Van Der Waerden Test

A Van Der Waerden Test is a non-parametric statistical hypothesis test used for analyzing the homogeneity of population distribution functions.

• • …
• Counter-Example(s)
  • Kruskal-Wallis Test
  • ANOVA.
• See: Quantile, Bartel Leendert Van Der Waerden, Statistical Test, Kruskal-Wallis One-Way Analysis of Variance.

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References

2017

• (ITL-SED, 2017) ⇒ Retrieved 2017-01-08 from NIST (National Intitute of Standards and Technology, US) website http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/vanderwa.htm
  • 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 Van Der Waerden test converts the ranks 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 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.

Let [math]\displaystyle{ n_i (i = 1, 2, \cdots, k) }[/math] represent the sample sizes for each of the [math]\displaystyle{ k }[/math] groups (i.e., samples) in the data. Let [math]\displaystyle{ N }[/math] denote the sample size for all groups. Let [math]\displaystyle{ X_{ij} }[/math] represent the ith value in the jth group. Then compute the normal scores as follows:

[math]\displaystyle{ A_{ij}=\frac{\phi^{−1}(R(Xij))}{N+1} }[/math]

with [math]\displaystyle{ R(X_{ij}) }[/math] and [math]\displaystyle{ \phi }[/math] denoting the rank of observation [math]\displaystyle{ X_{ij} }[/math] and the normal percent point function, respectively.

The average of the normal scores for each sample can then be computed as

[math]\displaystyle{ \overline{A_i}=\frac{1}{n_i}\sum^{n_i}_{j=1}A_{ij}\quad i=1,2,\cdots,k }[/math]

The variance of the normal scores can be computed as

[math]\displaystyle{ s^2=\frac{1}{N−1}\sum^k_{i=1}\sum^{ni}_{j=1}A^2_{ij} }[/math]

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

Test Statistic: [math]\displaystyle{ T1=\frac{1}{s^2}\sum{k}_{i=1} n_i\overline{A_i}^2 }[/math]

Significance Level: [math]\displaystyle{ \alpha }[/math]

Critical Region: [math]\displaystyle{ T1 \gt CHIPPF (\alpha ,k-1) }[/math] where CHIPPF is the chi-square percent point function.

Conclusion: Reject the null hypothesis if the test statistic is in the critical region.

2016

• (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Van_der_Waerden_test Retrieved:2016-12-17.
  • Named for 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).

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