Sample Size for a Nonparametric Prediction Interval for a Continuous Distribution
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-α)100\% for a continuous distribution.
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)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
|
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 (two-sided and lower prediction intervals only), or n.plus.one.minus.upl.rank > 1 (two-sided 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.
vector of positive integers indicating the required sample size(s) for the specified nonparametric prediction interval(s).
See the help file for predIntNpar.
Steven P. Millard (EnvStats@ProbStatInfo.com)
See the help file for predIntNpar.
# 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 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
# 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] 0.9Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.