R/twoSampleLinearRankTest.R
twoSampleLinearRankTest.RdTwo-sample linear rank test to detect a difference (usually a shift) between two
distributions. The Wilcoxon Rank Sum test is a special case of
a linear rank test. The function twoSampleLinearRankTest is part of
EnvStats mainly because this help file gives the necessary background to
explain two-sample linear rank tests for censored data (see twoSampleLinearRankTestCensored).
twoSampleLinearRankTest(x, y, location.shift.null = 0, scale.shift.null = 1,
alternative = "two.sided", test = "wilcoxon", shift.type = "location")Extended methods and derivations are available in
Nonparametric estimation and two-sample tests: twoSampleLinearRankTest.
Open the vignette locally with
vignette("nonparametric-methods", package = "EnvStats").
numeric vector of values for the first sample.
Missing (NA), undefined (NaN), and infinite (Inf,
-Inf) values are allowed but will be removed.
numeric vector of values for the second sample.
Missing (NA), undefined (NaN), and infinite (Inf,
-Inf) values are allowed but will be removed.
numeric scalar indicating the hypothesized value of \(\Delta\), the location
shift between the two distributions, under the null hypothesis. The default value is
location.shift.null=0. This argument is ignored if shift.type="scale".
numeric scalar indicating the hypothesized value of \(\tau\), the scale shift
between the two distributions, under the null hypothesis. The default value is scale.shift.null=1. This argument is ignored if shift.type="location".
character string indicating the kind of alternative hypothesis. The possible values
are "two.sided" (the default), "less", and "greater". See the
DETAILS section below for more information.
character string indicating which linear rank test to use. The possible values are:
"wilcoxon" (the default), "normal.scores", "moods.median", and
"savage.scores".
character string indicating which kind of shift is being tested. The possible values
are "location" (the default) and "scale".
a list of class "htestEnvStats" containing the results of the hypothesis test.
See the help file for htestEnvStats.object for details.
Conover, W.J. (1980). Practical Nonparametric Statistics. Second Edition. John Wiley and Sons, New York, Chapter 4.
Divine, G., H.J. Norton, R. Hunt, and J. Dinemann. (2013). A Review of Analysis and Sample Size Calculation Considerations for Wilcoxon Tests. Anesthesia & Analgesia 117, 699–710.
Hettmansperger, T.P. (1984). Statistical Inference Based on Ranks. John Wiley and Sons, New York, 323pp.
Hollander, M., and D.A. Wolfe. (1999). Nonparametric Statistical Methods, Second Edition. John Wiley and Sons, New York.
Millard, S.P., and S.J. Deverel. (1988). Nonparametric Statistical Methods for Comparing Two Sites Based on Data With Multiple Nondetect Limits. Water Resources Research, 24(12), 2087–2098.
Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, FL, pp.432–435.
Prentice, R.L. (1985). Linear Rank Tests. In Kotz, S., and N.L. Johnson, eds. Encyclopedia of Statistical Science. John Wiley and Sons, New York. Volume 5, pp.51–58.
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.
Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ.
The Wilcoxon Rank Sum test, also known as the Mann-Whitney U test, is the standard nonparametric test used to test for differences between two groups (e.g., Zar, 2010; USEPA, 2009, pp.16-14 to 16-20). Other possible nonparametric tests include linear rank tests based on scores other than the ranks, including the “normal scores” test and the “Savage scores” tests. The normal scores test is actually slightly more powerful than the Wilcoxon Rank Sum test for detecting small shifts in location if the underlying distribution is normal or lognormal. In general, however, there will be little difference between these two tests.
The results of calling the function twoSampleLinearRankTest with the
argument test="wilcoxon" will match those of calling the built-in
R function wilcox.test with the arguments exact=FALSE and
correct=FALSE. In general, it is better to use the built-in function
wilcox.test for performing the Wilcoxon Rank Sum test, since this
function can compute exact (rather than approximate) p-values.
