mk.test: Mann-Kendall Trend Test

Description

Performs the Mann-Kendall Trend Test

Usage

mk.test(x, alternative = c("two.sided", "greater", "less"), continuity = TRUE)

Value

A list with class "htest"

data.name: character string that denotes the input data
p.value: the p-value
statistic: the z quantile of the standard normal distribution
null.value: the null hypothesis
estimates: the estimates S, varS and tau
alternative: the alternative hypothesis
method: character string that denotes the test

Arguments

x: a vector of class "numeric" or a time series object of class "ts"
alternative: the alternative hypothesis, defaults to two.sided
continuity: logical, indicates whether a continuity correction should be applied, defaults to TRUE.

Details

The null hypothesis is that the data come from a population with independent realizations and are identically distributed. For the two sided test, the alternative hypothesis is that the data follow a monotonic trend. The Mann-Kendall test statistic is calculated according to:

$$ S = \sum_{k = 1}^{n-1} \sum_{j = k + 1}^n \mathrm{sgn}\left(x_j - x_k\right)$$

with $\mathrm{sgn}$ the signum function (see sign).

The mean of $S$ is $\mu = 0$. The variance including the correction term for ties is

$$ \sigma^2 = \left\{n \left(n-1\right)\left(2n+5\right) - \sum_{j=1}^p t_j\left(t_j - 1\right)\left(2t_j+5\right) \right\} / 18$$

where $p$ is the number of the tied groups in the data set and $t_j$ is the number of data points in the $j$-th tied group. The statistic $S$ is approximately normally distributed, with

$$z = S / \sigma$$

If continuity = TRUE then a continuity correction will be employed:

$$z = \mathrm{sgn}(S) ~ \left(|S| - 1\right) / \sigma$$

The statistic $S$ is closely related to Kendall's $\tau$: $$\tau = S / D$$

where

$$ D = \left[\frac{1}{2}n\left(n-1\right)- \frac{1}{2}\sum_{j=1}^p t_j\left(t_j - 1\right)\right]^{1/2} \left[\frac{1}{2}n\left(n-1\right) \right]^{1/2} $$

References

Hipel, K.W. and McLeod, A.I. (1994), Time Series Modelling of Water Resources and Environmental Systems. New York: Elsevier Science.

Libiseller, C. and Grimvall, A., (2002), Performance of partial Mann-Kendall tests for trend detection in the presence of covariates. Environmetrics 13, 71--84, http://dx.doi.org/10.1002/env.507.

Examples

Run this code

data(Nile)
mk.test(Nile, continuity = TRUE)

##
n <- length(Nile)
cor.test(x=(1:n),y=Nile, meth="kendall", continuity = TRUE)

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