Jonckheere-Terpstra Test for Detecting Ordered Differences Across Independent Groups

A nonparametric test for monotonic trends across ordered independent groups, assessing whether observations tend to increase (or decrease) systematically with group order.

Usage

jonckheereTerpstraTest(x, ...)

# S3 method for class 'formula'
jonckheereTerpstraTest(formula, data, subset, na.action, ...)

# Default S3 method
jonckheereTerpstraTest(
  x,
  g,
  alternative = c("two.sided", "increasing", "decreasing"),
  method = c("auto", "exact", "permutation", "asymptotic"),
  R = NULL,
  ...
)

Arguments

x: a numeric vector of observations, or a list of numeric vectors.
...: further arguments passed to methods.
formula: a formula of the form response ~ group.
data: an optional data frame containing the variables in formula.
subset: an optional expression specifying a subset of observations.
na.action: a function indicating how missing values should be handled.
g: a grouping variable corresponding to x, whose (factor) level order defines the hypothesised ordering; ignored when x is a list.
alternative: a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "increasing" or "decreasing".
method: a character string specifying the inference method, one of "auto" (default), "exact", "permutation" or "asymptotic". "auto" uses exact inference when possible (no ties, $n \le 100$), otherwise the asymptotic approximation.
R: the number of permutations, required when method = "permutation".

Value

A list with class "htest" containing the following components:

statistic: the value of the Jonckheere-Terpstra statistic.
parameter: the number of groups k and the total sample size n.
p.value: the p-value of the test.
alternative: a character string describing the alternative hypothesis.
method: a character string indicating the test performed and the inference method used.
data.name: a character string giving the names of the data.

Details

The Jonckheere-Terpstra statistic is $$JT = \sum_{k<l} \sum_{ij} \left[ I(X_{ik} < X_{jl}) + \frac{1}{2} I(X_{ik} = X_{jl}) \right]$$ where $i, j$ index observations from ordered groups $k, l$. Large values of the statistic indicate increasing trends across groups.

Exact p-values are computed from the exact permutation distribution using a dynamic programming recursion implemented in C++. Exact inference is only valid without ties and, for practical reasons, is offered for total sample sizes $n \le 100$.

When ties are present or sample sizes are large, permutation p-values can be computed by permuting group labels under the null hypothesis (method = "permutation"); the number of permutations is controlled by R, and the reported p-value uses the finite-sample correction $(m + 1)/(R + 1)$. This approach remains valid in the presence of ties.

With method = "asymptotic" (the fallback of "auto" when exact inference does not apply), a normal approximation with the tie-corrected variance of Hollander and Wolfe (1999, Eq. 6.19) is used.

References

Jonckheere, A. R. (1954) A distribution-free k-sample test against ordered alternatives. Biometrika, 41, 133–145.

Terpstra, T. J. (1952) The asymptotic normality and consistency of Kendall's test against trend, when ties are present in one ranking. Indagationes Mathematicae, 14, 327–333.

Hollander, M. and Wolfe, D. A. (1999) Nonparametric Statistical Methods, 2nd ed., New York: Wiley.

Examples

set.seed(1)
g <- ordered(rep(1:4, each = 10))
x <- rnorm(40) + 0.5 * as.numeric(g)

jonckheereTerpstraTest(x, g)
#> 
#> 	Jonckheere-Terpstra test for ordered alternatives (exact)
#> 
#> data:  x and g
#> JT = 437, k = 4, n = 40, p-value = 0.0007562
#> alternative hypothesis: two.sided
#> 

# with ties: permutation inference
x[1:2] <- mean(x[1:2])
jonckheereTerpstraTest(x, g, method = "permutation", R = 2000)
#> 
#> 	Jonckheere-Terpstra test for ordered alternatives (permutation, R =
#> 	2000)
#> 
#> data:  x and g
#> JT = 438, k = 4, n = 40, p-value = 0.0004998
#> alternative hypothesis: two.sided
#> 

coffee <- list(
  c_4 = c(447, 396, 383, 410),
  c_2 = c(438, 521, 468, 391, 504, 472),
  c_0 = c(513, 543, 506, 489, 407)
)
jonckheereTerpstraTest(coffee)
#> 
#> 	Jonckheere-Terpstra test for ordered alternatives (exact)
#> 
#> data:  x
#> JT = 59, k = 3, n = 15, p-value = 0.01973
#> alternative hypothesis: two.sided
#> 

# Hollander & Wolfe, Example 6.2:
# motivational effect of knowledge of performance
motiv <- list(
  no    = c(40, 35, 38, 43, 44, 41),
  rough = c(38, 40, 47, 44, 40, 42),
  acc   = c(48, 40, 45, 43, 46, 44))

jonckheereTerpstraTest(motiv, alternative = "increasing")
#> 
#> 	Jonckheere-Terpstra test for ordered alternatives (asymptotic)
#> 
#> data:  x
#> JT = 79, k = 3, n = 18, p-value = 0.02071
#> alternative hypothesis: increasing
#> 
## exact one-sided p-value 0.0379 as in Hollander & Wolfe

jonckheereTerpstraTest(motiv, alternative = "increasing",
                       method = "asymptotic")
#> 
#> 	Jonckheere-Terpstra test for ordered alternatives (asymptotic)
#> 
#> data:  x
#> JT = 79, k = 3, n = 18, p-value = 0.02071
#> alternative hypothesis: increasing
#> 

set.seed(42)
jonckheereTerpstraTest(motiv, alternative = "increasing",
                       method = "permutation", R = 10000)
#> 
#> 	Jonckheere-Terpstra test for ordered alternatives (permutation, R =
#> 	10000)
#> 
#> data:  x
#> JT = 79, k = 3, n = 18, p-value = 0.0194
#> alternative hypothesis: increasing
#>