Null Distribution of the Jonckheere-Terpstra k-Sample Test Statistic
The Jonckheere-Terpstra k-sample test statistic JT is defined as JT = ∑_{i<j} W_{ij} where W_{ij} is the Mann-Whitney statistic comparing samples i and j, indexed in the order of the stipulated increasing alternative. It is assumed that there are no ties in the pooled samples.
This function uses Harding's algorithm as far as computations are possible without becoming unstable.
djt(x, nn) pjt(x, nn) qjt(p, nn)
x |
a numeric vector, typically integers |
nn |
a vector of integers, representing the sample sizes in the order stipulated by the alternative |
p |
a vector of probabilities |
While Harding's algorithm is mathematically correct, it is problematic in its computing implementation. The counts become very large and normalizing them by combinatorials leads to significance loss. When that happens the functions return an error message: can't compute due to numerical instability. This tends to happen when the total number of sample values becomes too large. That depends also on the way the sample sizes are allocated.
For djt it is a vector
p = (p_1,…,p_n)
giving the values of
p_i = P(JT = x_i), where n is the length
of the input x.
For pjt it is a vector
P = (P_1,…,P_n)
giving the values of
P_i = P(JT <= x_i).
For qjt is a vecto r x = (x_1,…,x_n),where x_i
is the smallest x
such that P(JT <= x) >= p_i.
Harding, E.F. (1984), An Efficient, Minimal-storage Procedure for Calculating the Mann-Whitney U, Generalized U and Similar Distributions, Appl. Statist. 33 No. 1, 1-6.
Jonckheere, A.R. (1954), A Distribution Free k-sample Test against Ordered Alternatives, Biometrika, 41, 133-145.
Lehmann, E.L. (2006), Nonparametrics, Statistical Methods Based on Ranks, Revised First Edition, Springer Verlag.
Terpstra, T.J. (1952), The Asymptotic Normality and Consistency of Kendall's Test against Trend, when Ties are Present in One Ranking, Indagationes Math. 14, 327-333.
djt(c(-1.5,1.2,3), 2:4) pjt(c(2,3.4,7), 3:5) qjt(c(0,.2,.5), 2:4)
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