
VAN DER WAERDENName:
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 nonparametric test. The most common nonparametric test for the onefactor model is the KruskalWallis test. The KruskalWallis 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 KruskalWallis test when the normality assumptions are not satisfied. Let n_{i} (i = 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 ith value in the jth group. Then compute the normal scores as follows:
with R(X_{ij}) and \( \phi \) denoting the rank of observation X_{ij} and the normal percent point function, respectively. 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.
where <y> is the response (= dependent) variable; <x> is the factor (= independent) variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
VAN DER WAERDEN Y X SUBSET X = 1 TO 4
The populations i and j seem to be different if the following inequality is satisfied:
with t and T_{1} denoting the t percent point function and the Van Der Waerden test statistic, respectively. Dataplot writes all the pairwise multiple comparisons to the file "dpst1f.dat" in the current directory.
The word TEST is optional
SKIP 25 READ SPLETT2.DAT Y MACHINE SET WRITE DECIMALS 5 VAN DER WAERDEN Y MACHINEThe following output is generated. Van Der Waerden (Normal Scores) One Factor Test Response Variable: Y GroupID Variable: MACHINE H0: Samples Come From Identical Populations Ha: Samples Do Not Come From Identical Populations Summary Statistics: Total Number of Observations: 99 Number of Groups: 4 Variance of Normal Scores of Ranks 0.92890 Van Der Waerden Test Statistic Value: 39.78569 CDF of Test Statistic: 1.00000 PValue: 0.00000 Percent Points of the ChiSquare Reference Distribution  Percent Point Value  0.0 = 0.000 50.0 = 2.366 75.0 = 4.108 90.0 = 6.251 95.0 = 7.815 97.5 = 9.348 99.0 = 11.345 99.9 = 16.266 Conclusions (Upper 1Tailed Test)  Alpha CDF Critical Value Conclusion  10% 90% 6.251 Reject H0 5% 95% 7.815 Reject H0 2.5% 97.5% 9.348 Reject H0 1% 99% 11.345 Reject H0 Multiple Comparisons Table  I J Abar(i)Abar(j 90% CV 95% CV 99% CV  1 2 0.65906 0.35813 0.42803 0.56674 1 3 1.57550 0.35813 0.42803 0.56674 1 4 0.17816 0.35813 0.42803 0.56674 2 3 0.91644 0.35446 0.42364 0.56092 2 4 0.48089 0.35446 0.42364 0.56092 3 4 1.39733 0.35446 0.42364 0.56092  
Date created: 01/05/2006 Last updated: 12/11/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 