# Generate 15 observations from a normal distribution with parameters
# mean=3 and sd=1. Call these the observations from the reference group.
# Generate 10 observations from a normal distribution with parameters
# mean=3.5 and sd=1. Call these the observations from the treatment group.
# Compare the results of calling wilcox.test to those of calling
# twoSampleLinearRankTest with test="normal.scores".
# (The call to set.seed allows you to reproduce this example.)
set.seed(346)
x <- rnorm(15, mean = 3)
y <- rnorm(10, mean = 3.5)
print.htestEnvStats(wilcox.test(x, y))
#>
#> Results of Hypothesis Test
#> --------------------------
#>
#> Null Hypothesis: location shift = 0
#>
#> Alternative Hypothesis: True location shift is not equal to 0
#>
#> Test Name: Wilcoxon rank sum exact test
#>
#> Data: x and y
#>
#> Test Statistic: W = 32
#>
#> P-value: 0.0162759
#>
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: location shift = 0
#
#Alternative Hypothesis: True location shift is not equal to 0
#
#Test Name: Wilcoxon rank sum test
#
#Data: x and y
#
#Test Statistic: W = 32
#
#P-value: 0.0162759
twoSampleLinearRankTest(x, y, test = "normal.scores")
#>
#> Results of Hypothesis Test
#> --------------------------
#>
#> Null Hypothesis: Fy(t) = Fx(t)
#>
#> Alternative Hypothesis: Fy(t) != Fx(t) for at least one t
#>
#> Test Name: Two-Sample Linear Rank Test:
#> Normal Scores Test
#> Based on Normal Approximation
#>
#> Data: x = x
#> y = y
#>
#> Sample Sizes: nx = 15
#> ny = 10
#>
#> Test Statistic: z = -2.431099
#>
#> P-value: 0.01505308
#>
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: Fy(t) = Fx(t)
#
#Alternative Hypothesis: Fy(t) != Fx(t) for at least one t
#
#Test Name: Two-Sample Linear Rank Test:
# Normal Scores Test
# Based on Normal Approximation
#
#Data: x = x
# y = y
#
#Sample Sizes: nx = 15
# ny = 10
#
#Test Statistic: z = -2.431099
#
#P-value: 0.01505308
#----------
# Clean up
#---------
rm(x, y)
#==========
# Following Example 6.6 on pages 6.22-6.26 of USEPA (1994b), perform the
# Wilcoxon Rank Sum test for the TcCB data (stored in EPA.94b.tccb.df).
# There are m=47 observations from the reference area and n=77 observations
# from the cleanup unit. Then compare the results using the other available
# linear rank tests. Note that Mood's median test yields a p-value less
# than 0.10, while the other tests yield non-significant p-values.
# In this case, Mood's median test is picking up the residual contamination
# in the cleanup unit. (See the example in the help file for quantileTest.)
names(EPA.94b.tccb.df)
#> [1] "TcCB.orig" "TcCB" "Censored" "Area"
#[1] "TcCB.orig" "TcCB" "Censored" "Area"
summary(EPA.94b.tccb.df$Area)
#> Cleanup Reference
#> 77 47
# Cleanup Reference
# 77 47
with(EPA.94b.tccb.df,
twoSampleLinearRankTest(TcCB[Area=="Cleanup"], TcCB[Area=="Reference"]))
#>
#> Results of Hypothesis Test
#> --------------------------
#>
#> Null Hypothesis: Fy(t) = Fx(t)
#>
#> Alternative Hypothesis: Fy(t) != Fx(t) for at least one t
#>
#> Test Name: Two-Sample Linear Rank Test:
#> Wilcoxon Rank Sum Test
#> Based on Normal Approximation
#>
#> Data: x = TcCB[Area == "Cleanup"]
#> y = TcCB[Area == "Reference"]
#>
#> Sample Sizes: nx = 77
#> ny = 47
#>
#> Test Statistic: z = -1.171872
#>
#> P-value: 0.2412485
#>
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: Fy(t) = Fx(t)
#
#Alternative Hypothesis: Fy(t) != Fx(t) for at least one t
#
#Test Name: Two-Sample Linear Rank Test:
# Wilcoxon Rank Sum Test
# Based on Normal Approximation
#
#Data: x = TcCB[Area == "Cleanup"]
# y = TcCB[Area == "Reference"]
#
#Sample Sizes: nx = 77
# ny = 47
#
#Test Statistic: z = -1.171872
#
#P-value: 0.2412485
with(EPA.94b.tccb.df,
twoSampleLinearRankTest(TcCB[Area=="Cleanup"],
TcCB[Area=="Reference"], test="normal.scores"))$p.value
#> [1] 0.3399484
#[1] 0.3399484
with(EPA.94b.tccb.df,
twoSampleLinearRankTest(TcCB[Area=="Cleanup"],
TcCB[Area=="Reference"], test="moods.median"))$p.value
#> [1] 0.09707393
#[1] 0.09707393
with(EPA.94b.tccb.df,
twoSampleLinearRankTest(TcCB[Area=="Cleanup"],
TcCB[Area=="Reference"], test="savage.scores"))$p.value
#> [1] 0.2884351
#[1] 0.2884351