Van Der Waerden Test - Test Definition

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

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

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

\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 = \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 the k population distribution functions are identical
Ha: 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 = \frac{1}{s^2}\sum_{j=1}^kn_j\bar{A}_j^2

For significance level α, the critical region is

T_1 > \chi_{\alpha,k-1}^2

where Χα,k − 12 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 j1 and j2 seem to be different if the following inequality is satisfied:

\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 t1 − α/2 the (1 − α/2)-quantile of the t-distribution.

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