Nothing
inst/scripts/EPA.2009/Chapter16Examples.R
In EnvStats: Package for Environmental Statistics, Including US EPA Guidance
#####
# File: Script - EPA09, Chapter 16 Examples.R
#
# Purpose: Reproduce Examples in Chapter 16 of the EPA Guidance Document
#
# USEPA. (2009). "Statistical Analysis of Ground Water
# Monitoring Data at RCRA Facilities, Unified Guidance."
# EPA 530-R-09-007.
# Office of Resource Conservation and Recovery,
# Program Information and Implementation Division.
# March 2009.
#
# Errata are listed in:
# USEPA. (2010). "Errata Sheet - March 2009 Unified Guidance."
# EPA 530/R-09-007a.
# Office of Resource Conservation and Recovery,
# Program Information and Implementation Division.
# August 9, 2010.
#
# Author: Steven P. Millard
#
# Last
# Updated: September 15, 2012
#####
#NOTE: Unless otherwise noted, differences between what is shown in the
# EPA Guidance Document and results shown here are due to the
# EPA Guidance Document rounding intermediate results prior to
# the final result.
#######################################################################################
library(EnvStats)
# Figure 16-1, p. 16-2
#---------------------
df.vec <- c(1, 3, 7, 25)
lty.vec <- c(1, 5, 4, 6)
col.vec <- 1:4
n <- length(df.vec)
windows()
pdfPlot(dist = "t", param.list = list(df = df.vec[1]),
curve.fill = F, xaxt = "n",
left.tail.cutoff = pt(-5, df = df.vec[1]),
right.tail.cutoff = pt(-5, df = df.vec[1]),
xlab = "t-value", xlim = c(-5, 6),
ylim = c(0, 0.4), main = "")
axis(1, at = seq(-5, 5, by = 2.5))
for(i in 2:n) {
pdfPlot(dist = "t", param.list = list(df = df.vec[i]),
left.tail.cutoff = pt(-5, df = df.vec[i]),
right.tail.cutoff = pt(-5, df = df.vec[i]),
add = T, pdf.col = col.vec[i], pdf.lty = lty.vec[i])
}
legend(3, 0.4, rev(paste(format(df.vec), "df")), lty = rev(lty.vec),
col = rev(col.vec), bty = "n", lwd = 2)
mtext(expression(paste("Figure 16-1. Student's ",
italic(t), "-Distribtuion", sep = "")), cex = 1.25,
line = 2.5)
mtext("for Varying Degrees of Freedom", cex = 1.25,
line = 1)
rm(df.vec, lty.vec, col.vec, n, i)
#######################################################################################
# Example 16-1, pp. 16-6 to 16-7
#-------------------------------
Sulfate <- EPA.09.Ex.16.1.sulfate.df$Sulfate.ppm
Well.type <- EPA.09.Ex.16.1.sulfate.df$Well.type
fit <- lm(Sulfate ~ Well.type,
data = EPA.09.Ex.16.1.sulfate.df)
Residuals <- fit$residuals
# The function summaryStats will compute summary statistics and
# also the p-value for testing the equality of means between groups.
summaryStats(Sulfate.ppm ~ Well.type, data = EPA.09.Ex.16.1.sulfate.df,
digits = 2, p.value = TRUE, alternative = "greater",
stats.in.rows = TRUE)
#
# Background Downgradient Combined
#N 8 6 14
#Mean 536.25 608.33 567.14
#SD 26.69 18.35 43.4
#Median 540 605 565
#Min 490 590 490
#Max 570 630 630
#p.value.between.greater 0.000053
#95%.LCL.between 49.39
#95%.UCL.between Inf
#NA's 0 2 2
#N.Total 8 8 16
# The function stripChart will display the raw data by group,
# show means and SDs for each group, and compute the p-value for
# testing the equality of means between groups.
windows()
stripChart(Sulfate.ppm ~ Well.type, data = EPA.09.Ex.16.1.sulfate.df,
ylab = "Sulfate (ppm)", location.scale.digits = 2,
p.value = TRUE, alternative = "greater")
# Probability plot on residuals
windows()
qqPlot(Residuals, add.line = T,
ylab = "Sulfate residuals (ppm)",
main = "Figure 16-2. Probability Plot of \nCombined Sulfate Residuals")
