# Statistical Prediction Intervals for the Evaluation of Ground-Water Quality

by Robert D. Gibbons`1

ABSTRACT Factors for a norrnal distribution are given such that one may be 9996 conficlent that the two-sided prediction interval

## ac± rs or the one-sided prediction interval

## ac+ rs will contain all of the k future values, where x` and s are the sample mean and standard dcviation obtained from ii previous values. In the context of ground-water monitoring, the future samples may represent new monitoring values at cach of k downgradient wells, andthe nprevious values might be the historical monitoring results for one or morc upgradient wells. The 'I'ables provided in this paper allow the computation of one-sided and two-sided 9996 prediction intervals for previous sample sizes of n = 4 to 100 and future samples of k= 1 to 100. Modification of these intervals for log-normally distributed data is also presented.



## 'I. lNTRODUCTION 

# The principal objective of ground-water monitoring at hazardous waste disposal facilities is to empiricaIly determine if on-site chemical waste is migrating off-site in ground water. The principal method of ground water monitoring involves a conlparison of specific chemical constituents of ground water upgradient of the site with similarly collected and analyzed sarnples downgradieritof



## aUniversity of Iltinois, Biometric l.aboratory, Illinois State Psychiatric Institute, 1601 W. 'I'aylor St., Chicago, Illinois 60612. Received June 1986, revised April 1987, accepted April 1987. Discussion open until January 1, 1988.
Vol. 25, No. 4—GRaUND WATEIt---July-August 1987 

# the site. '1'he null hypothesis of intzrest in such monitoring programs is that of no difference in chemical composition of ground water upgradient and downgradient of the facility.

# 'I'he logicin such a comparison is that presumably, the upgradient wells monitor ground water that is unaffected bythe waste disposaI facility, whereas the downgradient wclls arc in the path of ground water that flows underneath the site, and therefore should provide an early warning system for any potential off-site migration of hazardous waste. 'I'he validity of this monitoring procedure

# is notthe focus of this paper. I'he development of a statistically rigorous hypothesis testing strategy is the focus of this paper.If ground water were in the form of a wellmixed underground river or lake th.tt spanned the entire arca of the site, the most direct approach to testing for upgradient versus downgradient differences would be to obtain estimates of ineansand variances for individual compounds for all upgradient wells and similar estimates for alI downgradient wells. lf the variances of these two distributions were reasonabiy similar, the measurements independent (i.e., not replicate measurements obtained from the same sample), and the compounds in question liad approximately normal (i.e., Gaussian) distributions, one could readily compare the two distributions using Student's t-statistic for each indicator paramet:er separately, or if a sufficient number of monitoring wells were available, all of the indicator parameters could be

455

# compared simultaneously using Hotelling's T2

statistic

# [i.e.,the multivariate extension of the t-statistic (Anderson, 1958)], Alternatively, if the assumption of homogeneity of variance could not be supported, one could use an unequal variance t-statistic such as Cochran's approximation to the 13ehrens-I-'ishcr t-statistic (5nedecor and Cochran, 1980). 

# I,'rom a statistical perspective it is unfortunate that the ground-water lake or river example falls considerably short of reality. Ground water oceurs in geologically discrete aquifers which generally move extremely slowly

# (i.e.,inches per year) and in rare cases can exhibit a plume-like projection, such that only localized areas of the downgra.dient geology niay exhibit contarnination. In light ofthis, pooling information obtained from downgradient wells, soine of which may be contaminated and some of which may be perfectly normal, does not scem sensiblc. If wc cannot estimate a mean and variance for a collection of downgradicnt well measurcnients on a particular occasion, we cannot compute a t-statistic. The reason for this is that a t-statistic is appropriate for the Comparison of two mean values; however, if each dUwngradient well is examincd individually, its mcasurcment is not themean of anything, it is sirnply a single new observation which rnust in turn hc cornparcd to a historical upgradient samplc of ineasurements in order to assess its significance. Stated statistically, we desire an interval or range of values that will have high probability of containing all new monitoring observations. if the monitoring program involves multiple wclls, wc must construct an interval that will include all of the k futurc obscrvations, where k is the number of wells that arc to bc evaluated on the next monitoring event. Intervals of this typearc called "prediction intervals" and necd to be distinguished both from a confidence interval on an unknown distribution paramcter (e.g., a confidence interval of a mean value), and from a tolerance interval which contains the values- of a specified proportion of the population with a certain level of confidence. I lahn (1.970) illustrates this distinction by providing an cxample of the typical astronaut's problem. "An astronaut who has bcen assigned to a limited number of spacc flights is not very interested in what will happen on the average in 

# the population of all space flights, of which his happens to be a random sample, or even in what will happen in 99% of the population of such space flights. lIis main interest is in thc worst that Inay happen in the onc, or three, or fivc• flights in which he will be personally involved." In this example, what wi1l happen on the average in the population of all space flights represcnts a confidencc intcrval, what wiIl happcn in 99% of the population of space flights is a tolerance interval, and wllat will happen in the next one, or three, or five space flights rcprescnts a prediction interval.

2. STATISTICAL DEVELOPME11tT 

# ()ur objective is to definc an interval

ac ±

# r(k, n;1-- a) s


# that will contain the values of all k ftiturc mcasurcments, where x and s arc the sample mean and standard deviation calculated froni the values


# xl,..., xnof n previous nieasurements; that is,

n 

# x = Z xi/n i=I

n 

# 52= Z(x; - ac)2/(n

~

### -1 

# The original n observations and the additional k samples are assumed to be independent random samplcs

# (i.e.,not analytical replicates) frorn thc.• same normal distribution. For the case of a single new observation, Wilks (1941) showed that when sampling from a N(p, (J2) with ,u and 02unknown, the coverage c of the interval

z+ rs ] 

## c= f 

# ~exp {-2(x -

# μ)

# 2 } ay
z

# -rs 2Q 

# will on the average be 1- cx, if and only if

r +1/n ttn_1;W21


# whcretln-1;W21is the 100(1 - a)% point of "Studcnt's" t-distribution with n-- 1 degrees of freedom. C:hcw(1968)

# extendcd this result to thc casc of k future observations using an approximate conservative method for obtaining prediction intcrvals bascd on thc Honferroni incquality (Miller, 1966). "I'his approximation leads to the 100 (1 - a)% two-sided prediction limit :

x±Nrl+ 1/n t(n—l;a/2k] s 

# and thc 100(1 - a)% one-sided upper prediction limit: x+ 1+ 1/n t(n-1;a/kl s flahn (1969 and 1970) compared results using

456

# this method and selected exact values obtained

using numerical quadrature and found the approximation to be satisfactory for practical purposes. Coniparison of values in Tables 1-3, which provide values of r(k,

# n;.99) for

# n=4 to 100 and k= 1 to 100 for one-sided intervals, and '1'ables 4-6 for the corresponding two-sided intervals reveal that on the average, the bias is less than 1% with a maximum deviation of 7% for n= 4 and k= 1. In fact, for n> 4 the bias is never greater than 1%►

# regardless of k. The small bias observed for n= 4 is always an underestilna.te; therefore, the approxilnation is conservative for very small n which should probably not be used in practice. For example, the exact one-tailed result for n= 15 and k= 4 is r(4, 15 ;.99) = 3.42, and the approxinlate value given in 'I'able 1 is 3.43. Usirng this example in a practical application, we wOuld conlpute the 99% upper prediction limit as x+ 3.43s, where in thisexainple, Y and s would be computed on 15 background (upgradient) nleasurenlents, and each ofthe four new monitoring (downgradient) rneasuremcnts wotlld siniply be compared to this Iimit value.

Table 1. Factors for Obtaining One-Sided 99% Prediction Limits for k 

## Additiona! Samples Given a Background Sample of Size n

Previvus

n 1 2 3 4 6 

### Number of new measurements (k) 6 7 8 9 10 11 12 13 14 15 4 4.70 6.81 6.51 7.06 7.49 7.86 8.17 8.45 8.71 8.94 9.15 9_36 9.53 9.70 9.86 6 3.96 4.77 6.28 5.66 5.96 6.22 6.44 6.63 6.81 6.97 7.11 7.24 7.37 7.41+ 7.59 6 3.67 4.23 4.63 4.93 5.17 6.37 6.54 5.69 6.83 5.96 6.06 6.16 6.26 6.3S 6.43 7 3.32 3.89 4.24 4,49 469 4.86 5.00 5.13 6.24 6.34 6.43 5.62 5.60 6.67 5.74 8 3.16 3.67 3.98 4,20 4.38 4.62 4.66 4.75 4.85 4.94 5.02 5.09 5.16 5.22 5.28 9 3.04 3.51 3.79 3.99 4.15 4.28 4.39 4.49 4.68 4.68 4.73 4.79 4,85 4.91 4.96 10 2.95 3.39 3,86 3.84 3.98 4.10 4.2 1 4.30 4.37 4.45 4.61 4.57 4.62 4.68 4.72 11 2.88 3.30 3.54 3.72 385 3.97 4.06 4.14 4.22 4.28 4.34 4.40 4.45 4.5

