Performs the Mann-Kendall Trend Test
mk.test(x, alternative = c("two.sided", "greater", "less"),
continuity = TRUE)a vector of class "numeric" or a time series object of class "ts"
the alternative hypothesis, defaults to two.sided
logical, indicates whether a continuity correction
should be applied, defaults to TRUE.
A list with class "htest"
character string that denotes the input data
the p-value
the z quantile of the standard normal distribution
the null hypothesis
the estimates S, varS and tau
the alternative hypothesis
character string that denotes the test
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
$$ D = \left[\frac{1}{2}n\left(n-1\right)- \frac{1}{2}\sum_{j=1}^p t_j\left(t_j - 1\right)\right]^{1/2} \left[\frac{1}{2}n\left(n-1\right) \right]^{1/2} $$
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.
# NOT RUN {
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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