# The R function t.test performs Student's t-test.
# By default, var.equal is set to FALSE, so you have to set it
# to TRUE for this example.
t.test(Sulfate[Well.type == "Downgradient"],
Sulfate[Well.type == "Background"],
alternative = "greater", var.equal = TRUE)
#
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: difference in means = 0
#
#Alternative Hypothesis: True difference in means is greater than 0
#
#Test Name: Two Sample t-test
#
#Estimated Parameter(s): mean of x = 608.3333
# mean of y = 536.2500
#
#Data: Sulfate[Well.type == "Downgradient"] and Sulfate[Well.type == "Background"]
#
#Test Statistic: t = 5.660985
#
#P-value: 5.271532e-05
#
#95% Confidence Interval: LCL = 49.38883
# UCL = Inf
rm(Sulfate, Well.type, fit, Residuals)
#####################################################################################################
# Example 16-2, pp. 16-9 to 16-10
#--------------------------------
longToWide(EPA.09.Ex.16.2.benzene.df, "Benzene.ppb", "Month", "Well.type")
# Background Downgradient
#Jan 0.5 0.5
#Feb 0.8 0.7
#Mar 1.6 4.6
#Apr 1.8 2.0
#May 1.1 16.7
#Jun 16.1 12.5
#Jul 1.6 26.3
#Aug 0.6 186.0
#NOTE: set var.equal=FALSE in the call to summaryStats
summaryStats(Benzene.ppb ~ Well.type, data = EPA.09.Ex.16.2.benzene.df,
digits = 2, p.value = TRUE, test.arg.list = list(var.equal = FALSE),
group.p.value.type = "between", alternative = "greater",
stats.in.rows = TRUE)
# Background Downgradient Combined
#N 8 8 16
#Mean 3.01 31.16 17.09
#SD 5.31 63.22 45.71
#Median 1.35 8.55 1.7
#Min 0.5 0.5 0.5
#Max 16.1 186 186
#Welch.p.value.between.greater 0.12
#95%.LCL.between -14.26
#95%.UCL.between Inf
Well.type <- EPA.09.Ex.16.2.benzene.df$Well.type
Benzene <- EPA.09.Ex.16.2.benzene.df$Benzene.ppb
t.test(Benzene[Well.type == "Downgradient"],
Benzene[Well.type == "Background"],
alternative = "greater", var.equal = FALSE)
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: difference in means = 0
#
#Alternative Hypothesis: True difference in means is greater than 0
#
#Test Name: Welch Two Sample t-test
#
#Estimated Parameter(s): mean of x = 31.1625
# mean of y = 3.0125
#
#Data: Benzene[Well.type == "Downgradient"] and Benzene[Well.type == "Background"]
#
#Test Statistic: t = 1.254938
#
#Test Statistic Parameter: df = 7.09878
#
#P-value: 0.1246182
#
#95% Confidence Interval: LCL = -14.25922
# UCL = Inf
set.seed(25)
windows()
stripChart(Benzene.ppb ~ Well.type, data = EPA.09.Ex.16.2.benzene.df,
method = "jitter", ylim = c(-25, 200),
ylab = "Benzene (ppb)",
p.value = TRUE, alternative = "greater",
test.arg.list = list(var.equal = FALSE))
rm(Well.type, Benzene)
#####################################################################################################
# Example 16-3, pp. 16-11 to 16-14
#---------------------------------
new.EPA.df <- EPA.09.Ex.16.2.benzene.df
new.EPA.df$Benzene.ppb <- log(new.EPA.df$Benzene.ppb)
names(new.EPA.df)[3] <- "log.Benzene.ppb"
table.df <- data.frame(
longToWide(EPA.09.Ex.16.2.benzene.df, "Benzene.ppb", "Month", "Well.type"),
longToWide(new.EPA.df, "log.Benzene.ppb", "Month", "Well.type")
)
names(table.df)[3:4] <- c("log.Background", "log.Downgradient")
round(table.df, 3)
# Background Downgradient log.Background log.Downgradient
#Jan 0.5 0.5 -0.693 -0.693
#Feb 0.8 0.7 -0.223 -0.357
#Mar 1.6 4.6 0.470 1.526
#Apr 1.8 2.0 0.588 0.693
#May 1.1 16.7 0.095 2.815
#Jun 16.1 12.5 2.779 2.526
#Jul 1.6 26.3 0.470 3.270
#Aug 0.6 186.0 -0.511 5.226
summaryStats(Benzene.ppb ~ Well.type, data = EPA.09.Ex.16.2.benzene.df)
# N Mean SD Median Min Max
#Background 8 3.0125 5.3108 1.35 0.5 16.1
#Downgradient 8 31.1625 63.2229 8.55 0.5 186.0
summaryStats(log(Benzene.ppb) ~ Well.type, data = EPA.09.Ex.16.2.benzene.df)
# N Mean SD Median Min Max
#Background 8 0.3719 1.0825 0.2827 -0.6931 2.7788
#Downgradient 8 1.8757 1.9847 2.0259 -0.6931 5.2257
#Step 1 - Test raw data for normality
#------------------------------------
gofGroupTest(Benzene.ppb ~ Well.type, data = EPA.09.Ex.16.2.benzene.df)
#Results of Group Goodness-of-Fit Test
-------------------------------------
#
#Test Method: Wilk-Shapiro GOF (Normal Scores)
#
#Hypothesized Distribution: Normal
#
#Data: Benzene.ppb
#
#Grouping Variable: Well.type
#
#Data Source: EPA.09.Ex.16.2.benzene.df
#
#Number of Groups: 2
#
#Sample Sizes: Background = 8
# Downgradient = 8
#
#Test Statistic: z (G) = -5.801397
#
#P-values for
#Individual Tests: Background = 1.192816e-05
# Downgradient = 3.459439e-05
#
#P-value for
#Group Test: 3.288234e-09
#
#Alternative Hypothesis: At least one group
# does not come from a
# Normal Distribution.
#Step 2 - Compute summary statistics and
# test log-transformed data for normality
#------------------------------------------------
summaryStats(log(Benzene.ppb) ~ Well.type,
data = EPA.09.Ex.16.2.benzene.df,
digits = 4, p.value = TRUE,
test.arg.list = list(var.equal = FALSE),
alternative = "greater", stats.in.rows = TRUE)
# Background Downgradient Combined
#N 8 8 16
#Mean 0.3719 1.8757 1.1238
#SD 1.0825 1.9847 1.7287
#Median 0.2827 2.0259 0.5289
#Min -0.6931 -0.6931 -0.6931
#Max 2.7788 5.2257 5.2257
#Welch.p.value.between.greater 0.044
#95%.LCL.between 0.0663
#95%.UCL.between Inf
gofGroupTest(log(Benzene.ppb) ~ Well.type, data = EPA.09.Ex.16.2.benzene.df)
#Results of Group Goodness-of-Fit Test
#-------------------------------------
#
#Test Method: Wilk-Shapiro GOF (Normal Scores)
#
#Hypothesized Distribution: Normal
#
#Data: log(Benzene.ppb)
