Environmental and Ecological Statistics 8, 91±107, 2001
G L E N D . J O H N S O N * and G A N A PAT I P. PAT I L
Center for Statistical Ecology and Environmental Statistics, Department of Statistics,
Pennsylvania State University, University Park, PA 16802
Received October 1998; Revised September 1999
When an environmental sampling objective is to classify all the sample units as contaminated or not, composite sampling with selective retesting can substantially reduce costs by reducing the number of units that require direct analysis. The tradeoff, however, is increased complexity that has its own hidden costs. For this reason, we propose a model for assessing the relative cost, expressed as the ratio of total expected cost with compositing to total expected cost without compositing (initial exhaustive testing). Expressions are derived for the following retesting protocols: (i) exhaustive, (ii) sequential and (iii) binary split. The effects of both false positive and false negative rates are also derived and incorporated.
The derived expressions of relative cost are illustrated for a range of values for various cost components that re¯ect typical costs incurred with hazardous waste site monitoring. Results allow those who are designing sampling plans to evaluate if any of these compositing/retesting protocols will be cost effective for particular applications.
Keywords : cost-effectiveness, environmental sampling, human health sampling, observational economy
1352-8505 # 2001 Kluwer Academic Publishers
When designing sampling plans for environmental and public health monitoring, the most expensive component is often quanti®cation of individual sample units, such as when the response variable may be a chemical concentration or presence of a pathogen that requires expensive laboratory analysis. With increasing demands for data of defensible quality, we are faced with a substantial societal expenditure. Actually the United States
Environmental Protection Agency (EPA) estimates that about $5 billion is spent annually by the EPA and the regulated community collecting data for research, regulatory decision making and regulatory compliance (USEPA, 1994).
Composite sampling, which is supported by the EPA (USEPA, 1985a, 1985b, 1989,
1994, 1995), offers a way to obtain desired information while staying within constraints of an analytical budget. Such economy is obtained by analyzing n composites of m individual
*Author for correspondence.
1352-8505 # 2001 Kluwer Academic Publishers

92 Johnson, Patil sample units, such that n 5 m , in order to reduce the number of expensive analyses. The theory and methods for estimation and inference with composite measurements have been developing for some time, resulting in extensive reviews of this topic for environmental monitoring (Gilbert, 1987; Lovison, Gore and Patil, 1994).
Compositing can usually reduce the number of required tests when estimating population parameters such as the mean (Edland and van Belle, 1994) or variance
(Fabrizio, Frank and Savino, 1995); however, many applications call for the classi®cation of each sample unit. Examples include the detection of pathogens in blood or water, characterizing the spatial distribution of contamination at a waste site or identifying ``hot spot'' locations. In these situations, compositing can be taken much further by infering values of the individual sample units comprising a composite based on initial composite sample measurements and selective retesting of individual sample units. A judiciously chosen retesting protocol can result in classi®cation of all the original individual sample units without having to perform an expensive test on each one.
While the various retesting protocols may reduce the number of required tests, they may involve costs that do not arise with exhaustive testing. For example, forming a composite of several soil samples may require mixing in a stainless steel container which has to be thoroughly washed and solvent rinsed between each use. Also, archiving aliquots of the individual samples for possible retesting might require extra refrigerator space. Then, of course, any extra labor costs must be factored in when deciding if a composite/retesting plan is truly cost-effective.
In this paper, a general cost function is developed for compositing with selective retesting for classifying all individual sample units and this function is applied to various retesting protocols for determining the conditions when a given retesting protocol is truly cost-effective.
Let us consider the whole process of obtaining the necessary data for classifying m distinct sample units, using composite sampling with selective retesting. We would adhere to the following steps:
1. Obtain m sample units, such as soil cores.
2. Obtain an aliquot from each sample unit, then archive the remaining material for each of the m units for possible retesting.
3. Form n composite samples, each consisting of aliquots of k original sample units, such that n ˆ m = k , rounded to the lowest integer.
4. Analyze the n composites.
5. Classify each composite as ``clean'' or ``contaminated''. If measuring a binary response ( presence/absence of a trait), then all sample units comprising a composite are classi®ed as clean (or negative) if the composite does not reveal the trait. If measuring a continuous response variable, then each original sample unit is classi®ed as ``clean'' if it's measured value is less than a numerical criterion, c .
Therefore, the composite sample measurement, y , is compared to c / k toavoid false negatives that can arise from diluting contaminated individual sample units with clean ones when forming composites. If y 5 c = k , then all units comprising the

