Perform a nonparametric test for a monotonic trend based on Kendall's tau statistic, and optionally compute a confidence interval for the slope.
kendallTrendTest(y, ...)
# S3 method for class 'formula'
kendallTrendTest(y, data = NULL, subset,
na.action = na.pass, ...)
# Default S3 method
kendallTrendTest(y, x = seq(along = y),
alternative = "two.sided", correct = TRUE, ci.slope = TRUE,
conf.level = 0.95, warn = TRUE, data.name = NULL, data.name.x = NULL,
parent.of.data = NULL, subset.expression = NULL, ...)Extended methods and derivations are available in
Trend tests and serial correlation: kendallTrendTest.
Open the vignette locally with
vignette("trend-and-correlation", package = "EnvStats").
an object containing data for the trend test. In the default method,
the argument y must be numeric vector of observations.
In the formula method, y must be a formula of the form y ~ 1 or
y ~ x. The form y ~ 1 indicates use the observations in the vector
y for the test for trend and use the default value of the argument x
in the call to kendallTrendTest.default. The form y ~ x indicates
use the observations in the vector y for the test for trend and use the
specified value of the argument x in the call to
kendallTrendTest.default. Missing (NA), undefined (NaN),
and infinite (Inf, -Inf) values are allowed but will be
removed.
specifies an optional data frame, list or environment (or object coercible by
as.data.frame to a data frame) containing the variables in the model.
If not found in data, the variables are taken from environment(formula),
typically the environment from which kendallTrendTest is called.
specifies an optional vector specifying a subset of observations to be used.
specifies a function which indicates what should happen when the data contain NAs.
The default is na.pass.
numeric vector of "predictor" values. The length of x must equal the length of y.
Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are
allowed but will be removed. The default value of x is the vector
of numbers \(1, 2, \dots, n\) where \(n\) is the number of elements in
y.
character string indicating the kind of alternative hypothesis. The
possible values are "two.sided" (tau not equal to 0; the default),
"less" (tau less than 0), and "greater" (tau greater than 0).
logical scalar indicating whether to use the correction for continuity in
computing the \(z\)-statistic that is based on the test statistic \(S\).
The default value is TRUE.
logical scalar indicating whether to compute a confidence interval for the
slope. The default value is TRUE.
numeric scalar between 0 and 1 indicating the confidence level associated
with the confidence interval for the slope. The default value is
0.95.
logical scalar indicating whether to print a warning message when
y does not contain at least two non-missing values,
or when x does not contain at least two unique non-missing values.
The default value is TRUE.
character string indicating the name of the data used for the trend test.
The default value is deparse(substitute(y)).
character string indicating the name of the data used for the predictor variable x.
If x is not supplied this argument is ignored. When x is supplied,
the default value is deparse(substitute(x)).
character string indicating the source of the data used for the trend test.
character string indicating the expression used to subset the data.
additional arguments affecting the test for trend.
A list of class "htestEnvStats" containing the results of the hypothesis
test. See the help file for htestEnvStats.object for details.
In addition, the following components are part of the list returned by
kendallTrendTest:
The value of the Kendall S-statistic.
The variance of the Kendall S-statistic.
A numeric vector of all possible two-point slope estimates.
This component is used by the function kendallSeasonalTrendTest.
Bradley, J.V. (1968). Distribution-Free Statistical Tests. Prentice-Hall, Englewood Cliffs, NJ.
Conover, W.J. (1980). Practical Nonparametric Statistics. Second Edition. John Wiley and Sons, New York, pp.256-272.
Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.
Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York, NY, Chapter 16.
Helsel, D.R. and R.M. Hirsch. (1988). Discussion of Applicability of the t-test for Detecting Trends in Water Quality Variables. Water Resources Bulletin 24(1), 201-204.
Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, NY.
Helsel, D.R., and R. M. Hirsch. (2002). Statistical Methods in Water Resources. Techniques of Water Resources Investigations, Book 4, chapter A3. U.S. Geological Survey. Available on-line at https://pubs.usgs.gov/tm/04/a03/tm4a3.pdf.
Hirsch, R.M., J.R. Slack, and R.A. Smith. (1982). Techniques of Trend Analysis for Monthly Water Quality Data. Water Resources Research 18(1), 107-121.
Hirsch, R.M. and J.R. Slack. (1984). A Nonparametric Trend Test for Seasonal Data with Serial Dependence. Water Resources Research 20(6), 727-732.
Hirsch, R.M., R.B. Alexander, and R.A. Smith. (1991). Selection of Methods for the Detection and Estimation of Trends in Water Quality. Water Resources Research 27(5), 803-813.
Hollander, M., and D.A. Wolfe. (1999). Nonparametric Statistical Methods, Second Edition. John Wiley and Sons, New York.
Kendall, M.G. (1938). A New Measure of Rank Correlation. Biometrika 30, 81-93.
Kendall, M.G. (1975). Rank Correlation Methods. Charles Griffin, London.
Mann, H.B. (1945). Nonparametric Tests Against Trend. Econometrica 13, 245-259.
Millard, S.P., and Neerchal, N.K. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, Florida.
Sen, P.K. (1968). Estimates of the Regression Coefficient Based on Kendall's Tau. Journal of the American Statistical Association 63, 1379-1389.
Theil, H. (1950). A Rank-Invariant Method of Linear and Polynomial Regression Analysis, I-III. Proc. Kon. Ned. Akad. v. Wetensch. A. 53, 386-392, 521-525, 1397-1412.
USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance. EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C.
USEPA. (2010). Errata Sheet - March 2009 Unified Guidance. EPA 530/R-09-007a, August 9, 2010. Office of Resource Conservation and Recovery, Program Information and Implementation Division. U.S. Environmental Protection Agency, Washington, D.C.
van Belle, G., and J.P. Hughes. (1984). Nonparametric Tests for Trend in Water Quality. Water Resources Research 20(1), 127-136.
Kendall's test for independence or trend is a nonparametric test. No assumptions are made about the distribution of the \(X\) and \(Y\) variables. Hirsch et al. (1982) introduced the "seasonal Kendall test" to test for trend within each season. They note that Kendall's test for trend is easy to compute, even in the presence of missing values, and can also be used with censored values.
van Belle and Hughes (1984) note that Kendall's test for trend is slightly less powerful than the test based on Spearman's rho, but it converges to normality faster. Also, Bradley (1968, p.288) shows that for the case of a linear model with normal (Gaussian) errors, the asymptotic relative efficiency of Kendall's test for trend versus the parametric test for a zero slope is 0.98.
The results of the function kendallTrendTest are similar to the
results of the built-in R function cor.test with the
argument method="kendall" except that cor.test
1) computes exact p-values when the number of pairs is less than 50 and
there are no ties, and 2) does not return a confidence interval for
the slope.
# Reproduce Example 17-6 on page 17-33 of USEPA (2009). This example
# tests for trend in sulfate concentrations (ppm) collected at various
# months between 1989 and 1996.
head(EPA.09.Ex.17.6.sulfate.df)
#> Sample.No Year Month Sampling.Date Date Sulfate.ppm
#> 1 1 89 6 89.6 1989-06-01 480
#> 2 2 89 8 89.8 1989-08-01 450
#> 3 3 90 1 90.1 1990-01-01 490
#> 4 4 90 3 90.3 1990-03-01 520
#> 5 5 90 6 90.6 1990-06-01 485
#> 6 6 90 8 90.8 1990-08-01 510
# Sample.No Year Month Sampling.Date Date Sulfate.ppm
#1 1 89 6 89.6 1989-06-01 480
#2 2 89 8 89.8 1989-08-01 450
#3 3 90 1 90.1 1990-01-01 490
#4 4 90 3 90.3 1990-03-01 520
#5 5 90 6 90.6 1990-06-01 485
#6 6 90 8 90.8 1990-08-01 510
# Plot the data
#--------------
dev.new()
with(EPA.09.Ex.17.6.sulfate.df,
plot(Sampling.Date, Sulfate.ppm, pch = 15, ylim = c(400, 900),
xlab = "Sampling Date", ylab = "Sulfate Conc (ppm)",
main = "Figure 17-6. Time Series Plot of \nSulfate Concentrations (ppm)")
)
Sulfate.fit <- lm(Sulfate.ppm ~ Sampling.Date,
data = EPA.09.Ex.17.6.sulfate.df)
abline(Sulfate.fit, lty = 2)
# Perform the Kendall test for trend
#-----------------------------------
kendallTrendTest(Sulfate.ppm ~ Sampling.Date,
data = EPA.09.Ex.17.6.sulfate.df)
#>
#> Results of Hypothesis Test
#> --------------------------
#>
#> Null Hypothesis: tau = 0
#>
#> Alternative Hypothesis: True tau is not equal to 0
#>
#> Test Name: Kendall's Test for Trend
#> (with continuity correction)
#>
#> Estimated Parameter(s): tau = 0.7667984
#> slope = 26.6666667
#> intercept = -1909.3333333
#>
#> Estimation Method: slope: Theil/Sen Estimator
#> intercept: Conover's Estimator
#>
#> Data: y = Sulfate.ppm
#> x = Sampling.Date
#>
#> Data Source: EPA.09.Ex.17.6.sulfate.df
#>
#> Sample Size: 23
#>
#> Test Statistic: z = 5.107322
#>
#> P-value: 3.267574e-07
#>
#> Confidence Interval for: slope
#>
#> Confidence Interval Method: Gilbert's Modification
#> of Theil/Sen Method
#>
#> Confidence Interval Type: two-sided
#>
#> Confidence Level: 95%
#>
#> Confidence Interval: LCL = 20.00000
#> UCL = 35.71182
#>
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: tau = 0
#
#Alternative Hypothesis: True tau is not equal to 0
#
#Test Name: Kendall's Test for Trend
# (with continuity correction)
#
#Estimated Parameter(s): tau = 0.7667984
# slope = 26.6666667
# intercept = -1909.3333333
#
#Estimation Method: slope: Theil/Sen Estimator
# intercept: Conover's Estimator
#
#Data: y = Sulfate.ppm
# x = Sampling.Date
#
#Data Source: EPA.09.Ex.17.6.sulfate.df
#
#Sample Size: 23
#
#Test Statistic: z = 5.107322
#
#P-value: 3.267574e-07
#
#Confidence Interval for: slope
#
#Confidence Interval Method: Gilbert's Modification
# of Theil/Sen Method
#
#Confidence Interval Type: two-sided
#
#Confidence Level: 95%
#
#Confidence Interval: LCL = 20.00000
# UCL = 35.71182
# Clean up
#---------
rm(Sulfate.fit)
graphics.off()