#
#Grouping Variable: Well.type
#
#Data Source: EPA.09.Ex.16.2.benzene.df
#
#Number of Groups: 2
#
#Sample Sizes: Background = 8
# Downgradient = 8
#
#Test Statistic: z (G) = -0.470135
#
#P-values for
#Individual Tests: Background = 0.04471158
# Downgradient = 0.84933304
#
#P-value for
#Group Test: 0.3191293
#
#Alternative Hypothesis: At least one group
# does not come from a
# Normal Distribution.
# Steps 3-5 - Compute Welch t-test
#---------------------------------
Month <- EPA.09.Ex.16.2.benzene.df$Month
Well.type <- EPA.09.Ex.16.2.benzene.df$Well.type
Benzene <- EPA.09.Ex.16.2.benzene.df$Benzene.ppb
log.Benzene <- log(Benzene)
t.test(log.Benzene[Well.type == "Downgradient"],
log.Benzene[Well.type == "Background"],
alternative = "greater", var.equal = FALSE)
#
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: difference in means = 0
#
#Alternative Hypothesis: True difference in means is greater than 0
#
#Test Name: Welch Two Sample t-test
#
#Estimated Parameter(s): mean of x = 1.8757293
# mean of y = 0.3718509
#
#Data: log.Benzene[Well.type == "Downgradient"] and log.Benzene[Well.type == "Background"]
#
#Test Statistic: t = 1.881485
#
#P-value: 0.04352518
#
#95% Confidence Interval: LCL = 0.06631003
# UCL = Inf
windows()
stripChart(log(Benzene.ppb) ~ Well.type, data = EPA.09.Ex.16.2.benzene.df,
ylab = "log [ Benzene (ppb) ]",
p.value = TRUE, alternative = "greater",
test.arg.list = list(var.equal = FALSE))
#Figure 16-3
#-----------
windows()
plot(as.numeric(Month), Benzene, type = "n", xaxt = "n",
xlab = "Month", ylab = "Benzene (ppb)",
main = "Figure 16-3. Benzene Time Series Plot")
axis(1, at = sort(unique(as.numeric(Month))), labels = levels(Month))
points(as.numeric(Month)[Well.type == "Background"],
Benzene[Well.type == "Background"],
type = "b", pch = 3, lty = 2)
points(as.numeric(Month)[Well.type == "Downgradient"],
Benzene[Well.type == "Downgradient"],
type = "b", pch = 15)
legend(1, 150, c("Downgradient", "Background"),
pch = c(15, 3), lty = 1:2)
rm(new.EPA.df, table.df, Month, Well.type, Benzene, log.Benzene)
#######################################################################################################
# Example 16-4, pp. 16-18 to 16-20
#---------------------------------
# NOTE: From the help file for wilcox.test:
# "The literature is not unanimous about the definitions of the
# Wilcoxon rank sum and Mann-Whitney tests. The two most common
# definitions correspond to the sum of the ranks of the first sample
# with the minimum value subtracted or not: R subtracts and S-PLUS does not,
# giving a value which is larger by m(m+1)/2 for a first sample of size m.
# (It seems Wilcoxon's original paper used the unadjusted sum of the ranks
# but subsequent tables subtracted the minimum.) "
# Step 1
#-------
longToWide(EPA.09.Ex.16.4.copper.df, "Copper.ppb", "Month", "Well", paste.row.name = TRUE)
# Well.1 Well.2 Well.3
#Month.1 4.2 5.2 9.4
#Month.2 5.8 6.4 10.1
#Month.3 11.3 11.3 14.5
#Month.4 7.0 11.5 16.1
#Month.5 7.0 10.1 21.5
#Month.6 8.2 9.7 17.6
new.EPA.df <- EPA.09.Ex.16.4.copper.df
new.EPA.df$Copper.ppb <- rank(new.EPA.df$Copper.ppb)
names(new.EPA.df)[4] <- "rank.Copper.ppb"
rank.df <- longToWide(new.EPA.df, "rank.Copper.ppb", "Month", "Well",
paste.row.name = TRUE)
rank.df
# Well.1 Well.2 Well.3
#Month.1 1.0 2.0 8.0
#Month.2 3.0 4.0 10.5
#Month.3 12.5 12.5 15.0
#Month.4 5.5 14.0 16.0
#Month.5 5.5 10.5 18.0
#Month.6 7.0 9.0 17.0
# Step 2
#-------
colSums(rank.df)["Well.3"]
#Well.3
# 84.5
# Step 3
#-------
Cu <- EPA.09.Ex.16.4.copper.df$Copper.ppb
Well.type <- EPA.09.Ex.16.4.copper.df$Well.type
wilcox.test(Cu[Well.type == "Compliance"], Cu[Well.type == "Background"],
alternative = "greater")
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: location shift = 0
#
#Alternative Hypothesis: True location shift is greater than 0
#
#Test Name: Wilcoxon rank sum test with continuity correction
#
#Data: Cu[Well.type == "Compliance"] and Cu[Well.type == "Background"]
#
#Test Statistic: W = 63.5
#
#P-value: 0.005659303
#
#Warning message:
#In wilcox.test.default(Cu[Well.type == "Compliance"], Cu[Well.type == :
# cannot compute exact p-value with ties
rm(new.EPA.df, rank.df, Cu, Well.type)
#######################################################################################################