~. 4.54 12 2.82 3.22 3.46 3.62 3.76 3.86 3.95 4.03 4.09 4.16 4.21 4.26 4.31 4.3E 4.40 13 2.78 3.16 3.39 3_54 3.67 3.77 3.85 3.93 3.99 4.05 4_11 4.16 4.20 4.24 4.28 14 2.74 3.11 3.33 3.48 3.60 3.70 3.78 3.85 3.91 3.97 4.02 4.07 4.11 4.15 4.19 15 2.71 3.07 3.28 3.43 3.64 3.63 3.71 3.78 3.84 4.90 3.95 3.99 4.03 4.07 4.11 16 2.68 3.03 3.24 3.38 3.49 3.68 3.66 3.72 3.78 3.84 3.88 3.93 3.97 4.00 4.04 17 2.66 3.00 3.20 3.34 3.46 3.64 3.61 3.68 3.73 3.78 3.83 3.87 3.91 3.95 3.98 18 2.64 2.97 3.17 3.31 3_41 3.60 3.67 3.63 3.69 3.74 3.78 3.82 3.236 3.88 3.93 19 2.62 2.96 3.14 3.28 3.38 3.46 3.63 3.60 3,65 3.70 3.74 3.78 3.82 3.85 3.88 20 2.60 2.93 3.12 3,25 3.36 3.43 3.50 3.56 3.61 3.66 3.70 3.74 3.78 3.81 3.84 21 2.69 2.91 3.10 3.22 3.32 3.41 3.47 3.53 3.58 3.63 3.67 3.71 3.75 3_78 3.81 22 2.57 2.89 3.08 3.20 3.30 3.38 3.45 3.61 3.66 3.60 3.64 3.68 3.71 3.75 3.7b 23 2.66 2.88 3.06 3.18 3.28 3.36 3.42 3.48 3.63 3.58 3.62 3.65 3.69 3.72 3.75 24 2.65 2.86 3.04 3.17 3.26 3.34 3.40 3.46 3,61 3.55 3.69 3.63 3.66 3.69 3.72 25 2.54 2.86 3.03 3.16 3.24 3.32 3,39 3.44 3.49 3.53 3.57 3.61 3.64 3.67 3.70 26 2.53 2.84 3.01 3.14 3.23 3.30 3.37 3.42 3,47 3.51 3.56 3.59 3.62 3.66 3.68 27 2.52 2.83 3.00 3.12 3.21 3.29 3.36 3.41 3.46 3.50 3.53 3.67 3.60 3.63 3.66 28 2.52 2.82 2.99 3 11 3.20 3.27 3.34 3.39 3.44 3 48 3.52 3.65 3.58 3.61 3.64 29 2_51 2.81 2.98 3.10 3.19 3.26 3.32 3.38 3.42 3.46 3.50 3.54 3.57 3.60 3.62 30 2.50 2.80 2.97 3.09 3.18 3.25 3.31 3.36 3.41 3.45 3.49 3.52 3.55 3.58 3.61 31 2.60 2.79 2.96 3.08 3.17 314 3.30 3.35 3.40 3.44 3.47 3.61 3.54 3.57 3.59 32 2.49 2.79 2.96 307 3.16 3.23 3.29 3.34 3.39 3 43 3.46 3.60 3.63 3.55 3.58 33 2.49 2.78 2.94 3.06 3.16 3.22 3.28 3.33 3.37 3.41 346 3.48 3.51 3.54 3.57 34 2.48 2.77 2.94 3.05 3.14 3.21 3.27 3.32 3.36 3.40 3.44 3.47 3.50 3.53 3.56 35 2.48 2.77 2.93 304 3.13 3.20 3.26 3.31 3.36 3.39 3.43 3.46 3.49 3.52 3,54 3g 2.47 2.76 2.92 3.04 3.12 3.19 3.26 3.30 3.35 3.39 3.42 3.46 3.48 3.51 3.53 37 2.47 2.76 2.92 3.03 3.12 3.19 3.24 3.29 3.34 3.38 3.41 3.44 3.47 3.50 3.53 38 2.46 2.76 2.91 3.02 3.11 3.18 3.24 3.29 3.33 3.37 3.40 3.44 3.46 3.49 3.52 39 2.46 2.76 2.91 3.02 3.10 3.17 3.23 3.28 3.32 3.36 3.40 3.43 3.46 3.48 3.51 40 2.46 2.74 2,90 3.01 310 3.17 3.22 3.27 3.31 3.36 3.39 3.42 3.45 3 47 3.50 41 2.46 2.74 2.90 3.01 3.09 3.16 3.22 3.27 3.31 3.35 338 3 41 3.44 3.47 3.49 42 2.45 2.73 2.89 3.00 3.09 3.15 3.21 3.26 3.30 3.34 3.37 3.41 3.43 3.46 3.48 43 2.45 2.73 2.89 3.00 3.08 3.16 3.20 3.26 3.30 3.33 3.37 3.40 3.43 3.45 3.48 44 2.44 2.73 2.88 299 3.08 3.14 3.20 5.25 3.29 3.33 3.36 3.39 3.42 3.45 3.47 45 2,44 2.72 2.88 2.99 3.07 3.14 3.19 3.24 3.28 3 32 336 3.39 3.41 3.44 3.46 46 2.44 2.72 2.88 2.98 3.07 3.13 3.19 3.24 3.28 3.32 3.35 3.38 3.41 3.43 3.46 47 2.44 2.72 2,87 2.98 3.06 3.13 3,18 3.23 3.27 3.31 3.34 3.38 3.40 3.43 3.45 48 2.43 2.71 2.87 2 98 3.06 3.12 3.18 3.23 3.27 3 31 3.34 3.37 3.40 3.42 3,45 49 2.43 2.71 2.86 297 3.06 3.12 3.18 3.22 3.26 3 30 3 34 3.37 3.39 3.42 3.44 50 2.43 211 2.86 2.97 3.06 3.12 3.17 3.22 3.26 3.30 3.33 3.36 3.39 3.41 3.44 60 2.41 2.68 2,84 2.94 3.01 3.08 3.14 3.18 3.22 3.26 3.29 3.32 3.35 3.37 3.40 70 2.40 2.67 2.82 2.92 3.00 3.06 3.12 3.16 3.20 3.24 3.27 3.30 3.32 3.3S 3.37 80 2.39 2.66 2.80 2.91 2.98 3.05 3,10 3.14 3.18 3 22 3.25 3.28 3.30 3.33 3.35 90 2.38 2.65 2.79 2.89 2.97 3.03 3.08 3.13 3.17 3.20 3.23 3.26 3.29 3.31 3.33 100 2.38 2.64 2.79 2.89 2.96 3.02 3.07 3.12 316 3.19 3.22 3.26 3.27 3.30 3.32


### Factor = t t.l



### _a~k~~l + 1/n


## 45i

### Tabie 2. Factors for Obtaining One-Sided 99% Prediction Limits for k Additional Samples Given a Background Sample of Size n

Previous Nucnber of new niessurernerrts (k)


