# Generate 20 observations from a lognormal mixture distribution with
# parameters mean1=1, cv1=0.5, mean2=5, cv2=1, and p.mix=0.1. Use
# predIntNpar to construct a two-sided prediction interval using the
# minimum and maximum observed values. Note that the associated confidence
# level is 90%. A larger sample size is required to obtain a larger
# confidence level (see the help file for predIntNparN).
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
dat <- rlnormMixAlt(n = 20, mean1 = 1, cv1 = 0.5,
mean2 = 5, cv2 = 1, p.mix = 0.1)
predIntNpar(dat)
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: None
#
#Data: dat
#
#Sample Size: 20
#
#Prediction Interval Method: Exact
#
#Prediction Interval Type: two-sided
#
#Confidence Level: 90.47619%
#
#Prediction Limit Rank(s): 1 20
#
#Number of Future Observations: 1
#
#Prediction Interval: LPL = 0.3647875
# UPL = 1.8173115
#----------
# Repeat the above example, but specify m=5 future observations should be
# contained in the prediction interval. Note that the confidence level is
# now only 63%.
predIntNpar(dat, m = 5)
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: None
#
#Data: dat
#
#Sample Size: 20
#
#Prediction Interval Method: Exact
#
#Prediction Interval Type: two-sided
#
#Confidence Level: 63.33333%
#
#Prediction Limit Rank(s): 1 20
#
#Number of Future Observations: 5
#
#Prediction Interval: LPL = 0.3647875
# UPL = 1.8173115
#----------
# Repeat the above example, but specify that a minimum of k=3 observations
# out of a total of m=5 future observations should be contained in the
# prediction interval. Note that the confidence level is now 98%.
predIntNpar(dat, k = 3, m = 5)
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: None
#
#Data: dat
#
#Sample Size: 20
#
#Prediction Interval Method: Exact
#
#Prediction Interval Type: two-sided
#
#Confidence Level: 98.37945%
#
#Prediction Limit Rank(s): 1 20
#
#Minimum Number of
#Future Observations
#Interval Should Contain: 3
#
#Total Number of
#Future Observations: 5
#
#Prediction Interval: LPL = 0.3647875
# UPL = 1.8173115
#==========
# Example 18-3 of USEPA (2009, p.18-19) shows how to construct
# a one-sided upper nonparametric prediction interval for the next
# 4 future observations of trichloroethylene (TCE) at a downgradient well.
# The data for this example are stored in EPA.09.Ex.18.3.TCE.df.
# There are 6 monthly observations of TCE (ppb) at 3 background wells,
# and 4 monthly observations of TCE at a compliance well.
# Look at the data
#-----------------
EPA.09.Ex.18.3.TCE.df
# Month Well Well.type TCE.ppb.orig TCE.ppb Censored
#1 1 BW-1 Background <5 5.0 TRUE
#2 2 BW-1 Background <5 5.0 TRUE
#3 3 BW-1 Background 8 8.0 FALSE
#...
#22 4 CW-4 Compliance <5 5.0 TRUE
#23 5 CW-4 Compliance 8 8.0 FALSE
#24 6 CW-4 Compliance 14 14.0 FALSE
longToWide(EPA.09.Ex.18.3.TCE.df, "TCE.ppb.orig", "Month", "Well",
paste.row.name = TRUE)