Cost analysis of composite sampling 93 composite are classi®ed as clean. Whether measuring a binary or continuous random variable, if the composite is not classi®ed as clean, then retesting must be done on individual sample units to identify which ones are contaminated.
6. Retest the archived original sample units as necessary, based on the results in step 5.
Now all original m sample units are classi®ed.
The retesting procedures can vary and have been the subject of investigation by many researchers over time, as summarized by Lovison, Gore and Patil (1994). Basically, more statistically ef®cient retesting procedures go hand-in-hand with increased complexity.
Therefore, in order to more accurately evaluate the true cost effectiveness, one must incorporate all of the costs involved with the protocol listed above. It is not enough to just compare the expected number of tests required with a compositing scheme to the number of tests required with initial exhaustive testing.
An expression of the relative cost … RC † for classifying m sample units as
``contaminated'' or not is proposed as:
RC ˆ cost of compositing with retesting cost of initial exhaustive testing
ˆ m … C s
‡ C a
† ‡ n … C c m … C s
‡
E ‰ Y k
C t
†
Š ‡ C t
E ‰ T k
Š†
1 † where
C s
C a
C c
E ‰ Y k
E ‰ T k m ˆ the number of individual sample units that must be classi®ed, n ˆ the number of composite sample units,
ˆ the cost of obtaining an individual sample from the ®eld or a specimen,
ˆ the cost of archiving an individual sample unit,
ˆ the cost of forming a composite sample, which is treated here as constant with respect tothe number of individual sample units used tocomprise
C t the composite,
ˆ the cost of testing a sample, which is the same cost whether we are testing an individual sample unit or a composite,
Š ˆ the expected number of composites to be formed, and
Š ˆ the expected number of tests to be done.
Equation (1) can be simpli®ed as
ˆ
C s
‡ C a
† ‡ 1 k
C s
C c
E ‰ Y
‡ C t
† k
Š ‡ C t
E ‰ T k
Š†
: … 2 †
Testing mechanisms, such as chromatographic or immunoassay techniques, are typically subject to some degree of error. In the case of monitoring for binary classi®cation, an error results in either a false positive or a false negative test. A false positive or false negative rate, denoted here as r p
, respectively, are actually statements of conditional probability such that and r n

94 Johnson, Patil r p
ˆ Pr … positive test result j sample is negative † and r n
ˆ Pr … negative test result j sample is positive † ; where Pr … ?
† means ``probability of … ?
† ''. The probability of a sample correctly testing negative or correctly testing positive is thus 1 r p and 1 r n
, respectively.
Composite sampling with retesting will actually reduce the overall false positive rate, which is obviously a very desirable property. A tradeoff, however, is that the false negative rate can be magni®ed with composite testing. For this reason, an investigator may wish to retest negative testing composites in order to control the false negative rate. Although this would increase the expected number of tests for a composite sampling protocol, one must consider that a signi®cant false negative rate would likewise cause an investigator to retest negative results when doing simple exhaustive testing.
For ®xed cost components, C s
; C a
®nd E ‰ Y k
Š and E ‰ T k
; C c and C t
, and composite sample size, k , we need to
Š for whichever retesting protocol we are considering. In the following sections, these expectations are provided, thus allowing construction of relative cost functions. For those situations where r p the overall design false positive rate, d p and r n cannot be ignored, methods for quantifying
, and false negative rate, d n
, are alsoderived.
This section considers the measurement of a binary response such as the presence or absence of a pathogen in drinking water or blood. Assuming that all initial m measurements are independent and identically distributed (iid) with probability p of testing positive, this case becomes one of testing a Bernoulli random variable with parameter p .
4.1
This is the simplest protocol, and was originally proposed in formal publication by
Dorfman (1943) when testing servicemen for syphilis by compositing aliquots of blood samples and testing the composite. If the composite sample tests negative, then all donors to that composite are classi®ed as negative. Meanwhile, if a composite tests positive, then aliquots of each sample which comprised that composite are retested to locate which donors are infected.
Since exhaustive retesting does not result in re-formation of composites, then E ‰ Y k
Š ˆ 1.
Meanwhile, the number of tests will either be 1 or k ‡ 1. Letting q ˆ 1 p , where p ˆ the probability of the trait in question occurring in any individual sample, the expected number of tests is described as:
E ‰ T k
Š ˆ 1 ?
q k ‡ … k ‡ 1 † ?
1 q k †
ˆ 1 ‡ k … 1 q k † : … 3 †
With exhaustive retesting, a false negative can occur when testing the initial composite

