Van Der Waerden Test

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



References

2017

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