The Jonckheere-Terpstra k-sample test statistic JT is defined as \(JT = \sum_{i<j} W_{ij}\) where \(W_{ij}\) is the Mann-Whitney statistic comparing samples \(i\) and \(j\), indexed in the order of the stipulated increasing alternative. There may be ties in the pooled samples.
jt.test(..., data = NULL, method=c("asymptotic","simulated","exact"),
dist = FALSE, Nsim = 10000)A list of class kSamples with components
"Jonckheere-Terpstra"
number of samples being compared
vector \((n_1,\ldots,n_k)\) of the \(k\) sample sizes
size of the pooled sample \(= n_1+\ldots+n_k\)
number of ties in the pooled sample
4 (or 5) vector containing the observed \(JT\), its mean and standard deviation and its asymptotic \(P\)-value, (and its simulated or exact \(P\)-value)
logical indicator, warning = TRUE when at least one
\(n_i < 5\)
simulated or enumerated null distribution
of the test statistic. It is NULL when dist = FALSE or when
method = "asymptotic".
the method used.
the number of simulations used.
Either several sample vectors, say \(x_1, \ldots, x_k\), with \(x_i\) containing \(n_i\) sample values. \(n_i > 4\) is recommended for reasonable asymptotic \(P\)-value calculation. The pooled sample size is denoted by \(N=n_1+\ldots+n_k\). The order of samples should be as stipulated under the alternative
or a list of such sample vectors,
or a formula y ~ g, where y contains the pooled sample values and g (same length as y) is a factor with levels identifying the samples to which the elements of y belong, the factor levels reflecting the order under the stipulated alternative,
= an optional data frame providing the variables in formula y ~ g.
= c("asymptotic","simulated","exact"), where
"asymptotic" uses only an asymptotic normal \(P\)-value approximation.
"simulated" uses Nsim simulated \(JT\) statistics based on random splits of the
pooled samples into samples of sizes
\(n_1, \ldots, n_k\), to estimate the \(P\)-value.
"exact" uses full enumeration of all sample splits with
resulting \(JT\) statistics to obtain the exact \(P\)-value.
It is used only when Nsim is at least as large as the number
$$ncomb = \frac{N!}{n_1!\ldots n_k!}$$
of full enumerations. Otherwise, method
reverts to "simulated" using the given Nsim. It also reverts
to "simulated" when \(ncomb > 1e8\) and dist = TRUE.
= FALSE (default) or TRUE. If TRUE, the
simulated or fully enumerated distribution vector null.dist
is returned for the JT test statistic. Otherwise, NULL is returned.
When dist = TRUE then Nsim <- min(Nsim, 1e8),
to limit object size.
= 10000 (default), number of simulation sample splits to use.
It is only used when method = "simulated",
or when method = "exact" reverts to method =
"simulated", as previously explained.
The JT statistic is used to test the hypothesis that the samples all come from the same but unspecified continuous distribution function \(F(x)\). It is specifically aimed at alternatives where the sampled distributions are stochastically increasing.
NA values are removed and the user is alerted with the total NA count. It is up to the user to judge whether the removal of NA's is appropriate.
The continuity assumption can be dispensed with, if we deal with independent random samples, or if randomization was used in allocating subjects to samples or treatments, and if we view the simulated or exact \(P\)-values conditionally, given the tie pattern in the pooled samples. Of course, under such randomization any conclusions are valid only with respect to the group of subjects that were randomly allocated to their respective samples. The asymptotic \(P\)-value calculation is valid provided all sample sizes become large.
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.
x1 <- c(1,2)
x2 <- c(1.5,2.1)
x3 <- c(1.9,3.1)
yy <- c(x1,x2,x3)
gg <- as.factor(c(1,1,2,2,3,3))
jt.test(x1, x2, x3,method="exact",Nsim=90)
# or
# jt.test(list(x1, x2, x3), method = "exact", Nsim = 90)
# or
# jt.test(yy ~ gg, method = "exact", Nsim = 90)
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