### n 16 17



### _ 18 19 ZO 21 ZS 23 24 26 26 27 28 29 30 4 10.01 10.16 10.30 10.43 10.55 10.6 10.79 10.90 11.00 11.11 11.31 11.30 11.40 11.49 11.67 5 7.70 7.79 7.89 7.97 8.06 8.14 8.22 8.29 8.36 8.43 8.50 8.56 8.62 8.68 8.74 6 6.51 6.58 6.65 6.74 6.79 6.85 6.90 6.96 7.02 7.07 7.12 7.17 7.41 7.26 7.30 7 5.80 5.86 6.94 5.98 6.03 6.08 6.13 6.17 6.21 6.26 6.30 6.34 6.38 6.41 6.45 8 5.34 6.39 6.44 5.49 5.63 6.67 5.61 6.66 5.69 6.73 5.76 6.80 6.83 6.86 5.89 9 6.01 5.06 5.10 6.14 6.18 6.22 6.26 6.39 6.32 6.36 619 6.42 6.44 6.47 5.50 10 4.77 4.81 4.86 4.89 4.92 4.96 4.99 6.02 5.06 6.08 6.11 5.14 6.16 5.19 5.21 11 4.68 4.62 4.66 4.69 4.73 4.76 4.79 4.82 4.84 4.87 4.90 4.92 4.94 4.97 4.99 12 4.44 4.47 4.61 4.64 4.57 4.60 4.63 4.66 4.68 4.71 4.73 4.76 4.77 4.80 4.82 13 4.32 4.36 418 4.42 4.44 4.47 4.60 4.62 4.55 4.67 4.59 4.62 4.64 4.66 4.68 14 4.22 4.26 4.28 4.31 4.34 4.37 4.39 4.42 4.44 4.46 4.48 4.50 4.62 4.54 4.56 15 4.14 4.17 4.20 4.23 4.25 4.78 4.30 4.33 4.36 4.37 4.39 4.41 4.43 4.45 4.46 16 4.07 4.10 4.13 4.15 4.18 4.40 4.23 4.25 4.27 4.29 4.31 4.33 4.35 4.37 4.38 17 4.01 4.04 4.07 4.09 4.12 4.14 4.16 4.18 4.20 4.22 4.24 4.26 4.28 4.30 4.31 18 3.96 3.99 4.01 4.04 4.06 4.08 4.11 4.13 4.16 4.17 4.18 4.20 4.22 4.24 4.25 19 3.91 3.94 3.97 3.99 4.01 4.04 4.06 4.08 4.10 4.11 4.13 4.15 4.17 4.18 4.20 20 3.87 3.90 3.92 3.96 3.97 3.99 4.01 4.03 4.06 4.07 4.09 4.10 4.12 4.14 4.15 21 3.84 3.86 3.89 3.91 3.93 3.96 3.97 3.99 4.01 4.03 4.05 4.06 4.08 4.09 4.11 ZZ 3.80 3.83 3.86 3.88 3.90 3.92 3.94 3.96 3.98 3.99 4.01 4.03 4.04 4.06 3 .07 23 3.77 3.80 3.82 3.86 3.87 3.89 3.91 3.93 3.94 3.96 3.98 3.99 4.01 4.02 4.04 24 3.75 3.77 3.80 3.82 3.84 3.86 3.88 3.90 3.95 3.93 3.95 3.96 3.98 3.99 4.01 25 3.72 3.76 3.77 3.79 3.82 3.84 3.85 3.87 3.89 3.91 3.92 5.94 3.95 3.97 3.98 26 370 3.73 3.75 3.77 3.79 3.81 3.83 3.86 3.87 3.88 3.90 3.91 3.93 5.94 3.95 27 3.68 3471 3.73 3.75 3.77 3.79 3.81 3.83 3.84 3.86 3.87 3.89 3.90 3.92 3.93 28 3.66 3.69 3.71 3.73 3.75 3.77 3.79 3.81 3.82 3.84 3.86 3.87 3.88 3.90 3.91 29 3.66 3.67 3.69 3.71 3.73 3.76 3.77 3.79 3.80 3.82 3.83 3.85 3.86 3.88 3.89 30 3.63 3.66 3.68 3.70 3.74 3.74 3.75 3.77 3.79 3.80 3.82 3.83 3.86 3.86 3.87 31 3.62 3.64 3.66 3.68 3.70 3.72 3.74 3.76 3.77 3.79 3.80 3.81 3.83 3.84 3.86 32 3.60 3.63 3.66 3.67 3.69 3.71 3.72 3.74 3.76 3.77 3.79 3.80 3.81 3.83 3.84 33 3.59 3.61 3.64 3.66 3.67 3.69 3.71 3.73 3.74 3.76 3.77 3.79 3.80 3.81 3.82 34 3.58 3.60 3.64 3.64 3.66 3.68 3.70 3.71 3.73 3.74 3.76 3.77 3.78 3.80 3.81 35 3.57 3.69 3.61 3.63 3.66 3.67 3.89 3.70 3.75 3.73 3.75 3.76 3.77 3.78 3.80 38 3.56 3.58 3.60 3.62 3.64 3.66 3.67 3.69 3.71 3.72 3.73 3.76 3.76 3.77 3.79 37 3.66 3.67 3.69 3.81 3.63 3.66 3.66 3.68 3.69 3.71 3.72 3.74 3.75 3.76 3.77 38 3.54 3.56 3.68 3.60 3.62 3.64 3.65 3.67 3.68 3.70 3.71 3.73 3.74 3.75 3.76 39 3.63 3.65 3.57 3.59 3.61 3.83 3.64 3.66 3.67 3.69 3.70 3.72 3.73 3.74 3.75 40 3.62 3.64 3.56 3.58 3.60 3.62 3.63 3.66 3.67 3.68 3.69 3.71 317 3.73 3.74 41 3.61 3.64 3.66 3.68 3.69 3.61 3.63 3,64 3.66 3.67 3.68 3.70 3.71 3.72 3.73 42 3.51 3.63 3.66 3.67 3.69 5.60 3.64 3.63 3.66 3.66 3.68 3.69 3.70 3.71 3.73 43 3.60 3.62 3.64 3.66 3.68 3.69 3.61 3.63 3.64 3.65 3.67 3.68 3.69 3.71 3.72 44 3.49 3.61 3.63 3.66 3.67 3.59 3.60 3.62 3.63 3.66 3.86 3.87 3.69 3.70 3.71 46 3.49 3.61 3.63 3.66 3.66 3.68 3.60 3.61 3.63 3.64 3.66 3.67 3.68 3.69 3.70 46 3.48 3.60 3.62 3.64 3.68 3.67 3.69 3.61 3.62 3.63 3.65 3.66 3.67 3.68 3.70 47 3.48 3.50 3.62 3.63 3.65 3.67 3.68 3.60 3.61 3.63 3.64 3.66 3.67 3.68 3.69 48 3.47 3.49 3.51 3.53 3.55 3.66 3.68 3.59 3.61 3.62 3.63 3.66 3.66 3.67 3.68 49 3.46 3.48 3.60 3.62 3.64 3.66 3.67 3.69 3.60 3.62 3.63 3.64 3.6b 3.66 3.68 b0 3.46 3.48 3.60 3.64 3.53 3.66 3.67 3.68 3.60 3.61 3.62 1.64 3.66 3.66 3.67 60 3.42 3.44 3.46 3.47 3.49 3.61 3.52 3.64 3.66 3.66 3.68 3.69 3.60 3.61 3.62 70 3.39 3.41 3.43 3.45 3.48 3.48 3.49 3.61 3.52 3.63 3.54 3.66 3.67 3.68 3.69 80 3.37 3.39 3.41 3.42 3.44 3.46 3.47 3.48 3.60 3.51 3.52 3.53 3.54 3.56 3.67 90 3.36 3.37 3.59 3.41 3.42 3.44 3.46 3.47 3.48 3,49 3.50 3.51 3.63 3.54 3.55 100 3.34 3.36 3.38 3.39 3.41 3.42 3.44 3.45 3.46 3.48 3.49 3.60 3.51 3.62 3.53

Factor =t(n-1,1-a/k)%/1f- 1/s


# 3. PREDICTION LIMITS FOR THE LOGNORMAL DISTRIBUTION 

# Whcn data exhibit a few elcvated values such that the frequency distrihution is "skewcd" with a. long right ta.il, estimates based oii the assuinptionof norniality do not apply. In practicc, transfornlations of data are used to stabilize variance and bring about normality ( t3ox and (:ox, 1964). Perhaps the most commonly Lised transformation in rhis situation is the natural log transformation (Aroian, 1941), wlhere x is a lognorrnal randorn variable, such that

# y=loge

# x-N(p,02

# ). Wlien jj and o2are unknown, we tnaysubstitut4

# the saniple estiniates y and sz wherc n I loge

# (x; )In i =-1

n ancl s= - Z[log, (x1) - y` J'I (n - 1).

i=1 

# Rased on a result obtained by Dahiya and Guttman (1982), thc 100(1 - a)96 prediction interval has thc familiar form

exp

# {y ±1 + ],/ntIn_l;a/21 s}. in light of this, the values in 'I'ables 1•3 may be used to obtain the one-sided 99% prediction limit:

exp { y` + r(k, n; .99) s}

458

# where y` and s are the mean and standard deviation

computed on natural log transformed data. Similarly, Tables 4-6 may be used to computc: the corresponding two-sided 99% prediction limit:

exp

# {y ± r(k, n;.99) s} 

# A note of caution must be raised regarding the use of log transformation in calculating prediction limits. 1'his point is best illustrated by an example. Consider the following data for 'fOX, and for sirnplicity let us assume that these measurements were collected from a single upgradicnt well, quarterly for two years so that sl is an unbiased estimate of 02. Upgradient wcll Quarter Ycar TOX 10ge(TOX)
Mw01 1 2985 48 3.87 Mwol 2 1985 5 1.61 

## MWoI 3 1985 44 3.78 Mw01 4 1985 24 3.18 MWoI 1 1986 17 2.83 Mw01 2 1986 ti 1.79 Mw01 3 1986 12 2.48 Mw01 4 1986 60 4.09 x=27.00 x'=2.96 s = 20.97 s = 0.95



## n-8 n=8

# Inspection of the original I'OX valucs reveals a distribution that is skewed to the right; thercfore, log

Table 3. Factors for Obtaining One-Sided 9996 Prediction Limits for k Additional Samples Given a Background Sample of Size n
Previoue Number of new measuremente (k)