# Example 16-5, pp. 16-22 to 16-23
#---------------------------------
# NOTE: The EnvStats for R function
# twoSampleLinearRankTestCensored()
# contains the Tarone-Ware test, along with several other
# similar tests.
PCE <- EPA.09.Ex.16.5.PCE.df$PCE.ppb
Well.type <- EPA.09.Ex.16.5.PCE.df$Well.type
Censored <- EPA.09.Ex.16.5.PCE.df$Censored
# Tarone-Ware test with hypergeometric variance
#----------------------------------------------
twoSampleLinearRankTestCensored(
x = PCE[Well.type == "Compliance"],
x.censored = Censored[Well.type == "Compliance"],
y = PCE[Well.type == "Background"],
y.censored = Censored[Well.type == "Background"],
test = "tarone.ware",
alternative = "greater",
variance = "hypergeometric")
#Results of Hypothesis Test
#Based on Censored Data
#--------------------------
#
#Null Hypothesis: Fy(t) = Fx(t)
#
#Alternative Hypothesis: Fy(t) > Fx(t) for at least one t
#
#Test Name: Two-Sample Linear Rank Test:
# Tarone-Ware Test
# with Hypergeometric Variance
#
#Censoring Side: left
#
#Data: x = PCE[Well.type == "Compliance"]
# y = PCE[Well.type == "Background"]
#
#Censoring Variable: x = Censored[Well.type == "Compliance"]
# y = Censored[Well.type == "Background"]
#
#Sample Sizes: nx = 8
# ny = 6
#
#Percent Censored: x = 12.5%
# y = 50.0%
#
#Test Statistics: nu = 8.458912
# var.nu = 20.912407
# z = 1.849748
#
#P-value: 0.03217495
# Logrank test with hypergeometric variance
#------------------------------------------
twoSampleLinearRankTestCensored(
x = PCE[Well.type == "Compliance"],
x.censored = Censored[Well.type == "Compliance"],
y = PCE[Well.type == "Background"],
y.censored = Censored[Well.type == "Background"],
test = "logrank",
alternative = "greater",
variance = "hypergeometric")
#Results of Hypothesis Test
#Based on Censored Data
#--------------------------
#
#Null Hypothesis: Fy(t) = Fx(t)
#
#Alternative Hypothesis: Fy(t) > Fx(t) for at least one t
#
#Test Name: Two-Sample Linear Rank Test:
# Logrank Test
# with Hypergeometric Variance
#
#Censoring Side: left
#
#Data: x = PCE[Well.type == "Compliance"]
# y = PCE[Well.type == "Background"]
#
#Censoring Variable: x = Censored[Well.type == "Compliance"]
# y = Censored[Well.type == "Background"]
#
#Sample Sizes: nx = 8
# ny = 6
#
#Percent Censored: x = 12.5%
# y = 50.0%
#
#Test Statistics: nu = 2.830406
# var.nu = 2.176723
# z = 1.918435
#
#P-value: 0.02752793
# Peto-Peto test with hypergeometric variance
#--------------------------------------------
twoSampleLinearRankTestCensored(
x = PCE[Well.type == "Compliance"],
x.censored = Censored[Well.type == "Compliance"],
y = PCE[Well.type == "Background"],
y.censored = Censored[Well.type == "Background"],
test = "peto.peto",
alternative = "greater",
variance = "hypergeometric")
#Results of Hypothesis Test
#Based on Censored Data
#--------------------------
#
#Null Hypothesis: Fy(t) = Fx(t)
#
#Alternative Hypothesis: Fy(t) > Fx(t) for at least one t
#
#Test Name: Two-Sample Linear Rank Test:
# Peto-Peto Test Using
# Prentice Survival Estimator
# with Hypergeometric Variance
#
#Censoring Side: left
#
#Data: x = PCE[Well.type == "Compliance"]
# y = PCE[Well.type == "Background"]
#
#Censoring Variable: x = Censored[Well.type == "Compliance"]
# y = Censored[Well.type == "Background"]
#
#Sample Sizes: nx = 8
# ny = 6
#
#Percent Censored: x = 12.5%
# y = 50.0%
#
#Test Statistics: nu = 1.888889
# var.nu = 1.028642
# z = 1.862406
#
#P-value: 0.03127296
# Gehan test with hypergeometric variance
#----------------------------------------
twoSampleLinearRankTestCensored(
x = PCE[Well.type == "Compliance"],
x.censored = Censored[Well.type == "Compliance"],
y = PCE[Well.type == "Background"],
y.censored = Censored[Well.type == "Background"],
test = "gehan",
alternative = "greater",
variance = "hypergeometric")
#Results of Hypothesis Test
#Based on Censored Data
#--------------------------
#
#Null Hypothesis: Fy(t) = Fx(t)
#
#Alternative Hypothesis: Fy(t) > Fx(t) for at least one t
#
#Test Name: Two-Sample Linear Rank Test:
# Gehan's Test
# with Hypergeometric Variance
#
#Censoring Side: left
#
#Data: x = PCE[Well.type == "Compliance"]
# y = PCE[Well.type == "Background"]
#
#Censoring Variable: x = Censored[Well.type == "Compliance"]
# y = Censored[Well.type == "Background"]
#
#Sample Sizes: nx = 8
# ny = 6
#
#Percent Censored: x = 12.5%
# y = 50.0%
#
#Test Statistics: nu = 27.000000
# var.nu = 227.000000
# z = 1.792053
#
#P-value: 0.03656224
rm(PCE, Well.type, Censored)
