Cox-Stuart Trend Test for Detecting Monotonic Trends in Ordered Data

Tests a sequence of observations for a monotone trend by pairing each observation in the first half of the series with the corresponding observation in the second half and applying a sign test to the differences.

Usage

coxStuartTest(x, alternative = c("two.sided", "increasing", "decreasing"))

Arguments

x: a numeric vector of observations in sequence order
alternative: a character string specifying the alternative hypothesis, one of "two.sided" (default), "increasing" or "decreasing"

Value

An object of class "htest" with components

statistic: the number of positive paired differences, with names attribute "S"
parameter: the number of untied pairs entering the test
p.value: the p-value
estimate: the proportion of increasing pairs
alternative: a character string describing the alternative hypothesis
method: the character string "Cox-Stuart trend test"
data.name: a character string giving the name of the data

Details

The series \(x_1, \ldots, x_N\) is split in the middle; with an odd \(N\) the central observation is discarded. Writing \(m = \lfloor N/2 \rfloor\), each of the first \(m\) remaining observations is paired with the one \(m\) places later in the reduced series, that is with \(x_{i+m}\) for even \(N\) and with \(x_{i+m+1}\) in the original indexing for odd \(N\). The statistic \(S\) counts how many of the \(m\) paired differences are positive. Pairs that are exactly tied carry no information about direction and are dropped.

Every observation enters at most one pair, so under independence and a continuous common distribution the signs are independent Bernoulli variables with probability one half. The p-value from binom.test() is then exact at any series length, however short. That exactness rests on the independence assumption: with serially dependent observations the signs need not be independent and the nominal level is no longer guaranteed.

The test is a special case of signTest() and inherits both its robustness and its low power: only the sign of each paired difference is used, and half the observations enter only as partners. It detects a monotone drift, not curvature or oscillation: a series that rises and then falls back can easily produce a p-value near one.

Unlike the other members of test.trend, which need a grouping factor or a contingency table, this test takes a bare series and therefore has no formula interface. Missing values are removed before the series is split, so the pairing refers to the observed values, not to their original positions.

References

Cox, D. R., Stuart, A. (1955) Some quick sign tests for trend in location and dispersion. Biometrika, 42(1/2), 80-95.

Examples

## a strictly increasing series
coxStuartTest(1:12)
#> 
#> 	Cox-Stuart trend test
#> 
#> data:  1:12
#> S = 6, n = 6, p-value = 0.03125
#> alternative hypothesis: two.sided
#> sample estimates:
#> proportion of increasing pairs 
#>                              1 
#> 
## [1] S = 6, n = 6, p-value = 0.03125

coxStuartTest(1:12, alternative = "increasing")$p.value
#> [1] 0.015625
## [1] 0.015625

## no trend
set.seed(1)
coxStuartTest(rnorm(50))$p.value
#> [1] 0.1077521