n 30 36 40 46 60 66 60 66 70 76 80 86 90 96 100 4 11.67 11.98 12.34 12.66 1.96 13.22 13.47 13.70 13.91 14.11 14.31 14.49 14.66 14.82 14.98 6 8.74 9.01 9.56 9.46 9.66 9.83 9.99 10.14 10.29 10.42 10.64 10.66 10.77 10.88 10.98 6 7.30 7.61 7.68 7.84 7.99 8.12 8.24 8.36 8.46 8.66 8.66 8.73 8.82 8.90 8.97 7 6.46 6.61 6.76 6.89 7.00 7.11 7.20 7.29 7.38 7.46 7.63 7.60 7.67 7.73 7.79 8 6.89 6.03 6.16 6.26 6.36 6.46 6.63 6.61 6.68 6.74 6.81 6.87 6.92 6.97 7.02 9 6.60 6.62 6.73 6.82 6.91 6.99 6.06 6.13 6.19 6.25 6.30 6.36 6.40 6.46 6.49 10 5.21 5.32 6.42 6.60 6.68 6.66 6.72 5.77 6.83 6.88 6.93 6.98 6.02 6.06 6.10 11 4.99 6.09 6.18 6.26 5.33 6.39 6.46 6.61 6.66 6.60 6.66 6.69 6.73 6.77 5.80 12 4.82 4.91 4.99 6.07 6.13 6.19 6.26 6.30 6.34 6.39 6.43 6.47 6.60 6.64 5.57 13 4.68 4.77 4.84 4.91 4.97 6.03 6.08 6.13 6.17 6.31 6.26 6.28 6.32 6.35 6.38 14 4.66 4.66 4.72 4.78 4.84 4.89 4.94 4.99 6.03 6.07 6.10 6.14 6.17 6.20 5.23 15 4.46 4.64 4.61 4.68 4.73 4.78 4.83 4.87 4.91 4.96 4.98 6.01 6.04 6.07 5.10 16 4.38 4.46 4.63 4.59 4.64 4.69 4.73 4.77 4.81 4.84 4.88 4.91 4.94 4.96 4.99 17 4.31 4.39 4.46 4.61 4.66 4.61 4.66 4.69 4.72 4.76 4.79 4.82 4.86 4.87 4.90 18 4.26 4.32 4.39 4.44 4.49 4.64 4.68 4.61 4.66 4.68 4.71 4.74 4.77 4.79 4.82 19 4.20 4.27 4.13 4.38 4.45 4.47 4.61 4.66 4.68 4.61 4.64 4.67 4.70 4.72 4.76 20 4.16 4.22 4.28 4.33 4.38 4.42 4.46 4.49 4.63 4.66 4.69 4.61 4.64 4.66 4.68 21 4.11 4.18 4.25 4.58 4.33 4.57 4.41 4.44 4.48 4.60 4.63 4.56 4.68 4,61 4.63 22 4.07 4.14 4.19 4.24 4.29 4.33 4.36 4.40 4.43 4.46 4.49 4.61 4.64 4.66 4.58 23 4.04 4.10 4.16 4.21 4.26 4.29 4.32 4.38 4.39 4.42 4.44 4.47 4.49 4.51 4.54 24 4.01 4.07 4.12 4.17 4.21 4.26 4.29 4.51 4.36 4.38 4.41 4.43 4.46 4.48 4.50 26 5.98 4.04 4.09 4.14 4.18 4.22 4.26 4.29 4.32 4.36 4.37 4.40 4.42 4.44 4.46 26 3.96 4.01 4.07 4.11 4.16 4.19 4.23 4.26 4.29 4.31 4.34 4.36 4.39 4.41 4.43 27 5.93 3.99 4.04 4.09 4.13 4.17 4.20 4.23 4.26 4.29 4.31 4.33 4.36 4.38 4.40 28 3.91 3.97 4.02 4.06 4.11 4.14 4.17 4.21 4.23 4.26 4.28 4.31 4.33 4.36 4.37 29 5.89 3.96 4.00 4.04 4.08 4.12 4.16 4.18 4.21 4.24 4.26 4.28 4.30 4.32 4.34 SO 5.87 3.93 3.98 4.02 4.06 4.10 4.13 4.16 4.19 4.21 4.24 4.26 4.28 4.30 4.32 31 5.86 3.91 3.96 4.00 4.04 4.08 4.11 4.14 4.17 4.19 4.22 4.24 4.26 4.28 4.30 32 3.84 3.90 3.94 3.99 4.03 4.06 4.09 4.12 4.16 4.17 4.20 4.22 4.24 4.26 4.28 33 3.82 3.88 3.93 3.97 4.01 4.04 4.08 4.10 4.13 4.16 4.18 4.30 4.22 4.24 4.26 34 1.81 3.87 3.91 3.96 3.99 4.03 4.06 4.09 4.11 4.14 4.16 4.18 4.20 4.22 4.24 36 3.80 3.86 3.90 3.94 3.98 4.01 4.04 4.07 4.10 4.12 4.16 4.17 4.19 4.21 4.22 36 3.79 3.84 3.89 3.93 3.97 4.00 4.03 4.06 4.08 4.11 4.13 1.16 4.17 4.19 4.21 37 3.77 3.83 3.88 3.92 3.96 3.99 4.02 4.04 4.07 4.09 4.12 4.14 4.16 4.18 4.19 38 3.76 3.82 3.86 3.90 3.94 3.97 4.00 4.03 4.06 4.08 4.10 4.12 4.14 4.16 4.18 39 3.75 3.81 3.85 3.89 5.93 3.96 5.99 4.02 4.06 4.07 4.09 4.11 4.13 4.15 4.17 40 5.74 3.80 3.84 3.88 3.92 3.96 3.98 4.01 4.03 4.06 4.08 4.10 4.12 4.14 4.15 41 3.73 3.79 3.83 3.87 3.91 3.94 3.97 4.00 4.02 4.05 4.07 4.09 4.11 4.12 4.14 42 3.73 3.78 3.82 3.86 3.90 3.93 3.96 3.99 4.01 4.04 4.06 4.08 4.10 4.11 4.13 43 3.72 3.77 3.82 3.86 3.89 3.92 3.96 3.98 4.00 4.03 4.06 4.07 4.09 4.10 4.12 44 3.71 3.76 3.81 3.86 3.88 3.91 3.94 5.97 5.99 4.02 4.04 4.06 4.08 4.09 4.11 46 5.70 3.76 3.80 3.84 3.87 5.91 3.95 3.96 3.98 4.01 4.05 4.06 4.07 4.08 4.10 46 3.70 3.76 3.79 3.83 3.87 3.90 3.93 3.95 3.98 4.00 4.02 4.04 4.06 4.08 4.09 47 3.09 3.74 3.78 3.82 3.86 3.89 3.92 3.94 3.97 3.99 4.01 4.03 4.05 4.07 4.08 48 5.68 3.73 3.78 3.82 3.86 3.88 3.91 3.94 5.96 3.98 4.00 4.02 4.04 4.06 4.07 49 3.88 3.73 3.77 3.81 3.84 3.88 3.90 3.93 3.96 3.98 4.00 4.02 4.03 4.05 4.07 60 3.67 3.73 3.76 3.80 3.84 3.87 5.90 3.92 3.96 3.97 3.99 4.01 4.03 4.04 4.06 60 3.62 3.67 3.71 3.76 3.78 3.81 3.84 3.87 3.89 3.91 3.93 3.96 3.97 3.98 4.00 70 5.69 3.64 5.68 3.72 3.76 3.78 3.80 3.83 3.86 3.87 3.89 3.91 3.93 3.94 3.96 80 3.67 3.61 3.66 3.69 3.72 3.76 3.78 1.80 3.82 3.84 3.86 3.88 3.89 3.91 3.93 90 3.66 3.69 5.63 3.67 3.70 3.75 3.76 3.78 3.80 3.82 3.84 3.86 3.87 3.89 3.90 100 3.63 5.58 5.62 3.66 3.68 3.71 3.74 3.76 3.78 5.60 3.82 3.84 5.86 3.87 3.88 

### Fsctor = t

~

### w-l.l-a/k)

## ~1 ♦I/R


# 459

### Table 4. Factors for Obtaining Two-Sided 99% Prediction Limits for k Additional Samples Given a Background Sample of Size n

Previous Number ot new measurements (k)