Cost analysis of composite sampling 95 or, if a positive test is obtained, a false negative occurs when retesting an individual sample. The overall design false negative rate therefore equals: d n
ˆ r n
‡ … 1 r n
† r n
ˆ 2 r n r n
2 : … 4 †
A false positive occurs when a composite tests positive, thus invoking retesting, and a retested individual sample falsely tests positive. The overall design false positive rate is expressed as follows.
d p
ˆ Pr … retest † r p
ˆ
"
Pr … retest j at least one out of k 1 is positive †… 1 q k 1
‡ Pr … retest j all k are negative †… q k 1 †
†
# r p
ˆ ‰… 1 r n
†… 1 q k 1 † ‡ r p q k 1 Š r p
ˆ r p
‰ 1 r n q k 1 … 1 r n r p
†Š : … 5 †
False positive and false negative rates can affect the expected number of tests, outcomes with exhaustive retesting. The probability of one test becomes
T k
, by affecting the probabilities of the different possible outcomes. Consider the two possible
Pr … T k
ˆ 1 † ˆ r n
1 q k † ‡ … 1 r p
† q k
ˆ r n
‡ q k … 1 r n r p
† ; and the probability of k ‡ 1 tests becomes
6 †
Pr … T k
ˆ k ‡ 1 † ˆ 1 Pr … T k
ˆ 1 †
ˆ 1 r n q k … 1 r n
Therefore the expected number of tests becomes r p
† :
E ‰ T k
Š ˆ k ‡ 1 k … r n
‡ q k … 1 r n r p
†† ; which can be used in Equation (2), along with E ‰ Y k
Š ˆ 1, toexpress relative cost.
7
8
†
†
4.2
The next alternative considered here is sequential retesting, as described by Sterrett (1957) and diagrammed in Fig. 1. For this protocol, if a composite tests positive, then the individual sample units that comprised the composite are tested until one tests positive.
The remaining untested individual units are then composited and tested. If the new subcomposite tests positive, the procedure is repeated for its contributors, whereas any time a composite tests negative, all of its contributors are declared ``clean'' and no new testing is performed.

96 Johnson, Patil
Figure 1.
Flow chart depicting the sequential retesting protocol for a single composite sample of size k .
The expected number of tests with sequential retesting has been shown elsewhere
(Lovison, Gore and Patil, 1994) to be:
E ‰ T k
Š ˆ 2 k … k 3 † q q 2
1 q k ‡ 1 p
; k ˆ 2 ; 3 ; . . .
: … 9 †
The expected number of composite samples that will need to be formed can be shown
(see Appendix) tobe:
E ‰ Y k
Š ˆ 1 ‡ … k 2 † p ; k ˆ 3 ; 4 ; . . .
: … 10 †
False negatives may occur when testing the initial composite or, if a positive test is obtained, a false negative may occur when either retesting an individual sample or a subcomposite the individual sample is a member of. The design false negative rate, d n
, is therefore the same as with exhaustive retesting as expressed in Equation (4).
The probability of a false positive decreases as the number of subcomposites that are tested increases. A simple and conservative way to overcome the complexity of calculating the overall design false positive rate, d p
, is toevaluate the worst case, which is equivalent toexhaustive retesting as expressed in Equation (5).

