NormalScore {SuppDists} | R Documentation |

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)

`x,q` |
vector of non-negative quantities |

`p` |
vector of probabilities |

`n` |
number of values to generate. If n is a vector, length(n) values will be generated |

`c` |
vector number of treatments |

`N` |
vector total number of observations |

`U` |
vector sum of reciprocals of the number of the c sample sizes |

`log, log.p` |
logical vector; if TRUE, probabilities p are given as log(p) |

`lower.tail` |
logical vector; if TRUE (default), probabilities are |

This is the Kruskal-Wallis statistic with ranks replaced by the expected values of normal order statistics. There are c treatments with sample sizes *nj, j=1 … c*. The total sample size is *N=Sum nj*. The distribution depends on c, N, and U, where *U=Sum (1/nj)*.

Let *eN(k)* be the expected value of the *kth* smallest observation in a sample of N independent normal variates. Rank all observations together, and let *Rij* denote the rank of observation *Xij*, *i=1 … nj* for treatment *j=1 … c*, then the normal scores test statistic is

*x=(N-1)[1/Sum(1 … N)(eN(k)^2)]Sum(1 … c)[(Sj^2)/nj]*

where *Sj=Sum(1 … nj)(eN(Rij))*.

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.

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.

Bob Wheeler bwheelerg@gmail.com

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)

[Package *SuppDists* version 1.1-9.4 Index]