n 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 4 6.81 7.06 7.86 8.46 8.94 9.35 9.70 10.01 10.30 10.66 10.79 11.00 11.21 11.40 11.57 5 4.77 5.66 6.22 6.63 6.97 7.24 7.48 7.70 7.89 8.06 8.22 8.36 8.50 8.62 8.74 6 4.23 4.93 5.37 5.69 5.96 6.16 6.36 6.61 6.65 6.79 6.90 7.02 7.12 7.21 7.30 7 3.89 4.49 4.86 5.13 6.34 5.62 6.67 6.80 6.92 6.03 6.13 6.21 6.30 6.38 6.45 8 3.674.20 4.62 4.75 4.94 6.09 6.22 6.34 5.44 6.53 5.61 6.69 5.76 5.83 5.89 9 3.51 3.99 4.28 4.49 4.66 4.79 4.91 6.01 5.10 6.18 5.25 6.32 6.39 6.44 6.60 10 3.39 3.84 4.10 4.304.45 4.674.68 4.77 4.85 4.92 4.99 6.06 5.11 6.16 5.21 11 3.30 3.72 3.974.14 4.28 4.40 4.60 4.68 4.66 4.73 4.79 4.84 4.90 4.94 4.99 12 3.22 3.62 3.86 4.03 4.16 4.26 4.36 4.44 4.61 4.67 4.63 4.68 4.73 4.77 4.82 13 3.16 5.64 3.773.93 4.05 4.16 4.24 4.32 4.38 4.44 4.50 4.56 4.69 4.64 4.68 14 3.11 3.48 3.70 3.86 3.97 4.074.15 4.22 4.28 4.34 4.39 4.44 4.48 4.52 4.56 15 3.073.43 3.63 3.78 3.90 3.99 4.07 4.14 4.20 4.26 4.30 416 4.39 4.43 4.46 16 3.03 3.38 3.68 3.72 3.84 3.93 4.00 4.07 4.13 4.18 4.23 4.27 4.31 4.35 4.38 17 3.00 3.34 3.54 3.68 3.78 3.873.95 4.01 4.07 4.15 4.16 4.40 4.24 4.28 4.31 18 2.97 311 3.60 3.63 3.74 3.82 3.89 3.96 4.01 4.06 4.11 4.16 4.18 4.22 4.26 19 2.96 3.28 3.46 3.603.70 3.78 3.85 3.91 3.97 4.01 4.06 4.10 4.13 4.17 4.20 20 2.93 3.25 3.43 3.663.66 3.74 3.81 3.87 3.92 3.97 4.01 4.05 4.09 4.12 4,15 21 2.91 3.22 3.41 3.65 3.63 3.71 3.78 3.84 3.89 3.93 3.97 4.01 4.06 4.08 4.11 22 2.89 3.20 3.38 3.61 3.60 3.68 3.76 3.80 3.86 3.90 3.94 3.98 4.01 4.04 4.07 23 2.88 5.18 3.36 3.48 3.68 3.66 3.72 3.77 3.82 3.87 3.91 3.94 3.98 4.01 4.04 24 2.86 3.173.54 3.46 3.65 5.63 3.69 3.76 3.80 3.84 3.88 3.92 3.95 3.98 4.01 25 2.85 3.15 3.32 3.44 3.53 3.61 3.67 3.72 3.77 3.82 3.85 3.89 3.92 3.95 3.98 26 2.84 3.14 3.30 3.473.61 3.59 3.65 3.70 3.76 3.79 3.83 3.87 3.90 3_93 3.95 27 2.83 3.12 3.59 3.41 3.50 3.673.63 3.68 3.73 3.77 3.81 3.84 3.87 3.90 3.93 28 2.82 3.11 3.273.39 3.48 3.65 3.61 3.66 3.71 3.76 3.79 3.82 3.85 3.88 3.91 29 2.81 3.10 3.26 3.38 3.46 3.54 3.60 3.66 3.69 3.73 3.77 3.80 3.83 3.86 3.89 30 2.80 3.09 3.25 3.36 3.46 3.62 3.68 3.63 3.88 3.72 3.75 3.79 3.82 3_86 317 31 2.79 3.08 3.24 3.36 3.44 3.61 3.67 1.65 3.66 3.70 3.74 3.77 3.80 3.83 346 32 2.79 3.073.23 3.34 3.43 3.50 3.65 3.60 3.66 3.69 3.72 3.76 1.79 3.81 3.84 33 2.78 3.06 3.22 3.33 3.41 3.48 3.54 3.69 3.64 3.67 3.71 3.74 3.77 3.80 3.82 34 2.773.05 3.21 3.52 3.40 3.473.63 3.68 3.62 3.66 3.70 3.73 3.76 3.78 3.81 35 2.773.04 3.20 3.31 3.39 3.46 3.65 317 3.e1 3.66 3.89 3.72 3.75 3.77 3.80 36 2.76 3.04 3.19 3.30 3.39 3.45 3.51 3.66 3.60 3.64 3.67 3.71 3.73 3.76 3.79 37 2.76 3.03 3.19 3.29 3.38 3.44 3.60 3.65 3.59 3.63 3.66 3.69 3.72 3.7L 3.77 38 2.76 3.02 3.18 3.29 3.373.44 3.49 3.64 5.58 3.62 3.65 3.68 3.71 3.74 3.76 39 2.75 3.02 3.173.28 3.36 3.43 3.48 3.63 3.57 3.61 3.64 3.67 3.70 3.73 3.75 40 2.74 3.01 3.173.273.36 3.42 3.47 3.65 3.66 3.60 3.63 3.67 3.69 3.72 3.74 41 2.74 3.01 3.16 3.273.35 3.41 3.47 3.51 3.66 3.59 3.63 3.66 3.68 3.71 3.73 42 2.73 3.00 3.15 3.263.34 3.41 3.46 3.51 3.56 3.59 3.62 3.65 3.68 3.70 3.73 43 2.73 3.00 3.15 3.26 3.33 3.40 3.45 3.60 3.64 3.58 3.61 3.64 3.67 3.69 3.72 44 2.732.99 3.14 3.26 3.33 3.39 3.46 3.49 3.63 3.67 3.60 3.63 3.66 3.69 3.71 45 2.72 2.99 3.14 3.54 3.32 3.39 3.44 3.49 313 3.66 3.60 3.63 3.66 3.68 3.70 46 2.72 2.98 3.13 3.24 3.32 3.38 3.43 3.48 3.62 3.66 3.69 3.62 3.66 3.67 3.70 47 2.72 2.98 3.13 3.23 3.51 3.38 3.43 3.48 3.62 3.65 3.68 3.61 S.64 3.67 3.69 48 2.71 2.98 3.12 3.23 3.31 3.373.42 3.47 3.61 3.56 3.58 5.61 3.63 3.66 3.68 49 2.71 2.973.12 3.22 3.30 3.373.42 3.46 3.60 3.64 3.57 3.60 3.63 3.65 3.68 60 2.71 2.97 3.12 3.22 3.30 3.36 3.41 3.46 3.60 3.63 3.57 3.60 3.62 3.66 3.67 60 2.682.94 3.08 3.18 3.28 3.32 3.37 3.42 3.46 3.49 3.62 3.55 3.68 5.60 3.62 70 2.672.92 3.06 3.16 3.24 3.30 3.55 3.39 3.43 3.46 3.49 3.52 3.54 3.57 3.69 80 2.66 2.91 3.05 3.14 3.21 3.28 3.33 3.37 3.41 3.44 3.17 3.60 3.52 3.54 3.57 90 2.66 2.89 3.03 3.133.20 3.26 3.31 3.36 3.39 3.42 3.45 3.48 3.50 3.53 3.55 100 2.642.89 3.02 3.12 3.19 3.55 3.30 3.34 3.38 3.41 3.44 3.46 3.49 3.51 3.53 Factor = t(R



# _1
,1



# _0/2k) 1+ 1

## fn 

# transformation appcars to be a natural choice for better approxima.ting the assumed normality of the statistical procedure. Indeed, the transforniation appears to have somewhat normali

-r.ed the distribution of these nieasurements. Assuming that we are interested in evaluating a singlc new downgradient measuremcnt, thc9996 

# prediction limit based on the original data is: 27.00 + 20.97(3.16) = 93.27 pph whereas, the 99% prediction limit for the log transformed data is: exp [2.96 + .95 (3.16)] = 388.39 ppb 

# Alternatively, if we directly estinlate the niean and standard deviation of the lognormal distribution(Aitchison, 1955), we obtain for the 99% prediction limit
28.16 + 27.71(3.1 6) =115.73 ppb 

# Inspection of these three prediction limits reveals that theestilnatebased on log transformed data is remarkablv elevated relative to the normal

. 

# and lognornial interval estimatcs. The lognorrnal prediction limit is, of course, incorrect because the use of thesc liniits assumcs a normal distribution for

# xt,whereas these estimates are based on log460

# normal assumptions; however, they do seem to give

quite reasonable estimates, at least, for practical purposes. Conversely, the interval estimatc obtained following log transformation of the data, a seemingly standard and benign practice, is extremely high and is likely to lcad to elevatedfalse negative rates even for liberal effect magnitudes. In fact, the limit value for the log transformed data is almost an entire order of magnitude higher than the maximum value in the background sample. It is for this reason that log transformation must be used very cautiously when constructing prediction intervals. 4. SAMPLING STRATEGIES 

# When using prediction intervals, we assume that both the original observations (xl,..., xn) 

# and the future observations (x„+t, ..., xn fk )

# arise from the same probability distribution, namely, N(,u, o2).

# Since, in practice, μ and o2are unknown quantities, we must estimate their value from a random sample of previous measurements by obtaining the sufficient statistics z and s, that is, the sample mean and standard deviation. If our background measurcments consist of multiplc quarterly measurements of a single well or muttiple upgradient wells at a single time-point,

Table 5. Factors for Obtaining Two-Sided 991 /o Prediction Limits for k Additional Samples Given a Background Sample of Size n

Previous IVumber of new messuraments (!t)