Cost analysis of composite sampling 97
The effects of false positive and false negative rates on the expected number of tests and composites are, respectively,
E ‰ T k
Š ˆ 2 k … k 3 †… r n p ‡ … 1 r p
† q † … r n p ‡ … 1 r p
† q † 2
1 … r n r p p ‡ … 1 r q ‡ … 1 r n p
†
† p q † k ‡ 1
; for k ˆ 2 ; 3 ; . . .
; and
11 †
E ‰ Y k
Š ˆ 1 ‡ … k 2 †… r p q ‡ … 1 r n
† p † for k ˆ 3 ; 4 ; . . .
; which can be used in Equation (2) for the relative cost expression.
12 †
4.3
Another alternative is binary split retesting, as proposed by Gill and Gottlieb (1974) and diagrammed in Fig. 2. For this protocol, a positive testing composite is divided into subcomposites of nearly equal size as possible, then each positive testing subcomposite is recursively divided and tested until all individual samples have been classi®ed. The advantage of binary split retesting is that one can reduce the number of tests by avoiding exhaustive retesting of all individual samples when a composite tests positive. With
Dorfman's protocol, one is committed to exhaustive retesting when a composite tests positive and with Sterrett's protocol, one can potentially retest all individual samples. The disadvantage is that Gill and Gottlieb's protocol is somewhat more complex and if the prevalence of a trait is high enough, the total number of tests can be much higher than with exhaustive or sequential retesting.
The expected number of tests with binary split retesting has been shown elsewhere
(Lovison, Gore and Patil, 1994) to be:
E ‰ T k
Š ˆ E ‰ T k
1
Š ‡ E ‰ T k
2
Š ‡ 1 2 q k ; k ˆ 2 ; 3 ; . . .
and the expected number of composites can be shown (see Appendix) to be:
13 †
E ‰ Y k
Š ˆ E ‰ Y k
1
Š ‡ E ‰ Y k
2
Š ‡ 1 2 q k ; k ˆ 4 ; 5 ; . . .
14 † where k
1
ˆ k
2
ˆ k = 2 if k is even and k
1
ˆ … k 1 † = 2 and k
2 k ˆ 3, either one composite is formed with probability q 3 with probability 1 q 3 , which implies that
ˆ … k ‡ 1 † = 2 if k is odd. For or two composites are formed
E ‰ Y
3
Š ˆ q 3 ‡ 2 … 1 q 3 †
ˆ 2 q 3 :
A false positive occurs when the initial k -size composite tests positive, and all subsequent smaller-size subcomposites … k
1
4 k
11
4 † and the resulting individual sample all falsely test positive. The design false positive rate can then be expressed as

98 Johnson, Patil
Figure 2.
Flow chart depicting the binary split retesting protocol for a single composite sample of size k .
d p
ˆ Pr … retesting composite of size k †
6 Pr … retesting composite of size k
1
†
6 Pr … retest composite of size k
11
† … r p
†
ˆ ‰… 1 r n
†… 1 q k † ‡ r p q k Š‰… 1 r n
†… 1 q k
1
† ‡ r p q k
1
Š
6 ‰… 1 r n
†… 1 q k
11
† ‡ r p q k
11
Š … r p
† ; … 15 † where k is split into sub-composites of size k
1 of size k
11 decreases d p
) is tolet k
ˆ … k p equals k p
1 † = 2.
c
, which are in turn split into sub-composites
, etc. At any stage of splitting, for any ``parent'' composite of size
``child'' composite of size k approach (that increases d p
) is tolet k c c
= 2 if k
ˆ … k p p is even. If k p
‡ 1 † = k p
, the is odd, then a conservative
2, whereas a liberal approach (that
A false negative can occur when testing the initial k -size composite or any subcomposite thereof or a resulting individual sample falsely tests negative. The design false negative rate, d n
, for an initial k -size composite can be expressed as

