Density, distribution function, quantile function, random generator and summary function for the normal scores test. A function to calculate expected values of normal order statistics is included.
dNormScore(x, c, N, U, log=FALSE)
pNormScore(q, c, N, U, lower.tail=TRUE, log.p=FALSE)
qNormScore(p, c, N, U, lower.tail=TRUE, log.p=FALSE)
rNormScore(n, c, N, U)
sNormScore(c, N, U)
normOrder(N)The output values conform to the output from other such functions in R. dNormScore() gives the density, pNormScore() the distribution function and qNormScore() its inverse. rNormScore() generates random numbers. sNormScore() produces a list containing parameters corresponding to the arguments -- mean, median, mode, variance, sd, third cental moment, fourth central moment, Pearson's skewness, skewness, and kurtosis. normOrder() gives the expected values of the normal order statistics for a sample of size N.
vector of non-negative quantities
vector of probabilities
number of values to generate. If n is a vector, length(n) values will be generated
vector number of treatments
vector total number of observations
vector sum of reciprocals of the number of the c sample sizes
logical vector; if TRUE, probabilities p are given as log(p)
logical vector; if TRUE (default), probabilities are \(P[X <= x]\), otherwise, \(P[X > x]\)
Bob Wheeler
This is the Kruskal-Wallis statistic with ranks replaced by the expected values of normal order statistics. There are c treatments with sample sizes \(n_j, j=1 \dots c\). The total sample size is \(N=\sum_1^c n_j\). The distribution depends on c, N, and U, where \(U=\sum_1^c (1/n_j)\).
Let \(e_N(k)\) be the expected value of the \(k_{th}\) smallest observation in a sample of N independent normal variates. Rank all observations together, and let \(R_{ij}\) denote the rank of observation \(X_{ij}\), \(i=1 \dots n_j\) for treatment \(j=1 \dots c\), then the normal scores test statistic is
$$x=(N-1)\frac{1}{\sum_{k=1}^{N} e_N(k)^2} \sum_{j=1}^{c}\frac{S_j^2}{n_j}$$
where \(S_j=\sum_{i=1}^{n_j}(e_N(R_{ij}))\).
See Lu and Smith (1979) for a thorough discussion and some exact tables for small r and n. The calculations made here use an incomplete beta approximation -- the same one used for Kruskal-Wallis, only differing in the calculation of the variance of the statistic.
The expected values of the normal order statistics use a modification of M.Maechler's C version of the Fortran algorithm given by Royston (1982). Spot checking the values against Harter (1969) confirms the accuracy to 4 decimal places as claimed by Royston.
Harter, H.L. (1969). Order statistics and their use in testing and estimation, volume 2. U.S. Supp. of Doc.
Lu, H.T. and Smith, P.J. (1979) Distribution of normal scores statistic for nonparametric one-way analysis of variance. Jour. Am Stat. Assoc. 74. 715-722.
Royston, J.P. (1982). Expected normal order statistics (exact and approximate) AS 177. Applied Statistics. 31. 161-165.
#Assuming three treatments, each with a sample size of 5
pNormScore(2, 3, 15, 0.6)
pNormScore(c(0.11,1.5,5.6), 3, 15, 0.6) ## approximately 5% 50% and 95%
sNormScore(3, 15, 0.6)
plot(function(x)dNormScore(x,c=5, N=15, U=0.6),0,5)
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