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van der waerden test wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 background 2 test definition 3 comparison with the kruskal wallis test 4 references toggle the table of contents van der waerden test 1 language français edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia statistical test named after the dutch mathematician bartel leendert van der waerden the van der waerden test is a statistical test that k population distribution functions are equal the van der waerden test converts the ranks from a standard kruskal wallis test to quantiles of the standard normal distribution details given below these are called normal scores and the test is computed from these normal scores the k population version of the test is an extension of the test for two populations published by van der waerden 1952 1953 background edit analysis of variance anova is a data analysis technique for examining the significance of the factors independent variables in a multi factor model the one factor model can be thought of as a generalization of the two sample t test that is the two sample t test is a test of the hypothesis that two population means are equal the one factor anova tests the hypothesis that k population means are equal the standard anova assumes that the errors i e residuals are normally distributed if this normality assumption is not valid an alternative is to use a non parametric test test definition edit let n j j 1 2 k represent the sample sizes for each of the k groups i e samples in the data let n denote the sample size for all groups let x ij represent the i th value in the j th group the normal scores are computed as a i j φ 1 r x i j n 1 displaystyle a_ ij phi 1 left frac r x_ ij n 1 right where r x ij denotes the rank of observation x ij and where φ 1 denotes the normal quantile function the average of the normal scores for each sample can then be computed as a j 1 n j i 1 n j a i j j 1 2 k displaystyle bar a _ j frac 1 n_ j sum _ i 1 n_ j a_ ij quad j 1 2 ldots k the variance of the normal scores can be computed as s 2 1 n 1 j 1 k i 1 n j a i j 2 displaystyle s 2 frac 1 n 1 sum _ j 1 k sum _ i 1 n_ j a_ ij 2 the van der waerden test can then be defined as follows h 0 all of the k population distribution functions tend to yield the same observation h a at least one of the populations tends to yield larger observations than at least one of the other populations the test statistic is t 1 1 s 2 j 1 k n j a j 2 displaystyle t_ 1 frac 1 s 2 sum _ j 1 k n_ j bar a _ j 2 for significance level α the critical region is t 1 χ α k 1 2 displaystyle t_ 1 chi _ alpha k 1 2 where χ α k 1 2 is the α quantile of the chi squared distribution with k 1 degrees of freedom the null hypothesis is rejected if the test statistic is in the critical region if the hypothesis of identical distributions is rejected one can perform a multiple comparisons procedure to determine which pairs of populations tend to differ the populations j 1 and j 2 seem to be different if the following inequality is satisfied a j 1 a j 2 s t 1 α 2 n 1 t 1 n k 1 n j 1 1 n j 2 displaystyle left vert bar a _ j_ 1 bar a _ j_ 2 right vert s t_ 1 alpha 2 sqrt frac n 1 t_ 1 n k sqrt frac 1 n_ j_ 1 frac 1 n_ j_ 2 with t 1 α 2 the 1 α 2 quantile of the t distribution comparison with the kruskal wallis test edit the most common non parametric test for the one factor model is the kruskal wallis test the kruskal wallis test is based on the ranks of the data the advantage of the van der waerden test is that it provides the high efficiency of the standard anova analysis when the normality assumptions are in fact satisfied but it also provides the robustness of the kruskal wallis test when the normality assumptions are not satisfied references edit conover w j 1999 practical nonparameteric statistics third ed wiley pp 396 406 van der waerden b l 1952 order tests for the two sample problem and their power indagationes mathematicae 14 453 458 van der waerden b l 1953 order tests for the two sample problem ii iii proceedings of the koninklijke nederlandse akademie van wetenschappen serie a 564 303 310 311 316 this article incorporates public domain material from the national institute of standards and technology v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional study natural experiment quasi experiment statistical inference statistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model model specification l p space parameter location scale shape parametric family likelihood monotone location scale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness sensitivity analysis frequentist inference point estimation estimating equations maximum likelihood method of moments m estimator minimum distance unbiased estimators mean unbiased minimum variance rao blackwellization lehmann scheffé theorem median unbiased plug in interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2 tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihood ratio g test score lagrange multiplier wald z test normal specific tests parametric student s t test f test goodness of fit chi squared kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodges lehmann estimator rank sum mann whitney nonparametric anova 1 way kruskal wallis 2 way friedman ordered alternative jonckheere terpstra van der waerden test bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlation regression analysis correlation pearson product moment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model bayesian regression non standard predictors nonlinear regression nonparametric semiparametric isotonic robust homoscedasticity and heteroscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova analysis of variance ancova manova degrees of freedom categorical multivariate time series survival analysis categorical cohen s kappa contingency table graphical model log linear model mcnemar s test cochran mantel haenszel statistics multivariate regression manova principal components canonical correlation discriminant analysis cluster analysis classification structural equation model factor analysis multivariate distributions elliptical distributions normal time series general decomposition trend stationarity seasonal adjustment exponential smoothing cointegration structural break granger causality specific tests dickey fuller johansen q statistic ljung box durbin watson breusch godfrey time domain autocorrelation acf partial pacf cross correlation xcf arma model arima model box jenkins autoregressive conditional heteroskedasticity arch vector autoregression var autoregressive model ar frequency domain spectral density estimation fourier analysis least squares spectral analysis wavelet whittle likelihood survival survival function kaplan meier estimator product limit proportional hazards models accelerated failure time aft model first hitting time hazard function nelson aalen estimator test log rank test applications biostatistics bioinformatics clinical trials studies epidemiology medical statistics engineering statistics chemometrics methods engineering probabilistic design process quality control reliability system identification social statistics actuarial science census crime statistics demography econometrics jurimetrics national accounts official statistics population statistics psychometrics spatial statistics cartography environmental statistics geographic information system geostatistics kriging category mathematics portal commons wikiproject retrieved from https en wikipedia org w index php title van_der_waerden_test oldid 1206154089 categories statistical tests nonparametric statistics hidden categories articles with short description short description matches wikidata wikipedia articles incorporating text from the national institute of standards and technology this page was last edited on 11 february 2024 at 09 31 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of 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