predIntNormSimultaneous: Simultaneous Prediction Interval for a Normal Distribution

Description

Estimate the mean and standard deviation of a normal distribution, and construct a simultaneous prediction interval for the next $r$ sampling “occasions”, based on one of three possible rules: $k$ -of- $m$ , California, or Modified California.

Usage

predIntNormSimultaneous(x, n.mean = 1, k = 1, m = 2, r = 1, rule = "k.of.m", 
    delta.over.sigma = 0, pi.type = "upper", conf.level = 0.95, 
    K.tol = .Machine$double.eps^0.5)

Arguments

a numeric vector of observations, or an object resulting from a call to an estimating function that assumes a normal (Gaussian) distribution (e.g., enorm, eqnorm, enormCensored, etc.). If x is a numeric vector, missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are allowed but will be removed.

n.mean

positive integer specifying the sample size associated with the future averages. The default value is n.mean=1 (i.e., individual observations). Note that all future averages must be based on the same sample size.

for the $k$ -of- $m$ rule (rule="k.of.m"), a positive integer specifying the minimum number of observations (or averages) out of $m$ observations (or averages) (all obtained on one future sampling “occassion”) the prediction interval should contain with confidence level conf.level. The default value is k=1. This argument is ignored when the argument rule is not equal to "k.of.m".

positive integer specifying the maximum number of future observations (or averages) on one future sampling “occasion”. The default value is m=2, except when rule="Modified.CA", in which case this argument is ignored and m is automatically set equal to 4.

positive integer specifying the number of future sampling “occasions”. The default value is r=1.

rule

character string specifying which rule to use. The possible values are "k.of.m" ( $k$ -of- $m$ rule; the default), "CA" (California rule), and "Modified.CA" (modified California rule). See the DETAILS section below for more information.

delta.over.sigma

numeric scalar indicating the ratio $Δ / σ$ . The quantity $Δ$ (delta) denotes the difference between the mean of the population that was sampled to construct the prediction interval, and the mean of the population that will be sampled to produce the future observations. The quantity $σ$ (sigma) denotes the population standard deviation for both populations. See the DETAILS section below for more information. The default value is delta.over.sigma=0.

pi.type

character string indicating what kind of prediction interval to compute. The possible values are pi.type="upper" (the default), and pi.type="lower".

conf.level

a scalar between 0 and 1 indicating the confidence level of the prediction interval. The default value is conf.level=0.95.

K.tol

numeric scalar indicating the tolerance to use in the nonlinear search algorithm to compute $K$ . The default value is K.tol=.Machine$double.eps^(1/2). For many applications, the value of $K$ needs to be known only to the second decimal place, in which case setting K.tol=1e-4 will speed up computation a bit.

Value

If x is a numeric vector, predIntNormSimultaneous returns a list of class "estimate" containing the estimated parameters, the prediction interval, and other information. See the help file for estimate.object for details.

If x is the result of calling an estimation function, predIntNormSimultaneous returns a list whose class is the same as x. The list contains the same components as x, as well as a component called interval containing the prediction interval information. If x already has a component called interval, this component is replaced with the prediction interval information.

Details

What is a Simultaneous Prediction Interval? A prediction interval for some population is an interval on the real line constructed so that it will contain $k$ future observations from that population with some specified probability $(1 - α) 100 %$ , where $0 < α < 1$ and $k$ is some pre-specified positive integer. The quantity $(1 - α) 100 %$ is called the confidence coefficient or confidence level associated with the prediction interval. The function predIntNorm computes a standard prediction interval based on a sample from a normal distribution.

The function predIntNormSimultaneous computes a simultaneous prediction interval that will contain a certain number of future observations with probability $(1 - α) 100 %$ for each of $r$ future sampling “occasions”, where $r$ is some pre-specified positive integer. The quantity $r$ may refer to $r$ distinct future sampling occasions in time, or it may for example refer to sampling at $r$ distinct locations on one future sampling occasion, assuming that the population standard deviation is the same at all of the $r$ distinct locations.

The function predIntNormSimultaneous computes a simultaneous prediction interval based on one of three possible rules:

For the $k$ -of- $m$ rule (rule="k.of.m"), at least $k$ of the next $m$ future observations will fall in the prediction interval with probability $(1 - α) 100 %$ on each of the $r$ future sampling occasions. If obserations are being taken sequentially, for a particular sampling occasion, up to $m$ observations may be taken, but once $k$ of the observations fall within the prediction interval, sampling can stop. Note: When $k = m$ and $r = 1$ , the results of predIntNormSimultaneous are equivalent to the results of predIntNorm.
For the California rule (rule="CA"), with probability $(1 - α) 100 %$ , for each of the $r$ future sampling occasions, either the first observation will fall in the prediction interval, or else all of the next $m - 1$ observations will fall in the prediction interval. That is, if the first observation falls in the prediction interval then sampling can stop. Otherwise, $m - 1$ more observations must be taken.
For the Modified California rule (rule="Modified.CA"), with probability $(1 - α) 100 %$ , for each of the $r$ future sampling occasions, either the first observation will fall in the prediction interval, or else at least 2 out of the next 3 observations will fall in the prediction interval. That is, if the first observation falls in the prediction interval then sampling can stop. Otherwise, up to 3 more observations must be taken.

