Van der Waerden 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 one-way analysis of variance 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).

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.

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

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

The variance of the normal scores can be computed as

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

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