n 16 17 18 19 20 21 22 23 24 26 26 27 28 29 30 4 11. 74 11,90 12 . 05 12.20 12.34 12.47 12.60 12.72 12.84 12.95 13.06 13.17 13.27 13.37 13.47 5 8.85 8.96 9.06 9.16 9.25 9.34 9.42 9.50 9.58 9.65 9.73 9.80 9.86 9.93 9.99 6 7.39 7.47 7.64 7.61 7.68 7.76 7.81 7.87 7.93 7.99 8.04 8.09 8.14 8.19 8.24 7 6.62 6.68 6.64 6.70 6.76 6.81 6.86 6.91 6.96 7.00 7.04 7.09 7.13 7.17 7.20 8 5.95 6.00 6.06 6.10 6.16 6.20 6.24 6.28 6.32 6.36 6.40 6.43 6.47 6.50 6.53 9 5.55 5.60 5.64 6.69 5.73 5.77 5.81 6.84 5.88 6.91 6.94 5.97 6.00 6.03 6.06 10 6.26 6.30 614 6.38 6.42 5.45 5.49 5.52 5.55 5.58 5.61 5.64 5.66 5.69 5.72 11 6.03 6.07 6.11 6.15 5.18 6.21 5.24 6.27 6.30 6,33 6.36 5.38 5.41 5.43 6.45 12 4.86 4.89 4.93 4.96 4.99 5.02 5.05 6.08 6.11 5.13 5.16 5.18 5.20 5.22 5.25 13 4.71 4.75 4.78 4.81 4.84 4.87 4.90 4.92 4.96 4.97 5.00 5.02 6.04 5.06 5.08 14 4.60 4.63 4.66 4.69 4.72 4.76 4.77 4.80 4.82 4.84 4.86 4.88 4.90 4.92 4.94 15 4.60 4.53 4.56 4.59 4.61 4.64 4.66 4.69 4.71 4.73 4_75 4.77 4.79 4.81 4.83 16 4.41 4.45 4.47 4.50 4.63 4.56 4.67 4.60 4.62 4.64 4.66 4.68 4.70 4.71 4.73 17 4.34 4.37 4.40 4.43 4.45 4.47 4.60 4.62 4.54 4.66 4.58 4.60 4.61 4.63 4.65 l8 4.28 4.31 4.34 4.36 4.39 4.41 4.43 4.46 4.47 4.49 4.61 4.63 4.54 4.56 4.58 19 4.23 4.26 4.28 4.31 4.33 4.35 4.37 4.39 4.41 4.43 4.46 4.47 4.48 4.50 4_51 20 4.18 4.21 4.23 4.26 4.28 4.30 4.32 4.34 4.36 4.38 4.39 4.41 4.43 4.44 4.46 21 4.14 4.16 4.19 4.21 4.23 4.25 4.27 4.29 4.31 4.33 4_35 4.36 4.38 4.39 4.41 22 4.10 4.12 4.15 4.17 4.19 4.21 4.53 4.25 4.27 4.29 4.30 4.32 4.34 4.35 4.36 23 4.06 4.09 4.11 4.14 4.16 4.18 4.20 4.21 4.23 4.26 4.27 4.28 4.30 4.31 4.32 24 4.03 4.06 4.08 4.10 4.12 4.14 4.16 4.18 4.20 4.21 4.23 4.26 4.26 4.28 4.29 26 4.00 4.03 4.05 4.07 4.09 4.11 4.13 4.15 4.17 4.18 4.20 4.21 4.23 4.24 4.26 26 3.98 4.00 4.03 4.06 4.07 4.09 4.10 4.12 4.14 4.16 4.17 4.19 4.20 4.21 4.23 27 3.96 3.98 4.00 4.02 4.04 4.06 4.08 4.10 4.11 4.13 4.14 4.16 4.17 4.19 4.20 28 3.93 3.96 3.98 4.00 4.02 4.04 4.06 4.07 4.09 4.11 4.12 4.13 4.16 4.16 4.17 29 3.91 3.94 3.96 3.98 4.00 4.02 4.03 4.05 4.07 4.08 4.10 4.11 4.13 4.14 4.15 30 3.90 3.92 3.94 3.96 3.98 4.00 4.01 4.03 4.06 4.06 4.08 4.09 4.10 4.12 4.13 31 3.88 3.90 3.92 3.94 3.96 3.98 4.00 4.01 4.03 4.04 4.06 4.07 4.09 4.10 4.11 32 3.86 3.88 3.91 3.93 3.94 3.96 3.98 4.00 4.01 4.03 4.04 4.06 4.07 4.08 4.09 33 3.85 3.87 3.89 3.91 3.93 3.95 3.96 3.98 3.99 4.01 4.02 4.04 4.05 4.06 4.08 34 3.83 3.86 3.88 3.90 3.91 3.93 3.96 3.96 3.98 3.99 4.01 4.02 4.03 4.05 4.06 36 3.82 3.84 3.86 3.88 3.90 3.92 3.93 3.96 3.97 3.98 3.99 4.01 4.02 4.03 4.04 S6 3.81 3.83 3.86 3.87 3.89 3.90 3.92 3.94 3.96 3.97 3.98 3.99 4.01 4.02 4.03 37 3.80 3.82 3.84 3.86 3.88 3.89 3.91 3.92 S.94 3.96 3.97 3.98 3.99 4.00 4.02 38 3.79 3.81 3.83 3.85 3.86 3.88 3.90 3.91 3.93 3.94 3.95 3.97 3.98 3.99 4.00 39 3.78 3.80 3.82 3.84 3.86 3.87 3.89 3.90 3.92 3.93 3.94 3.96 3.97 3.98 3.99 40 3.77 3.79 3.81 3.83 3.84 3.86 3.88 3.89 3.91 3.92 3.93 3.96 3.96 3497 3.98 41 3.76 3.78 3.80 3.82 3.83 3.86 3.87 3.88 3.90 3.91 3.92 3.94 3.95 3.96 3.97 42 3.75 3.77 3.79 3.81 3.82 3.84 3.86 3.87 3.89 3.90 3.91 3.93 3.94 3.95 3.96 43 3.74 3.76 3.78 3.80 3.82 3.83 3.85 3.86 3.88 3.89 3.90 3.92 3.93 3.94 3.95 44 3.73 3.76 3.77 3.79 3.81 3.82 3.84 3.85 3.87 3.88 3.89 3.91 3.92 3.93 3.94 46 3.72 3.74 3.76 3.78 3.80 3.82 3.83 3.86 3.86 3.87 3.89 3.90 3.91 3.92 3.93 46 3.72 3.74 3.76 3.77 3.79 3.81 3.82 3.84 3.86 3.87 3.88 3.89 3.90 3.91 3.93 47 3.71 3.73 3.75 3.77 3.78 3.80 3.82 3.83 3.84 3.86 3.87 3.88 3.90 3.91 3.92 48 3.70 3.72 3.74 3.76 3.78 3.79 3.81 3.82 3.84 3.86 3.86 3.88 3.89 3.90 3.91 49 3.70 3.72 3.74 3.76 3.77 3.79 3.80 3.82 3.83 3.84 3.86 3.87 3.88 3.89 3.90 50 3.69 3.71 3.73 3.76 3.76 3.78 3.80 3.81 3.82 3.84 3.86 3.86 3.87 3.89 3.90 60 3.64 3.66 3.68 3.70 3.71 3.73 3.74 3.76 3.77 3.78 3.80 3.81 3.82 3.83 3.84 70 3.61 3.63 3.65 3.66 3.68 3.69 3.71 3.72 3.74 3.76 3.76 3.77 3.78 3.79 3.80 80 3.59 3.60 3.62 3.64 3.65 3.67 3.68 3.70 3.71 3.72 3.73 3.74 3.76 3.76 3.78 90 3.57 3.58 3.60 3.62 3.63 3.65 3.66 3.67 3.69 3.70 3.71 3.72 3.73 3.74 3.75 100 3.56 3.57 3.59 3.60 3.62 3.63 3.65 3.66 3.67 3.68 3.69 3.71 3.72 3.73 3.74

Factor = t~„-~,~-~/~k)~l + 1/n



## 461

### Table 6. Factors for Obtaining Two•Sided 9996 Prediction Limits for k Additional Samples Given a Background Sample of Sixe n

Previoue Number ot now measurements (k)


