Named after the Dutch mathematicianBartel Leendert van der Waerden, theVan der Waerden test is astatistical test thatk population distribution functions are equal. The Van der Waerden test converts the ranks from a standardKruskal-Wallis test toquantiles of the standard normal distribution (details given below). These are called normal scores and the test is computed from these normal scores.

Thek population version of the test is an extension of the test for two populations published by Van der Waerden (1952,1953).

Analysis of Variance (ANOVA) is adata 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 thetwo 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 thatk population means are equal. The standard ANOVA assumes that the errors (i.e., residuals) arenormally distributed. If this normality assumption is not valid, an alternative is to use anon-parametric test.

Letn_j (j = 1, 2, ...,k) represent the sample sizes for each of thek groups (i.e., samples) in the data. LetN denote the sample size for all groups. LetX_ij represent thei^th value in thej^th group. The normal scores are computed as

${\displaystyle A_{ij}={\mathrm{\Phi }}^{-1}\left(\frac{R(X_{ij})}{N+1}\right)}$

whereR(X_ij) denotes the rank of observationX_ij and whereΦ⁻¹ denotes the normalquantile function. The average of the normal scores for each sample can then be computed as

${\displaystyle {\overline{A}}_j=\frac{1}{n_j}\sum _{i=1}^{n_j}{A}_{ij}j=1,2,\dots ,k}$

The variance of the normal scores can be computed as

${\displaystyle s^2=\frac{1}{N-1}\sum _{j=1}^k\sum _{i=1}^{n_j}{A}_{ij}^2}$

The Van der Waerden test can then be defined as follows:

H₀: All of thek 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

${\displaystyle T_1=\frac{1}{s^2}\sum _{j=1}^k{n}_j{\overline{A}}_j^2}$

Forsignificance level α, the critical region is

${\displaystyle T_1>{\chi }_{\alpha ,k-1}^2}$

where Χ_{α,k − 1}² is the α-quantile of thechi-squared distribution withk − 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 amultiple comparisons procedure to determine which pairs of populations tend to differ. The populationsj₁ andj₂ seem to be different if the following inequality is satisfied:

${\displaystyle \left|{\overline{A}}_{j_1}-{\overline{A}}_{j_2}\right|>s{t}_{1-\alpha /2}\sqrt{\frac{N-1-T_1}{N-k}}\sqrt{\frac{1}{n_{j_1}}+\frac{1}{n_{j_2}}}}$

witht_{1 − α/2} the (1 − α/2)-quantile of thet-distribution.

The most common non-parametric test for the one-factor model is theKruskal-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.

• 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.

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