############################################################################
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#####
# File: Script - EPA09, Chapter 16 Examples.R
#
# Purpose: Reproduce Examples in Chapter 16 of the EPA Guidance Document
#
# USEPA. (2009). "Statistical Analysis of Ground Water
# Monitoring Data at RCRA Facilities, Unified Guidance."
# EPA 530-R-09-007.
# Office of Resource Conservation and Recovery,
# Program Information and Implementation Division.
# March 2009.
#
# Errata are listed in:
# USEPA. (2010). "Errata Sheet - March 2009 Unified Guidance."
# EPA 530/R-09-007a.
# Office of Resource Conservation and Recovery,
# Program Information and Implementation Division.
# August 9, 2010.
#
# Author: Steven P. Millard
#
# Last
# Updated: September 15, 2012
#####
#NOTE: Unless otherwise noted, differences between what is shown in the
# EPA Guidance Document and results shown here are due to the
# EPA Guidance Document rounding intermediate results prior to
# the final result.
#######################################################################################
library(EnvStats)
# Figure 16-1, p. 16-2
#---------------------
df.vec <- c(1, 3, 7, 25)
lty.vec <- c(1, 5, 4, 6)
col.vec <- 1:4
n <- length(df.vec)
windows()
pdfPlot(dist = "t", param.list = list(df = df.vec[1]),
curve.fill = F, xaxt = "n",
left.tail.cutoff = pt(-5, df = df.vec[1]),
right.tail.cutoff = pt(-5, df = df.vec[1]),
xlab = "t-value", xlim = c(-5, 6),
ylim = c(0, 0.4), main = "")
axis(1, at = seq(-5, 5, by = 2.5))
for(i in 2:n) {
pdfPlot(dist = "t", param.list = list(df = df.vec[i]),
left.tail.cutoff = pt(-5, df = df.vec[i]),
right.tail.cutoff = pt(-5, df = df.vec[i]),
add = T, pdf.col = col.vec[i], pdf.lty = lty.vec[i])
}
legend(3, 0.4, rev(paste(format(df.vec), "df")), lty = rev(lty.vec),
col = rev(col.vec), bty = "n", lwd = 2)
mtext(expression(paste("Figure 16-1. Student's ",
italic(t), "-Distribtuion", sep = "")), cex = 1.25,
line = 2.5)
mtext("for Varying Degrees of Freedom", cex = 1.25,
line = 1)
rm(df.vec, lty.vec, col.vec, n, i)
#######################################################################################
# Example 16-1, pp. 16-6 to 16-7
#-------------------------------
Sulfate <- EPA.09.Ex.16.1.sulfate.df$Sulfate.ppm
Well.type <- EPA.09.Ex.16.1.sulfate.df$Well.type
fit <- lm(Sulfate ~ Well.type,
data = EPA.09.Ex.16.1.sulfate.df)
Residuals <- fit$residuals
# The function summaryStats will compute summary statistics and
# also the p-value for testing the equality of means between groups.
summaryStats(Sulfate.ppm ~ Well.type, data = EPA.09.Ex.16.1.sulfate.df,
digits = 2, p.value = TRUE, alternative = "greater",
stats.in.rows = TRUE)
#
# Background Downgradient Combined
#N 8 6 14
#Mean 536.25 608.33 567.14
#SD 26.69 18.35 43.4
#Median 540 605 565
#Min 490 590 490
#Max 570 630 630
#p.value.between.greater 0.000053
#95%.LCL.between 49.39
#95%.UCL.between Inf
#NA's 0 2 2
#N.Total 8 8 16
# The function stripChart will display the raw data by group,
# show means and SDs for each group, and compute the p-value for
# testing the equality of means between groups.
windows()
stripChart(Sulfate.ppm ~ Well.type, data = EPA.09.Ex.16.1.sulfate.df,
ylab = "Sulfate (ppm)", location.scale.digits = 2,
p.value = TRUE, alternative = "greater")
# Probability plot on residuals
windows()
qqPlot(Residuals, add.line = T,
ylab = "Sulfate residuals (ppm)",
main = "Figure 16-2. Probability Plot of \nCombined Sulfate Residuals")