# BW-1 BW-2 BW-3 CW-4
#Month.1 <5 7 <5
#Month.2 <5 6.5 <5
#Month.3 8 <5 10.5 7.5
#Month.4 <5 6 <5 <5
#Month.5 9 12 <5 8
#Month.6 10 <5 9 14
# Construct the prediction limit based on the background well data
# using the maximum value as the upper prediction limit.
# Note that since all censored observations are censored at one
# censoring level and the censoring level is less than all of the
# uncensored observations, we can just supply the censoring level
# to predIntNpar.
#-----------------------------------------------------------------
with(EPA.09.Ex.18.3.TCE.df,
predIntNpar(TCE.ppb[Well.type == "Background"],
m = 4, pi.type = "upper", lb = 0))
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: None
#
#Data: TCE.ppb[Well.type == "Background"]
#
#Sample Size: 18
#
#Prediction Interval Method: Exact
#
#Prediction Interval Type: upper
#
#Confidence Level: 81.81818%
#
#Prediction Limit Rank(s): 18
#
#Number of Future Observations: 4
#
#Prediction Interval: LPL = 0
# UPL = 12
# Since the value of 14 ppb for Month 6 at the compliance well exceeds
# the upper prediction limit of 12, we might conclude that there is
# statistically significant evidence of an increase over background
# at CW-4. However, the confidence level associated with this
# prediction limit is about 82%, which implies a Type I error level of
# 18%. This means there is nearly a one in five chance of a false positive.
# Only additional background data and/or use of a retesting strategy
# (see predIntNparSimultaneous) would lower the false positive rate.
#==========
# Example 18-4 of USEPA (2009, p.18-19) shows how to construct
# a one-sided upper nonparametric prediction interval for the next
# median of order 3 of xylene at a downgradient well.
# The data for this example are stored in EPA.09.Ex.18.4.xylene.df.
# There are 8 monthly observations of xylene (ppb) at 3 background wells,
# and 3 montly observations of TCE at a compliance well.
# Look at the data
#-----------------
EPA.09.Ex.18.4.xylene.df
# Month Well Well.type Xylene.ppb.orig Xylene.ppb Censored
#1 1 Well.1 Background <5 5.0 TRUE
#2 2 Well.1 Background <5 5.0 TRUE
#3 3 Well.1 Background 7.5 7.5 FALSE
#...
#30 6 Well.4 Compliance <5 5.0 TRUE
#31 7 Well.4 Compliance 7.8 7.8 FALSE
#32 8 Well.4 Compliance 10.4 10.4 FALSE
longToWide(EPA.09.Ex.18.4.xylene.df, "Xylene.ppb.orig", "Month", "Well",
paste.row.name = TRUE)
# Well.1 Well.2 Well.3 Well.4
#Month.1 <5 9.2 <5
#Month.2 <5 <5 5.4
#Month.3 7.5 <5 6.7
#Month.4 <5 6.1 <5
#Month.5 <5 8 <5
#Month.6 <5 5.9 <5 <5
#Month.7 6.4 <5 <5 7.8
#Month.8 6 <5 <5 10.4
# Construct the prediction limit based on the background well data
# using the maximum value as the upper prediction limit.
# Note that since all censored observations are censored at one
# censoring level and the censoring level is less than all of the
# uncensored observations, we can just supply the censoring level
# to predIntNpar.
#
# To compute a prediction interval for a median of order 3 (i.e.,
# a median based on 3 observations), this is equivalent to
# constructing a nonparametric prediction interval that must hold
# at least 2 of the next 3 future observations.
#-----------------------------------------------------------------
with(EPA.09.Ex.18.4.xylene.df,
predIntNpar(Xylene.ppb[Well.type == "Background"],
k = 2, m = 3, pi.type = "upper", lb = 0))
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: None
#
#Data: Xylene.ppb[Well.type == "Background"]
#
#Sample Size: 24
#
#Prediction Interval Method: Exact
#
#Prediction Interval Type: upper
#
#Confidence Level: 99.1453%
#
#Prediction Limit Rank(s): 24
#
#Minimum Number of
#Future Observations
#Interval Should Contain: 2
#
#Total Number of
#Future Observations: 3
#
#Prediction Interval: LPL = 0.0
# UPL = 9.2
# The Month 8 observation at the Complance well is 10.4 ppb of Xylene,
# which is greater than the upper prediction limit of 9.2 ppb, so
# conclude there is evidence of contamination at the
# 100% - 99% = 1% Type I Error Level
#==========
# Cleanup
#--------
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