## n 30 35 40 46 60 55 60 65 70 76 80 85 90 95 100 4 13.47 13.91 14.31 14

. 14.98 15.27 16.54 16.80 16.03 16.26 16.46 16.66 16.85 17.03 17.20 5 9.99 10.29 10.64 10.77 10.98 11.17 11.36 11.61 11.67 11.81 11.94 12.07 12.19 12.31 12.42 6 8.24 8.46 8.66 8.82 8.97 9.11 9.24 9.36 9.48 9.68 9.68 9.77 9.86 9.96 10.03 7 7.20 7.38 7.53 7.67 7.79 7.90 8.01 8.10 8.19 8.28 8.35 8.43 8.50 8.57 8.63 8 6.53 6.68 6.81 6.92 7.02 7.12 7.20 7.29 7.36 7.43 7.50 7.56 7.62 7.67 7.72 9 6.06 6.19 6.30 6.40 6.49 6.57 6.66 6.72 6.78 6.84 6.90 6.95 7.00 7.05 7.10 10 5.72 5.83 5.93 6.02 6.10 6.17 6.24 6.30 6.36 6.41 6.46 6.61 6.56 6.60 6.64 11 5.45 5.56 5.65 5.73 5.80 5.87 5.93 6.99 6.04 6.09 6.13 6.18 6.22 6.25 6.29 12 5.25 6.34 5.43 5.50 5.57 5.63 5.69 5.74 5.79 6.83 5.87 5.91 5.95 5.99 6.02 13 5.08 6.17 6.26 5.32 6.38 6.44 6.49 5.64 5.68 6.63 5.66 5.70 6.74 5.77 5.80 14 4.94 6.03 5.10 6.17 5.23 5.28 633 6.38 6.42 6.46 6.49 6.53 5.56 5.69 5.62 15 4.83 4.91 4.98 5.04 5.10 5.16 6.20 6.24 6.28 5.32 6.36 6.38 5.42 5.45 5.47 16 4.73 4.81 4.88 4.94 4.99 5.04 6.08 5.12 6.16 6.20 5.23 6.26 5.29 5.32 5.35 17 4.65 4.72 4.79 4.85 4.90 4.94 4.99 6.03 6.06 5.10 6.13 6.16 6.19 6.21 5,24 18 4.58 4.65 4.71 4.77 4.82 4.86 4.90 4.94 4.98 6.01 5.04 5.07 5.10 6.12 5.15 19 4.51 4.58 4.64 4.70 4.75 4.79 4.83 4.87 4.90 4.93 4.96 4.99 6.02 5.04 5.06 20 4.46 4.53 4.69 4.64 4.68 4.73 4.77 4.80 4.83 4.86 4.89 4.92 4.95 4.97 4.99 21 4.41 4.48 4.53 4.58 4.63 4.67 4.71 4.74 4.77 4.80 4.83 4.86 4.88 4.01 4.93 22 4.36 4.43 4.49 4.54 4.58 4.62 4.66 4.69 4.72 4.76 4.78 4.80 4.83 4.85 4.87 23 4.32 4.39 4.44 4.49 4.64 4.68 4.61 4.64 4.67 4.70 4.73 4.75 4.78 4.80 4.82 24 4.29 4.35 4.41 4.46 4.50 4.63 4.57 4.60 4.63 4.66 4.69 4.71 4.73 4.76 4.78 26 4.26 4.32 4.37 4.42 4.48 4.50 4.53 4.66 4.59 4.62 4.65 4.67 4.69 4.71 4.73 26 4.23 4.29 4.34 4.39 4.43 4.46 4.50 4.53 4.56 4.58 4.61 4.63 4.66 4.68 4.70 27 4.20 4.26 4.31 4.36 4.40 4.43 4.47 4.60 4.63 4.65 4.58 4.60 4.62 4.64 4.66 28 4.17 4.25 4.28 4.33 4.37 4.40 4.44 4.47 4.60 4.52 4.S5 4.57 4.59 4.61 4.63 29 4.15 4.21 4.26 4.30 4.34 4.38 4.41 4.44 4.47 4.49 4.52 4.54 4.56 4.58 4.60 30 4.13 4.19 4.24 4.28 4.32 4.35 4.39 4.42 4.44 4.47 4.49 4.61 4.63 4.65 4.57 31 4.11 4.17 4.22 4.26 4.30 4.33 4.86 4.39 4.42 4.44 4.47 4.49 4.51 4.53 4.55 32 4.09 4.15 4.20 4.24 4.28 4.31 4.34 4.37 4.40 4.42 4.45 4.47 4.49 4.51 4.52 33 4.08 4.13 4.18 4.22 4.26 4.29 4.32 4.35 4.38 4.40 4.42 4.45 4.47 4.48 4.50 34 4.06 4.11 4.16 4.20 4.24 4.27 4.30 4.33 4.36 4.38 4.40 4.43 4.45 4.46 4.48 35 4.04 4.10 4.16 4.19 4.22 4.26 4.29 4.31 4.34 4.36 4.39 4.41 4.43 4.45 4.46 36 4.03 4.08 4.13 4.17 4.21 4.24 4.27 4.30 4.32 4.35 4.37 4.39 4.41 4.43 4.45 37 4.02 4.07 4.12 4.16 4.19 4.23 4.26 4.28 4.31 4.33 4.35 4.37 4.39 4.41 4.43 38 4.00 4,06 4.10 4.14 4.18 4.21 4.24 4.27 4.29 4.32 4.34 4.36 4.38 4.40 4.41 39 3.99 4.06 4.09 4.13 4.17 4.20 4.23 4.26 4.28 4.30 4.32 4.34 4.36 4.38 4.40 40 3.98 4.03 4.08 4.12 4.15 4.19 4.41 4.24 4.27 4.29 4.31 4.33 4.36 4.37 4.38 41 3.97 4.02 4.07 4.11 4.14 447 4.20 4.23 4.25 4.28 4.30 4.32 4.34 4.35 4.37 42 3.96 4.01 4.06 4.10 4.15 4.16 4.19 4.22 4.24 4.26 4.29 4.31 4.32 4.34 

### 4.36 43 3.96 4.00 4.05 4.09 4.12 4.15 4.18 4.51 4.23 4.25 4.27 4.29 4.31 4.33 4.35 44 3.94 3.99 4.04 4.08 4.11 4.14 4.17 4.20 4.22 4.24 4.26 4.28 4.30 4.32 4.33 45 3.93 3.98 4.03 4.07 4.10 4.13 4.16 4.19 4.21 4.23 4.26 4.27 4.29 4.31 4.32 46 3.93 3.98 4.02 4.06 4.09 4.12 4.16 4.18 4.20 4.22 4.24 4.28 4.28 4.30 4.31 47 3.92 3.97 4.01 4.06 4.08 4.11 4.14 4.17 4.19 4.21 4.23 4.25 4-27 4.29 4.30 48 3.91 3.96 4.00 4.04 4.07 4.11 4.13 4.16 4.18 4.20 4.22 4.24 4.26 4.28 4.29 49 3.90 3.95 4.00 4.03 4.07 4.10 4.12 4.16 4.17 4.20 4.22 4.23 4.25 4.27 4.29 5o 3.90 3.96 3.99 4.03 4.06 4.09 4.12 4.14 4.17 4.19 4.21 4.23 4.24 4.26 4.28 60 3.84 3.89 3.93 3.97 4.00 4.03 4.05 4.08 4.10 4.12 4.14 4.16 4.18 4.19 4. 21 70 3.80 3.86 3.89 3.93 3.96 3.98 4.01 4.03 4.06 4.08 4.09 4.11 4.13 4.14 4.16 80 3.78 3.82 3.86 3.89 3.93 3.95 3.98 4.00 4.02 4.04 4.06 4.08 4.09 4.11 4.12 90 3.75 3.80 3.84 3.87 3.90 3.93 3.95 3.98 4.00 4.02 4.03 4.05 4.07 4.08 4.10 100 3.74 3.78 3.82 3.86 3.88 3.91 3.93 3.96 3.98 4.00 4.01 4.03 4.05 4.06 4.08


### F:ttor

# =f(n-1,1-a/1R)\/1+ 11/1



# s2

# = a z. Howcver, when multiplc upgradient wells are evaluated on several quarters, the previous equation yields biased results unless we make the demonstrably false assumption that temporal and spatial variability are identical. 'I'he unbiased variance estimate for the case of multiple upgradient wells and multiple samplings of each well (selected at sufficicnt intervals to insure independence, for example, quarterly sampling) can be obtained froni the

# random effects xi1-μ+wt+T I

## +Ei1 

# where

# x;jis the measurement on background (upgradient) well

# ion sampling event

# j; pis the unknown mean level for background measurements (i. e., averaged across wells and time); w 1 is a random variable distributed N(0,

# a',,)that describes the n 

# analysis of variance model [see Winer (1971), L

# (xr -ac)z/(n - 1) is an unbiased estiniatc of page 427, for a similar examplc] . l iere, we are i=1

. 

# modeling thc background measurernents in terms of two random factors corresponding to wells and time-points. 'I'he general form of this model is given by :

462

# deviation of well i fromμ; rj

# is a random variable

distributed N(0,o; )

# that describes the deviation attime j from μ; and Frj is a random residual distributed 1V(0, vE). Assuming independence of thex=j, 

# the expected value of a measurement is:

E(xij)=μ 

# with varianceV(xij) =QW + O

# ~+ QE 

# 1'his model corresporids to what is termed a randomized bdock designin the statistical literature,and the expected mean square for the wells, timepoints, and residuals are given, respectively, as:

MSW

# = tow + oF

w 

# _ t £ (

~

# 'i

# - i)2/(u
,_ 1) i=f


## MST= woT+ QE



# - z~

# 2; (acj- ac)2l(t - 1) 

# j=1

~~



## w t

_ _ £ £ (xij — xi — xj + x)2l (W —~

## ) (t — .i* ) 

# i-1 j=1


# wlherexij

# is the measurement for welli

# on occasion j; w

# is the number of wells;t

# is the number of quarterly measurements per well; ac; is the niean for all measurernents from welli; acj

# is the mean for all nleasurements from tirne j; and x is the mean of all measurements across wells and tiines. Solving for the individual variance components Q2,, , oT ,

# and o

~, we find that :

2

# - MSw- MSE

~W


## t

2

# - MSr-1WSE

W7 ~


# and o f = MSC

The unbiased estimate of the total variance is therefore:
V(xtj) _'QGJ+pT +QF



# =MSw- MSE+



# Msr - MSE+ mS t w E When the time-points corresponding to the differentwells cannot be assumed to be equal, then the timepoint factor is said to be "nested" within the well factor [see Winer (1971), page 360, for a detailed description of nested factor designs) . This would occur, for instance, if the background observations for well 1 were made quarterly in 1.972, while the obscrvations for well 2 were made quarterly in 1976, and those for well 3 were made in 1975, etc. '1'his nested model is given by:

xzj = /,( + c.Wt

# + Trj+ EiJ 

# where xzj, A. wI, andE.ijare as before, whileTijis a random variable disrributedN'(0, QT)

# that describes the deviation of well i at time j fronl the overall mean value for well z(μ -• w i) averaged over time. If satnpling events are "nested" within wells, the expected mean squares must be nrtodified to reflect this design. Specifically, the expected mean square for the wells is: 

# mSW= tnW+ Qr

+ v

# ~

zu

~

## t


# while the mean square for time is:

MSr = QT + oE

w t 

# _ £Z (.'aCtj — ,XI )2/

# [w{t A i=1 j

~

# I 

# With tirne ncstcct within wc:lls, the error term QE cannot be scparately cstimatcd since there is only one observatiotn per well x tiiile cell. "i'hus, solving for the individual variance components 02; and

# aT ,we f ind r.hat :

AISW

# ._ MST

t 

# and v7+ v£

=

~

# YIST
I'he tinbiased estimatc: of the total variance is now:

Qw+ oT + Qc 

# MSw- MST

= t + :ti'S7


# w(xp — x)2 z

~

# : t

# (x=j — x}z

t

## ;

# ~1 (w

-- 1

# ) i

# ~I j

# ~I

## [w(t



## t 

# The above model assumes that all wells are rneasured at the same time-points, and thus, the two factors are referred to as "crossed" factors. w t (xij '" xi)