# The R function t.test performs Student's t-test.
# By default, var.equal is set to FALSE, so you have to set it
# to TRUE for this example.
t.test(Sulfate[Well.type == "Downgradient"],
Sulfate[Well.type == "Background"],
alternative = "greater", var.equal = TRUE)
#
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: difference in means = 0
#
#Alternative Hypothesis: True difference in means is greater than 0
#
#Test Name: Two Sample t-test
#
#Estimated Parameter(s): mean of x = 608.3333
# mean of y = 536.2500
#
#Data: Sulfate[Well.type == "Downgradient"] and Sulfate[Well.type == "Background"]
#
#Test Statistic: t = 5.660985
#
#P-value: 5.271532e-05
#
#95% Confidence Interval: LCL = 49.38883
# UCL = Inf
rm(Sulfate, Well.type, fit, Residuals)
#####################################################################################################
# Example 16-2, pp. 16-9 to 16-10
#--------------------------------
longToWide(EPA.09.Ex.16.2.benzene.df, "Benzene.ppb", "Month", "Well.type")
# Background Downgradient
#Jan 0.5 0.5
#Feb 0.8 0.7
#Mar 1.6 4.6
#Apr 1.8 2.0
#May 1.1 16.7
#Jun 16.1 12.5
#Jul 1.6 26.3
#Aug 0.6 186.0
#NOTE: set var.equal=FALSE in the call to summaryStats
summaryStats(Benzene.ppb ~ Well.type, data = EPA.09.Ex.16.2.benzene.df,
digits = 2, p.value = TRUE, test.arg.list = list(var.equal = FALSE),
group.p.value.type = "between", alternative = "greater",
stats.in.rows = TRUE)
# Background Downgradient Combined
#N 8 8 16
#Mean 3.01 31.16 17.09
#SD 5.31 63.22 45.71
#Median 1.35 8.55 1.7
#Min 0.5 0.5 0.5
#Max 16.1 186 186
#Welch.p.value.between.greater 0.12
#95%.LCL.between -14.26
#95%.UCL.between Inf
Well.type <- EPA.09.Ex.16.2.benzene.df$Well.type
Benzene <- EPA.09.Ex.16.2.benzene.df$Benzene.ppb
t.test(Benzene[Well.type == "Downgradient"],
Benzene[Well.type == "Background"],
alternative = "greater", var.equal = FALSE)
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: difference in means = 0
#
#Alternative Hypothesis: True difference in means is greater than 0
#
#Test Name: Welch Two Sample t-test
#
#Estimated Parameter(s): mean of x = 31.1625
# mean of y = 3.0125
#
#Data: Benzene[Well.type == "Downgradient"] and Benzene[Well.type == "Background"]
#
#Test Statistic: t = 1.254938
#
#Test Statistic Parameter: df = 7.09878
#
#P-value: 0.1246182
#
#95% Confidence Interval: LCL = -14.25922
# UCL = Inf
set.seed(25)
windows()
stripChart(Benzene.ppb ~ Well.type, data = EPA.09.Ex.16.2.benzene.df,
method = "jitter", ylim = c(-25, 200),
ylab = "Benzene (ppb)",
p.value = TRUE, alternative = "greater",
test.arg.list = list(var.equal = FALSE))
rm(Well.type, Benzene)
#####################################################################################################
# Example 16-3, pp. 16-11 to 16-14
#---------------------------------
new.EPA.df <- EPA.09.Ex.16.2.benzene.df
new.EPA.df$Benzene.ppb <- log(new.EPA.df$Benzene.ppb)
names(new.EPA.df)[3] <- "log.Benzene.ppb"
table.df <- data.frame(
longToWide(EPA.09.Ex.16.2.benzene.df, "Benzene.ppb", "Month", "Well.type"),
longToWide(new.EPA.df, "log.Benzene.ppb", "Month", "Well.type")
)
names(table.df)[3:4] <- c("log.Background", "log.Downgradient")
round(table.df, 3)
# Background Downgradient log.Background log.Downgradient
#Jan 0.5 0.5 -0.693 -0.693
#Feb 0.8 0.7 -0.223 -0.357
#Mar 1.6 4.6 0.470 1.526
#Apr 1.8 2.0 0.588 0.693
#May 1.1 16.7 0.095 2.815
#Jun 16.1 12.5 2.779 2.526
#Jul 1.6 26.3 0.470 3.270
#Aug 0.6 186.0 -0.511 5.226
summaryStats(Benzene.ppb ~ Well.type, data = EPA.09.Ex.16.2.benzene.df)
# N Mean SD Median Min Max
#Background 8 3.0125 5.3108 1.35 0.5 16.1
#Downgradient 8 31.1625 63.2229 8.55 0.5 186.0
summaryStats(log(Benzene.ppb) ~ Well.type, data = EPA.09.Ex.16.2.benzene.df)
# N Mean SD Median Min Max
#Background 8 0.3719 1.0825 0.2827 -0.6931 2.7788
#Downgradient 8 1.8757 1.9847 2.0259 -0.6931 5.2257
#Step 1 - Test raw data for normality
#------------------------------------
gofGroupTest(Benzene.ppb ~ Well.type, data = EPA.09.Ex.16.2.benzene.df)
#Results of Group Goodness-of-Fit Test
-------------------------------------
#
#Test Method: Wilk-Shapiro GOF (Normal Scores)
#
#Hypothesized Distribution: Normal
#
#Data: Benzene.ppb
#
#Grouping Variable: Well.type
#
#Data Source: EPA.09.Ex.16.2.benzene.df
#
#Number of Groups: 2
#
#Sample Sizes: Background = 8
# Downgradient = 8
#
#Test Statistic: z (G) = -5.801397
#
#P-values for
#Individual Tests: Background = 1.192816e-05
# Downgradient = 3.459439e-05
#
#P-value for
#Group Test: 3.288234e-09
#
#Alternative Hypothesis: At least one group
# does not come from a
# Normal Distribution.
#Step 2 - Compute summary statistics and
# test log-transformed data for normality
#------------------------------------------------
summaryStats(log(Benzene.ppb) ~ Well.type,
data = EPA.09.Ex.16.2.benzene.df,
digits = 4, p.value = TRUE,
test.arg.list = list(var.equal = FALSE),
alternative = "greater", stats.in.rows = TRUE)
# Background Downgradient Combined
#N 8 8 16
#Mean 0.3719 1.8757 1.1238
#SD 1.0825 1.9847 1.7287
#Median 0.2827 2.0259 0.5289
#Min -0.6931 -0.6931 -0.6931
#Max 2.7788 5.2257 5.2257
#Welch.p.value.between.greater 0.044
#95%.LCL.between 0.0663
#95%.UCL.between Inf
gofGroupTest(log(Benzene.ppb) ~ Well.type, data = EPA.09.Ex.16.2.benzene.df)
#Results of Group Goodness-of-Fit Test
#-------------------------------------
#
#Test Method: Wilk-Shapiro GOF (Normal Scores)
#
#Hypothesized Distribution: Normal
#
#Data: log(Benzene.ppb)