Simultaneous prediction intervals can be extended to using averages (means) in place of single observations (USEPA, 2009, Chapter 19). That is, you can create a simultaneous prediction interval that will contain a specified number of averages (based on which rule you choose) on each of $r$ future sampling occassions, where each each average is based on $w$ individual observations. For the function predIntNormSimultaneous, the argument n.mean corresponds to $w$ .

The Form of a Prediction Interval Let $\underset{―}{x} = x_{1}, x_{2}, \dots, x_{n}$ denote a vector of $n$ observations from a normal distribution with parameters mean= $μ$ and sd= $σ$ . Also, let $w$ denote the sample size associated with the future averages (i.e., n.mean= $w$ ). When $w = 1$ , each average is really just a single observation, so in the rest of this help file the term “averages” will replace the phrase “observations or averages”.

For a normal distribution, the form of a two-sided $(1 - α) 100 %$ prediction interval is: $[\bar{x} - K s, \bar{x} + K s] (1)$ where $\bar{x}$ denotes the sample mean: $\bar{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i} (2)$ $s$ denotes the sample standard deviation: $s^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (x_{i} - \bar{x})^{2} (3)$ and $K$ denotes a constant that depends on the sample size $n$ , the confidence level, the number of future sampling occassions $r$ , and the sample size associated with the future averages, $w$ . Do not confuse the constant $K$ (uppercase K) with the number of future averages $k$ (lowercase k) in the $k$ -of- $m$ rule. The symbol $K$ is used here to be consistent with the notation used for tolerance intervals (see tolIntNorm).

Similarly, the form of a one-sided lower prediction interval is: $[\bar{x} - K s, \infty] (4)$ and the form of a one-sided upper prediction interval is: $[- \infty, \bar{x} + K s] (5)$

Note: For simultaneous prediction intervals, only lower (pi.type="lower") and upper (pi.type="upper") prediction intervals are available.

The derivation of the constant $K$ is explained in the help file for predIntNormSimultaneousK.

Prediction Intervals are Random Intervals A prediction interval is a random interval; that is, the lower and/or upper bounds are random variables computed based on sample statistics in the baseline sample. Prior to taking one specific baseline sample, the probability that the prediction interval will perform according to the rule chosen is $(1 - α) 100 %$ . Once a specific baseline sample is taken and the prediction interval based on that sample is computed, the probability that that prediction interval will perform according to the rule chosen is not necessarily $(1 - α) 100 %$ , but it should be close. See the help file for predIntNorm for more information.

References

Barclay's California Code of Regulations. (1991). Title 22, Section 66264.97 [concerning hazardous waste facilities] and Title 23, Section 2550.7(e)(8) [concerning solid waste facilities]. Barclay's Law Publishers, San Francisco, CA.

Davis, C.B. (1998a). Ground-Water Statistics \& Regulations: Principles, Progress and Problems. Second Edition. Environmetrics \& Statistics Limited, Henderson, NV.

Davis, C.B. (1998b). Personal Communication, September 3, 1998.

Davis, C.B., and R.J. McNichols. (1987). One-sided Intervals for at Least $p$ of $m$ Observations from a Normal Population on Each of $r$ Future Occasions. Technometrics 29, 359--370.

Fertig, K.W., and N.R. Mann. (1977). One-Sided Prediction Intervals for at Least $p$ Out of $m$ Future Observations From a Normal Population. Technometrics 19, 167--177.

Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.

Hahn, G.J. (1969). Factors for Calculating Two-Sided Prediction Intervals for Samples from a Normal Distribution. Journal of the American Statistical Association 64(327), 878-898.

Hahn, G.J. (1970a). Additional Factors for Calculating Prediction Intervals for Samples from a Normal Distribution. Journal of the American Statistical Association 65(332), 1668-1676.

Hahn, G.J. (1970b). Statistical Intervals for a Normal Population, Part I: Tables, Examples and Applications. Journal of Quality Technology 2(3), 115-125.

Hahn, G.J. (1970c). Statistical Intervals for a Normal Population, Part II: Formulas, Assumptions, Some Derivations. Journal of Quality Technology 2(4), 195-206.