, iz i j= t t zv (t -

# 01

463

# It is of considerable interest to compare the

biased variance estimate s2with the unbiased variance cstimate

# V(xtj )given above. The simple variance estimate that assumes equal spatial and temporal variability has expcctation (the author is grateful to an anonymous reviewer for raising this 

# point):

w t



# F. [ EE

# (xtj - xi)ll(n - 1)) i=1 j=1



# n-t

= O

# ~+O7

+ Qf

' 1 ~ 

# wherc n is the totai number of ineasurcments. '1'his cquation revcals that s2will always underestimate 

# V(xtj ).For cxample, with two wclls and four quarterly measurements, the contribution of the well variance Q2, is underestimated by 43%. '['he addition of two upgradient wells decreases this biasto 20%, and the addition of a second years background increases the bias only slightly to 2 3%. When multiple upgradient wells comprise the background saniple, the unbiased estimate

# V(x=j ) should

# alwaysbe uscd in place of s2. "fhe importance of using multiple upgradieint wells is dueboth to the large spatial variability that is comnionly observed for naturally occurring constituents of ground watcr and bccause it eliminates theconfound between spatial variability and contamination. In the casc of a single upgradicnt well, an upgradient versus downgradient differcncc may reflect either spatial variability or contamination (z.e., there is no estimate oi' Qt,, ). In c;ontrast, when multiplc upgradient wells are sampled, an estimateof sratial variability Qw is available, and the resulting prediction limit will provide for spatial cf'fects. A critical assumption, however, is that both spatial and temporal variability are the same for t.ipgradient and downgradient locations. 'Chis is a cornplex problcm, and the fcasibility of tllis assumption should always be cvaluatcd on a site-specific basis to hring into consideration hydrogeology as well as problems with previous conditions and sources of off-site contamination.

5. ILLUSTRATIaN 

# Considcr thc following data for pli obtained from two upgradient wells eac:h with four quarterly ,measurements.
Quarter Upgradient well 1 Upgradientwell 2


## 1 7.1 7.7 2 7.4 7.8 

## 3 7.7 7.9 4 8.0 8.0 

# The summary statistics for the nested design
are: ac= (7.1+7.4+7.7+8.0+7.7+7.8+7.9+8.0)/8=7.70 

# xwelll = (7.1 + 7.4 + 7.7 + 8.0)/4 = 7.55 xwc112 =(7.7 + 7.8 + 7.9 + 8.0)/4 = 7.85

M.SW= 4 [(7.5 5-7.70)2+(7.85-7.70)Z] /(2-1)=0.180 

# 1lgST=[(7.1--7.55)2+(7.4-7.55)1+(7.7-7.55)2+ (8.0--7.55)1+(7.7-7.85)2+(7.8

-7.85)2+ (7.9- 7.85)2+(8.0 - 7.85)a] /(2(4- 1)]

~

## 0.08 3 

# The resulting variance estimates are thereforc::

Z

# Msw - MST Qw= 

# t = (.180 -- .08 3 )/4 = 0.024



# o2=MST=0.083 V (x,j ) =0.108 

# As expectcd, the biased variancc estintate


# s2

-

# .097 underestima.tesV(xtj)

# since, in this

s2= 4/7u

~

2

# ,,+ vT + aE .

# The variance example, estimate for the randomired blocks design 

# V(xij )=.120 was somewhat larger than the result obtained for the nested model, because a unique estimate of the re5idual error is available 

# (ow= .037,

# QT= .050, and QE = .033). 

# In ternls of establishing prediction intervals for future observations, we obtain thc.

~

# following estiniates from Tables 4-6 assuming the more general nested sampling design (Z.e., the time-points 

# are not the same for all wells). Number ef downgradient wells Prediction interval

1 7.70 f 3.67(.108) =6.49 - 8.91 2 7.70 i 4.20(.108)=6.32 - 9.08 3 7.70 t 4.52(.108) =6.22 - 9.19 

## 4 7.70 t 4.75 (108)=6.13 - 9.26 5 7.70 f 4.94(V.108)=6.08 - 9.32 6 7.70 f 5.09(V_ .108)=6.03 - 9.37 7 7.70 -.t- 5.22(V_ .108)=5.98 - 9.42 8 7.70 ± 5.34(v'108)-5.94 - 9.46 9 7.70 f 5.44 (108)=5.91 - 9.49 10 7.70 f 5.53 (.108)=5.88 - 9.52 20 7.70 f 6.15(.108)=5.68 - 9.72 30 7.70 1 6.53 ( 108)=5.55 - 9.85 40 7.70 ± 6. 81(.108 )--5. 46 - 9. 94 50 7.70 ± 7.02(V_._1O9)=5.39 - 10.01 100 7.70 f 7.72(.108)=5.16 - 10.24


# 464

# With a doubling of sample size from 8 to 16 (e.g.,

four upgradient wells with quarterly sampling for one year), the prediction intervals would range from 6.70 to 8.70 for a single downgradient well to 5.94 to 9.46 for a site with 100 downgradient wells, assuming that

# xand

# V(x;j )remained constant.

6. DISCUSSlON 

# Although the statistical monitoring strategy developed here may appear comprehensive, it is only a small first step. For example, none of the methodology presented here is directly appropriate for the analysis of indicator compounds that exhibit values below method detection iimits, or go undetected in a proportion of the sample. Perhaps, the simplest solution to this problem is to use the prediction limits developed here with estimates ofthe mean and variance of a censored normal distribution (Cohen, 1961) or censored lognormal distribution (Aitchison, 1955). Using these procedures, however, it is not at all clear how unbiased variance estimates would be obtained, as aredescribed here for the case of complcte information. As an even more extreme omission, the strategy presented here is completely inappropriate for the analysis of volatile organic compounds that occur in less than 5% of all measurements obtained from clean upgradient wells, trip blanks, and field blanks (Plumb and Parolini, 1986; Hurd, 1987). Inspections of these potentially useful indicator compounds suggest that they may arise from a Poisson process (Gibbons, 1987) or possibly even a censored Poisson distribution. Statistical methods for monitoring infrequcntly detected compounds,such as the volatile organic priority pollutant compounds are sorely needed in ground-water monitoring, and statistical research in this area is strongly encouraged.

ACKNODULEDGMENTS Z'he work reported herein was funded by a grant from Waste Management lncorporated. Waste Management also provided access to an extensive national data base that proved to be critical in the development and illustration of these statistical methods. I am indebted to Donald I iecdekcr of theUniversity of Chicago for his excellent cornputa

# tional assistance and for reviewing the manuscript. I also wish to thank an anonyrnous reviewer for clarifying a number of points in an earlier draft. REFERENCES Aitchison, ,l . 1955. On the distribution of a positive random variable having a discrete probability mass at the origin. Journal of the American Statistical Association, v. 50, pp. 901-908. Anderson, T. W. 1958. An Introduction to Multivariate Statistical Analysis. Wiley, New York. Aroian, L. A. 1941. A study of Fisher's z distribution and the related F distribution. Ann. Math. Statist. v. 12, pp. 429-448.I3ox, G.E.P. and D. R. Cox. 1964. An analysis of transforrnations (with discussion.) J. Roy. Statist. Soc. Ser. B. v. 26, pp. 211-252.Chcw, V. 1968. Simultaneous prediction intervals. 1'echnornetrics. v. 10, pp. 323-331. Cohen, A. C. 1961. Tables for maximum likelihood 

## estimates from singly truncated and singly censored samples. Technometrics. v. 3, pp. 5 35-541. Dahiya, R. C. and I. Guttman. 1982. Shortest confidence and prediction intervals for the log-normal. The Canadian Journal of Statistics. v. 10, pp. 277-291. Gibbons, R. D. 1987. Statistical models for the analysis of rare volatile organic compounds in waste disposal sites. Ground Water. ln press.Hahn, G. J. 1969. Facturs for calculating two-sided prediction intervals for samples from a normal distribution. Journal of the American Statistical Association. v. 64, pp. 878-888.tiahn, G. J. 1970. Additional factors for calculating prediction intervals for samplcs from a normal distribution. }ournal of the Amcrican Statistical Association. v. 65, pp. 1668-1676. 

## Hurd, M. 1987. Determining the impact of land disposal: The review of organic analytical data. Unpublished manuscript, Environmental 1'esting and Certification Corp. 

## ,tililler, R. G. 1966. Simultaneous Statistical Inferenee. Springer-Verlag, New York. 

## Plumb, R. H. and J. R. Parolini. 1986. Urgailic contamination of groundwater near hazardous waste disposal sites. 'I'echnical Report: EPA Las Vegas Laboratory. Snedecor, G, W. and W. G. Cochran. 1980. Statistical Methods: Seventh Edition. iowa State University Press, Ames, Iowa. Wilks, S. S. 1941. Determination of sample sizes for setting tolerance limits. Ann. Math. Statist. v. 12, pp 91-96. Winer, B. J. 1971. Statistieal Principles in Experimental 

## Design. Second E

~

## ;dition. McGraw

-

## Hill, New York.

Robert D. Gibbons received bis Pb,U. in Statistics from the University of Chicagoin1981, andiscurrently an Associate Professorof Biostatistics atthe University of lllinois. !n 1985, he received a YoungScientistAward from the C)f'fice of rVaval Researcb for studies in Theoretical Statistics.

## A mcijor area of his applied statistical work over the past f our years bas been related to problems irr the statistical

## monitoring of ground water at hazardous waste disposal facilities.



## 465