#
#Grouping Variable: Well.type
#
#Data Source: EPA.09.Ex.16.2.benzene.df
#
#Number of Groups: 2
#
#Sample Sizes: Background = 8
# Downgradient = 8
#
#Test Statistic: z (G) = -0.470135
#
#P-values for
#Individual Tests: Background = 0.04471158
# Downgradient = 0.84933304
#
#P-value for
#Group Test: 0.3191293
#
#Alternative Hypothesis: At least one group
# does not come from a
# Normal Distribution.
# Steps 3-5 - Compute Welch t-test
#---------------------------------
Month <- EPA.09.Ex.16.2.benzene.df$Month
Well.type <- EPA.09.Ex.16.2.benzene.df$Well.type
Benzene <- EPA.09.Ex.16.2.benzene.df$Benzene.ppb
log.Benzene <- log(Benzene)
t.test(log.Benzene[Well.type == "Downgradient"],
log.Benzene[Well.type == "Background"],
alternative = "greater", var.equal = FALSE)
#
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: difference in means = 0
#
#Alternative Hypothesis: True difference in means is greater than 0
#
#Test Name: Welch Two Sample t-test
#
#Estimated Parameter(s): mean of x = 1.8757293
# mean of y = 0.3718509
#
#Data: log.Benzene[Well.type == "Downgradient"] and log.Benzene[Well.type == "Background"]
#
#Test Statistic: t = 1.881485
#
#P-value: 0.04352518
#
#95% Confidence Interval: LCL = 0.06631003
# UCL = Inf
windows()
stripChart(log(Benzene.ppb) ~ Well.type, data = EPA.09.Ex.16.2.benzene.df,
ylab = "log [ Benzene (ppb) ]",
p.value = TRUE, alternative = "greater",
test.arg.list = list(var.equal = FALSE))
#Figure 16-3
#-----------
windows()
plot(as.numeric(Month), Benzene, type = "n", xaxt = "n",
xlab = "Month", ylab = "Benzene (ppb)",
main = "Figure 16-3. Benzene Time Series Plot")
axis(1, at = sort(unique(as.numeric(Month))), labels = levels(Month))
points(as.numeric(Month)[Well.type == "Background"],
Benzene[Well.type == "Background"],
type = "b", pch = 3, lty = 2)
points(as.numeric(Month)[Well.type == "Downgradient"],
Benzene[Well.type == "Downgradient"],
type = "b", pch = 15)
legend(1, 150, c("Downgradient", "Background"),
pch = c(15, 3), lty = 1:2)
rm(new.EPA.df, table.df, Month, Well.type, Benzene, log.Benzene)
#######################################################################################################
# Example 16-4, pp. 16-18 to 16-20
#---------------------------------
# NOTE: From the help file for wilcox.test:
# "The literature is not unanimous about the definitions of the
# Wilcoxon rank sum and Mann-Whitney tests. The two most common
# definitions correspond to the sum of the ranks of the first sample
# with the minimum value subtracted or not: R subtracts and S-PLUS does not,
# giving a value which is larger by m(m+1)/2 for a first sample of size m.
# (It seems Wilcoxon's original paper used the unadjusted sum of the ranks
# but subsequent tables subtracted the minimum.) "
# Step 1
#-------
longToWide(EPA.09.Ex.16.4.copper.df, "Copper.ppb", "Month", "Well", paste.row.name = TRUE)
# Well.1 Well.2 Well.3
#Month.1 4.2 5.2 9.4
#Month.2 5.8 6.4 10.1
#Month.3 11.3 11.3 14.5
#Month.4 7.0 11.5 16.1
#Month.5 7.0 10.1 21.5
#Month.6 8.2 9.7 17.6
new.EPA.df <- EPA.09.Ex.16.4.copper.df
new.EPA.df$Copper.ppb <- rank(new.EPA.df$Copper.ppb)
names(new.EPA.df)[4] <- "rank.Copper.ppb"
rank.df <- longToWide(new.EPA.df, "rank.Copper.ppb", "Month", "Well",
paste.row.name = TRUE)
rank.df
# Well.1 Well.2 Well.3
#Month.1 1.0 2.0 8.0
#Month.2 3.0 4.0 10.5
#Month.3 12.5 12.5 15.0
#Month.4 5.5 14.0 16.0
#Month.5 5.5 10.5 18.0
#Month.6 7.0 9.0 17.0
# Step 2
#-------
colSums(rank.df)["Well.3"]
#Well.3
# 84.5
# Step 3
#-------
Cu <- EPA.09.Ex.16.4.copper.df$Copper.ppb
Well.type <- EPA.09.Ex.16.4.copper.df$Well.type
wilcox.test(Cu[Well.type == "Compliance"], Cu[Well.type == "Background"],
alternative = "greater")
#Results of Hypothesis Test
#--------------------------
#
#Null Hypothesis: location shift = 0
#
#Alternative Hypothesis: True location shift is greater than 0
#
#Test Name: Wilcoxon rank sum test with continuity correction
#
#Data: Cu[Well.type == "Compliance"] and Cu[Well.type == "Background"]
#
#Test Statistic: W = 63.5
#
#P-value: 0.005659303
#
#Warning message:
#In wilcox.test.default(Cu[Well.type == "Compliance"], Cu[Well.type == :
# cannot compute exact p-value with ties
rm(new.EPA.df, rank.df, Cu, Well.type)
#######################################################################################################
