Van der Waerden test
Named after the Dutch mathematician
The k population version of the test is an extension of the test for two populations published by Van der Waerden (1952,1953).
Background
Test definition
Let nj (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 Xij represent the ith value in the jth group. The normal scores are computed as
where R(Xij) denotes the rank of observation Xij 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:
- H0: All of the k population distribution functions tend to yield the same observation
- Ha: At least one of the populations tends to yield larger observations than at least one of the other populations
The test statistic is
For
where Χα,k − 12 is the α-
with t1 − α/2 the (1 − α/2)-quantile of the t-distribution.
Comparison with the Kruskal-Wallis test
The most common non-parametric test for the one-factor model is the
References
- 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