set of balance diagnostics that contribute towards an evolving concept of 'best practice' when using IPTW to estimate causal treatment effects using observational data.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/sim.6607

Moving towards best practice when using inverse probability of treatment weighting (IPTW) using the propensity score to estimate causal treatment effects in observational studies

With more than 500 pages of new material, the Handbook of Parametric and Nonparametric Statistical Procedures, Fourth Edition carries on the esteemed tradition of the previous editions, providing up-to-date, in-depth coverage of now more than 160 statistical procedures. The book also discusses both theoretical and practical statistical topics, such as experimental design, experimental control, and statistical analysis. New to the Fourth Edition Multivariate statistics including matrix algebra, multiple regression, Hotellings T2, MANOVA, MANCOVA, discriminant function analysis, canonical correlation, logistic regression, and principal components/factor analysis Clinical trials, survival analysis, tests of equivalence, analysis of censored data, and analytical procedures for crossover design Regression diagnostics that include the Durbin-Watson test Log-linear analysis of contingency tables, Mantel-Haenszel analysis of multiple 2 2 contingency tables, trend analysis, and analysis of variance for a Latin square design Levene and Brown-Forsythe tests for evaluating homogeneity of variance, the Jarque-Bera test of normality, and the extreme studentized deviate test for identifying outliers Confidence intervals for computing the population median and the difference between two population medians The relationship between exponential and Poisson distribution Eliminating the need to search across numerous books, this handbook provides you with everything you need to know about parametric and nonparametric statistical procedures. It helps you choose the best test for your data, interpret the results, and better evaluate the research of others.

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David J. Sheskin

Papers

Handbook of Parametric and Nonparametric Statistical Procedures

The Kruskal–Wallis One-Way Analysis of Variance by Ranks

The Friedman Two-Way Analysis of Variance by Ranks

The Wilcoxon Matched-Pairs Signed-Ranks Test

Inferential Statistical Tests Employed with Two Dependent Samples (and Related Measures of Association/Correlation)