Hahn, G.J., and W.Q. Meeker. (1991). Statistical Intervals: A Guide for Practitioners. John Wiley and Sons, New York.

Hahn, G., and W. Nelson. (1973). A Survey of Prediction Intervals and Their Applications. Journal of Quality Technology 5, 178-188.

Hall, I.J., and R.R. Prairie. (1973). One-Sided Prediction Intervals to Contain at Least $m$ Out of $k$ Future Observations. Technometrics 15, 897--914.

Millard, S.P. (1987). Environmental Monitoring, Statistics, and the Law: Room for Improvement (with Comment). The American Statistician 41(4), 249--259.

Millard, S.P., and Neerchal, N.K. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, Florida.

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.

Examples

Run this code

# NOT RUN {
  # Generate 8 observations from a normal distribution with parameters 
  # mean=10 and sd=2, then use predIntNormSimultaneous to estimate the 
  # mean and standard deviation of the true distribution and construct an 
  # upper 95% prediction interval to contain at least 1 out of the next 
  # 3 observations. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(479) 
  dat <- rnorm(8, mean = 10, sd = 2) 

  predIntNormSimultaneous(dat, k = 1, m = 3) 

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Normal
  #
  #Estimated Parameter(s):          mean = 10.269773
  #                                 sd   =  2.210246
  #
  #Estimation Method:               mvue
  #
  #Data:                            dat
  #
  #Sample Size:                     8
  #
  #Prediction Interval Method:      exact
  #
  #Prediction Interval Type:        upper
  #
  #Confidence Level:                95%
  #
  #Minimum Number of
  #Future Observations
  #Interval Should Contain:         1
  #
  #Total Number of
  #Future Observations:             3
  #
  #Prediction Interval:             LPL =    -Inf
  #                                 UPL = 11.4021

  #----------

  # Repeat the above example, but do it in two steps.  First create a list called 
  # est.list containing information about the estimated parameters, then create the 
  # prediction interval.

  est.list <- enorm(dat) 
  est.list 

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Normal
  #
  #Estimated Parameter(s):          mean = 10.269773
  #                                 sd   =  2.210246
  #
  #Estimation Method:               mvue
  #
  #Data:                            dat
  #
  #Sample Size:                     8


  predIntNormSimultaneous(est.list, k = 1, m = 3) 

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Normal
  #
  #Estimated Parameter(s):          mean = 10.269773
  #                                 sd   =  2.210246
  #
  #Estimation Method:               mvue
  #
  #Data:                            dat
  #
  #Sample Size:                     8
  #
  #Prediction Interval Method:      exact
  #
  #Prediction Interval Type:        upper
  #
  #Confidence Level:                95%
  #
  #Minimum Number of
  #Future Observations
  #Interval Should Contain:         1
  #
  #Total Number of
  #Future Observations:             3
  #
  #Prediction Interval:             LPL =    -Inf
  #                                 UPL = 11.4021 

  #----------

  # Compare the 95% 1-of-3 upper prediction interval to the California and 
  # Modified California prediction intervals.  Note that the upper prediction 
  # bound for the Modified California rule is between the bound for the 
  # 1-of-3 rule bound and the bound for the California rule. 

  predIntNormSimultaneous(dat, k = 1, m = 3)$interval$limits["UPL"] 
  #    UPL 
  #11.4021 
 
  predIntNormSimultaneous(dat, m = 3, rule = "CA")$interval$limits["UPL"]  
  #     UPL 
  #13.03717 

  predIntNormSimultaneous(dat, rule = "Modified.CA")$interval$limits["UPL"]  
  #     UPL 
  #12.12201

  #----------

  # Show how the upper bound on an upper 95% simultaneous prediction limit increases 
  # as the number of future sampling occasions r increases.  Here, we'll use the 
  # 1-of-3 rule.

  predIntNormSimultaneous(dat, k = 1, m = 3)$interval$limits["UPL"]
  #    UPL 
  #11.4021

  predIntNormSimultaneous(dat, k = 1, m = 3, r = 10)$interval$limits["UPL"]
  #     UPL 
  #13.28234

  #----------

  # Compare the upper simultaneous prediction limit for the 1-of-3 rule 
  # based on individual observations versus based on means of order 4.

  predIntNormSimultaneous(dat, k = 1, m = 3)$interval$limits["UPL"]
  #    UPL 
  #11.4021

  predIntNormSimultaneous(dat, n.mean = 4, k = 1, 
    m = 3)$interval$limits["UPL"]
  #     UPL 
  #11.26157

  #==========

  # Example 19-1 of USEPA (2009, p. 19-17) shows how to compute an 
  # upper simultaneous prediction limit for the 1-of-3 rule for 
  # r = 2 future sampling occasions.  The data for this example are 
  # stored in EPA.09.Ex.19.1.sulfate.df.

