predIntNparN {EnvStats}  R Documentation 
Sample Size for a Nonparametric Prediction Interval for a Continuous Distribution
Description
Compute the sample size necessary for a nonparametric prediction interval to
contain at least k
out of the next m
future observations with
probability (1\alpha)100\%
for a continuous distribution.
Usage
predIntNparN(k = m, m = 1, lpl.rank = ifelse(pi.type == "upper", 0, 1),
n.plus.one.minus.upl.rank = ifelse(pi.type == "lower", 0, 1),
pi.type = "two.sided", conf.level = 0.95, n.max = 5000, maxiter = 1000)
Arguments
k 
vector of positive integers specifying the minimum number of future
observations out of 
m 
vector of positive integers specifying the number of future observations.
The default value is 
lpl.rank 
vector of positive integers indicating the rank of the order statistic to use for
the lower bound of the prediction interval. If 
n.plus.one.minus.upl.rank 
vector of positive integers related to the rank of the order statistic to use for
the upper bound of the prediction interval. A value of

pi.type 
character string indicating what kind of prediction interval to compute.
The possible values are 
conf.level 
numeric vector of values between 0 and 1 indicating the confidence level
associated with the prediction interval. The default value is

n.max 
positive integer greater than 1 indicating the maximum possible sample size. The
default value is 
maxiter 
positive integer indicating the maximum number of iterations to use in the

Details
If the arguments k
, m
, lpl.rank
, and
n.plus.one.minus.upl.rank
are not all the same length, they are replicated
to be the same length as the length of the longest argument.
The function predIntNparN
initially computes the required sample size n
by solving Equation (11) or (12) in the help file for predIntNpar
for
n
, depending on the value of the argument pi.type
. If k < m
,
lpl.rank > 1
(twosided and lower prediction intervals only), or
n.plus.one.minus.upl.rank > 1
(twosided and upper prediction intervals only),
then this initial value of n
is used as the upper bound in a binary search
based on Equation (8) in the help file for predIntNpar
and is
implemented via the R function uniroot
with the argument
tolerance
set to 1
.
Value
vector of positive integers indicating the required sample size(s) for the specified nonparametric prediction interval(s).
Note
See the help file for predIntNpar
.
Author(s)
Steven P. Millard (EnvStats@ProbStatInfo.com)
References
See the help file for predIntNpar
.
See Also
predIntNpar
, predIntNparConfLevel
,
plotPredIntNparDesign
.
Examples
# Look at how the required sample size for a nonparametric prediction interval
# increases with increasing confidence level:
seq(0.5, 0.9, by = 0.1)
#[1] 0.5 0.6 0.7 0.8 0.9
predIntNparN(conf.level = seq(0.5, 0.9, by = 0.1))
#[1] 3 4 6 9 19
#
# Look at how the required sample size for a nonparametric prediction interval
# increases with number of future observations (m):
1:5
#[1] 1 2 3 4 5
predIntNparN(m = 1:5)
#[1] 39 78 116 155 193
#
# Look at how the required sample size for a nonparametric prediction interval
# increases with minimum number of observations that must be contained within
# the interval (k):
predIntNparN(k = 1:5, m = 5)
#[1] 4 7 13 30 193
#
# Look at how the required sample size for a nonparametric prediction interval
# increases with the rank of the lower prediction limit:
predIntNparN(lpl.rank = 1:5)
#[1] 39 59 79 100 119
#==========
# Example 183 of USEPA (2009, p.1819) shows how to construct
# a onesided 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 BW1 Background <5 5.0 TRUE
#2 2 BW1 Background <5 5.0 TRUE
#3 3 BW1 Background 8 8.0 FALSE
#...
#22 4 CW4 Compliance <5 5.0 TRUE
#23 5 CW4 Compliance 8 8.0 FALSE
#24 6 CW4 Compliance 14 14.0 FALSE
longToWide(EPA.09.Ex.18.3.TCE.df, "TCE.ppb.orig", "Month", "Well",
paste.row.name = TRUE)
# BW1 BW2 BW3 CW4
#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
# If we construct the prediction limit based on the background well
# data using the maximum value as the upper prediction limit,
# the associated confidence level is only 82%.
#
predIntNparConfLevel(n = 18, m = 4, pi.type = "upper")
#[1] 0.8181818
# We would have to collect an additional 18 observations to achieve a
# confidence level of at least 90%:
predIntNparN(m = 4, pi.type = "upper", conf.level = 0.9)
#[1] 36
predIntNparConfLevel(n = 36, m = 4, pi.type = "upper")
#[1] 0.9