# Example 16-5, pp. 16-22 to 16-23
#---------------------------------
# NOTE: The EnvStats for R function
# twoSampleLinearRankTestCensored()
# contains the Tarone-Ware test, along with several other
# similar tests.
PCE <- EPA.09.Ex.16.5.PCE.df$PCE.ppb
Well.type <- EPA.09.Ex.16.5.PCE.df$Well.type
Censored <- EPA.09.Ex.16.5.PCE.df$Censored
# Tarone-Ware test with hypergeometric variance
#----------------------------------------------
twoSampleLinearRankTestCensored(
x = PCE[Well.type == "Compliance"],
x.censored = Censored[Well.type == "Compliance"],
y = PCE[Well.type == "Background"],
y.censored = Censored[Well.type == "Background"],
test = "tarone.ware",
alternative = "greater",
variance = "hypergeometric")
#Results of Hypothesis Test
#Based on Censored Data
#--------------------------
#
#Null Hypothesis: Fy(t) = Fx(t)
#
#Alternative Hypothesis: Fy(t) > Fx(t) for at least one t
#
#Test Name: Two-Sample Linear Rank Test:
# Tarone-Ware Test
# with Hypergeometric Variance
#
#Censoring Side: left
#
#Data: x = PCE[Well.type == "Compliance"]
# y = PCE[Well.type == "Background"]
#
#Censoring Variable: x = Censored[Well.type == "Compliance"]
# y = Censored[Well.type == "Background"]
#
#Sample Sizes: nx = 8
# ny = 6
#
#Percent Censored: x = 12.5%
# y = 50.0%
#
#Test Statistics: nu = 8.458912
# var.nu = 20.912407
# z = 1.849748
#
#P-value: 0.03217495
# Logrank test with hypergeometric variance
#------------------------------------------
twoSampleLinearRankTestCensored(
x = PCE[Well.type == "Compliance"],
x.censored = Censored[Well.type == "Compliance"],
y = PCE[Well.type == "Background"],
y.censored = Censored[Well.type == "Background"],
test = "logrank",
alternative = "greater",
variance = "hypergeometric")
#Results of Hypothesis Test
#Based on Censored Data
#--------------------------
#
#Null Hypothesis: Fy(t) = Fx(t)
#
#Alternative Hypothesis: Fy(t) > Fx(t) for at least one t
#
#Test Name: Two-Sample Linear Rank Test:
# Logrank Test
# with Hypergeometric Variance
#
#Censoring Side: left
#
#Data: x = PCE[Well.type == "Compliance"]
# y = PCE[Well.type == "Background"]
#
#Censoring Variable: x = Censored[Well.type == "Compliance"]
# y = Censored[Well.type == "Background"]
#
#Sample Sizes: nx = 8
# ny = 6
#
#Percent Censored: x = 12.5%
# y = 50.0%
#
#Test Statistics: nu = 2.830406
# var.nu = 2.176723
# z = 1.918435
#
#P-value: 0.02752793
# Peto-Peto test with hypergeometric variance
#--------------------------------------------
twoSampleLinearRankTestCensored(
x = PCE[Well.type == "Compliance"],
x.censored = Censored[Well.type == "Compliance"],
y = PCE[Well.type == "Background"],
y.censored = Censored[Well.type == "Background"],
test = "peto.peto",
alternative = "greater",
variance = "hypergeometric")
#Results of Hypothesis Test
#Based on Censored Data
#--------------------------
#
#Null Hypothesis: Fy(t) = Fx(t)
#
#Alternative Hypothesis: Fy(t) > Fx(t) for at least one t
#
#Test Name: Two-Sample Linear Rank Test:
# Peto-Peto Test Using
# Prentice Survival Estimator
# with Hypergeometric Variance
#
#Censoring Side: left
#
#Data: x = PCE[Well.type == "Compliance"]
# y = PCE[Well.type == "Background"]
#
#Censoring Variable: x = Censored[Well.type == "Compliance"]
# y = Censored[Well.type == "Background"]
#
#Sample Sizes: nx = 8
# ny = 6
#
#Percent Censored: x = 12.5%
# y = 50.0%
#
#Test Statistics: nu = 1.888889
# var.nu = 1.028642
# z = 1.862406
#
#P-value: 0.03127296
# Gehan test with hypergeometric variance
#----------------------------------------
twoSampleLinearRankTestCensored(
x = PCE[Well.type == "Compliance"],
x.censored = Censored[Well.type == "Compliance"],
y = PCE[Well.type == "Background"],
y.censored = Censored[Well.type == "Background"],
test = "gehan",
alternative = "greater",
variance = "hypergeometric")
#Results of Hypothesis Test
#Based on Censored Data
#--------------------------
#
#Null Hypothesis: Fy(t) = Fx(t)
#
#Alternative Hypothesis: Fy(t) > Fx(t) for at least one t
#
#Test Name: Two-Sample Linear Rank Test:
# Gehan's Test
# with Hypergeometric Variance
#
#Censoring Side: left
#
#Data: x = PCE[Well.type == "Compliance"]
# y = PCE[Well.type == "Background"]
#
#Censoring Variable: x = Censored[Well.type == "Compliance"]
# y = Censored[Well.type == "Background"]
#
#Sample Sizes: nx = 8
# ny = 6
#
#Percent Censored: x = 12.5%
# y = 50.0%
#
#Test Statistics: nu = 27.000000
# var.nu = 227.000000
# z = 1.792053
#
#P-value: 0.03656224
rm(PCE, Well.type, Censored)
############################################################################
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