  # We will pool data from 4 background wells that were sampled on 
  # a number of different occasions, giving us a sample size of 
  # n = 25 to use to construct the prediction limit.

  # There are 50 compliance wells and we will monitor 10 different 
  # constituents at each well at each of the r=2 future sampling 
  # occasions.  To determine the confidence level we require for 
  # the simultaneous prediction interval, USEPA (2009) recommends 
  # setting the individual Type I Error level at each well to 
 
  # 1 - (1 - SWFPR)^(1 / (Number of Constituents * Number of Wells))
  
  # which translates to setting the confidence limit to 

  # (1 - SWFPR)^(1 / (Number of Constituents * Number of Wells))

  # where SWFPR = site-wide false positive rate.  For this example, we 
  # will set SWFPR = 0.1.  Thus, the confidence level is given by:

  nc <- 10
  nw <- 50
  SWFPR <- 0.1
  conf.level <- (1 - SWFPR)^(1 / (nc * nw))

  conf.level
  #[1] 0.9997893

  #----------

  # Look at the data:

  names(EPA.09.Ex.19.1.sulfate.df)
  #[1] "Well"                 "Month"                "Day"                 
  #[4] "Year"                 "Date"                 "Sulfate.mg.per.l"    
  #[7] "log.Sulfate.mg.per.l"

  EPA.09.Ex.19.1.sulfate.df[, 
    c("Well", "Date", "Sulfate.mg.per.l", "log.Sulfate.mg.per.l")]

  #    Well       Date Sulfate.mg.per.l log.Sulfate.mg.per.l
  #1  GW-01 1999-07-08             63.0             4.143135
  #2  GW-01 1999-09-12             51.0             3.931826
  #3  GW-01 1999-10-16             60.0             4.094345
  #4  GW-01 1999-11-02             86.0             4.454347
  #5  GW-04 1999-07-09            104.0             4.644391
  #6  GW-04 1999-09-14            102.0             4.624973
  #7  GW-04 1999-10-12             84.0             4.430817
  #8  GW-04 1999-11-15             72.0             4.276666
  #9  GW-08 1997-10-12             31.0             3.433987
  #10 GW-08 1997-11-16             84.0             4.430817
  #11 GW-08 1998-01-28             65.0             4.174387
  #12 GW-08 1999-04-20             41.0             3.713572
  #13 GW-08 2002-06-04             51.8             3.947390
  #14 GW-08 2002-09-16             57.5             4.051785
  #15 GW-08 2002-12-02             66.8             4.201703
  #16 GW-08 2003-03-24             87.1             4.467057
  #17 GW-09 1997-10-16             59.0             4.077537
  #18 GW-09 1998-01-28             85.0             4.442651
  #19 GW-09 1998-04-12             75.0             4.317488
  #20 GW-09 1998-07-12             99.0             4.595120
  #21 GW-09 2000-01-30             75.8             4.328098
  #22 GW-09 2000-04-24             82.5             4.412798
  #23 GW-09 2000-10-24             85.5             4.448516
  #24 GW-09 2002-12-01            188.0             5.236442
  #25 GW-09 2003-03-24            150.0             5.010635

  # Construct the upper simultaneous prediction limit for the 
  # 1-of-3 plan based on the log-transformed sulfate data

  log.Sulfate <- EPA.09.Ex.19.1.sulfate.df$log.Sulfate.mg.per.l

  pred.int.list.log <- 
    predIntNormSimultaneous(x = log.Sulfate, k = 1, m = 3, r = 2, 
      rule = "k.of.m", pi.type = "upper", conf.level = conf.level)

  pred.int.list.log

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Normal
  #
  #Estimated Parameter(s):          mean = 4.3156194
  #                                 sd   = 0.3756697
  #
  #Estimation Method:               mvue
  #
  #Data:                            log.Sulfate
  #
  #Sample Size:                     25
  #
  #Prediction Interval Method:      exact
  #
  #Prediction Interval Type:        upper
  #
  #Confidence Level:                99.97893%
  #
  #Minimum Number of
  #Future Observations
  #Interval Should Contain
  #(per Sampling Occasion):         1
  #
  #Total Number of
  #Future Observations
  #(per Sampling Occasion):         3
  #
  #Number of Future
  #Sampling Occasions:              2
  #
  #Prediction Interval:             LPL =     -Inf
  #                                 UPL = 5.072355


  # Now exponentiate the prediction interval to get the limit on 
  # the original scale

  exp(pred.int.list.log$interval$limits["UPL"])
  #     UPL 
  #159.5497 

  #==========

  # Cleanup
  #--------

  rm(dat, est.list, nc, nw, SWFPR, conf.level, log.Sulfate, 
    pred.int.list.log)
# }

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