Cost analysis of composite sampling 99 d n
ˆ r n or r ‡ … d n
† k = 2 if k is even ;
ˆ r n r ‡ … d n
† k ‡ 1
2 if k is odd ; … 16 † where the subscripts of d size k n re¯ect the subcomposite sample sizes. For any initial composite
, the overall design false negative rate, d n
, is solved for by iterative application of
Equation (16).
The expected number of tests and composites are, respectively, affected in the following way.
E ‰ T k
Š ˆ E ‰ T k
1
Š ‡ E ‰ T k
2
Š ‡ 1 2 …… 1 r p
† q ‡ r n p † k
; for k ˆ 2 ; 3 ; . . .
; and
17 †
E ‰ Y k
Š ˆ E ‰ Y k
1
Š ‡ E ‰ Y k
2
Š ‡ 1 2 …… 1 r p
† q ‡ r n p † k ; … 18 † for k ˆ 4 ; 5 ; . . .
. Equations (17) and (18) can then be used recursively for any value of k in
Equation (2) for expressing relative cost.
The discussion thus far has been based on testing a binary random variable, such as the presence/absence of a trait. This section extends the theory to measurement of nonnegative continuous random variables, such as the concentration (mass/mass or mass/ volume) of a chemical contaminant in some environmental media.
With a continuous random variable, a sample unit is classi®ed as negative if its measured value is less than some numerical criterion, such as an action level or cleanup standard. However, a k -point composite can not be classi®ed as negative simply because its measured value is less than the criterion because this observation alone does not guarantee that each contributor to the composite is less than the criterion. The only way to assure that all the individual values, x i for i ˆ 1 ; . . .
; k , that contributed to the measured composite value, y , are less than the criterion, c , is to classify a composite as negative only when y 5 c = k . Therefore, the probability of declaring that a composite sample is negative equals q k
ˆ Pr … X
1
‡ X
2
‡ ‡ X k
5 c † : … 19 †
X
1
The only way to compute such a probability is to know the distribution of
‡ X ‡ ‡ X k
, which is the k -fold convolution of the distribution of X if all the X i are independent. Therefore, q
Since the event then
2
‰ X
1
‡ X
2 k depends on k , c and the distribution of X .
‡ ‡ X k
5 c Š is contained in the event ‰ X
1
5 c ; . . .
; X k
5 c Š , q k
ˆ Pr … X
1
‡ ‡ X k
5 c †
ˆ Pr … X
1
‡ ‡ X k
5 c j X
1
5 c ; . . .
; X k
5 c † Pr … X
1
5 c ; . . .
; X k
5 c †
ˆ Pr … X
1
‡ ‡ X k
5 c j X
1
5 c ; . . .
; X k
5 c † q k : … 20 †

100 Johnson, Patil
Equation (20) implies that q k q k ; therefore, the relative cost for measuring a continuous random variable will always have a lower bound that is equivalent to the relative cost obtained when measuring a binary random variable.
Calculating the expected number of composites and tests is mainly limited by dependence on the underlying distribution of the characteristic of interest, X . Since the distribution is typically not known, relative cost can be approximated for design purposes using the expressions for a ``presence/absence'' response variable.
On one hand, this leads to over-optimistic results since the probability of declaring a sizek composite as negative has an upper bound equal to the probability of declaring the same size composite as negative when measuring a binary ``presence/absence'' random variable (as shown by Equation (20)). On the other hand, however, the ``presence/ absence'' expressions are based on complete random mixing of individual sample units when forming composites which is less cost effective than when composites are purposely made as homogeneous as possible (the converse is true when compositing to estimate the mean). For example, when the spatial distribution of a contaminant in soil is highly autocorrelated, compositing spatially proximate soil cores increases the chances of mixing clean samples with clean ones and contaminated samples with contaminated ones. This results in more ef®cient retesting than when soil cores are grabbed at random from across the site for compositing (Boswell and Patil, 1990).
In order to visualize relative cost of the compositing/retesting protocols described in this paper, values for the cost functions were realized over ranges of the various cost components. Although these composite sampling protocols may be applied to a wide variety of applications, cost ranges were selected to re¯ect realistic costs encountered with hazardous waste site cleanups. These values were based on the ®rst author's experience and solicitation of current prices from the United States Environmental Protection Agency for SUPERFUND contract laboratories, the Pennsylvania Department of Environmental
Protection for the Pennsylvania Hazardous Site Cleanup Program, a private laboratory and a remediation company.
Since we did not have typical false positive or false negative rates for this illustration, we present the situation where values for r p and r n are assumed tobe negligible and are therefore set equal to zero. To evaluate multiple hypothetical values for each of r p and r n would substantially complicate this illustration anyhow.
For simpli®cation, all input costs are expressed relative to the cost of obtaining one sample unit, C used were C c
= s
, which in turn is set equal to one monetary unit. The compositing costs
C s
ˆ 0 : 1, 0.5 and 1.0, and the testing costs were
1000. Furthermore, the cost of archiving, C
C t a usually factored into laboratory testing costs, C
= C s
ˆ 1, 10, 100 and
, was set equal tozerofor all cases since this is t
; however, archiving cost was left in the relative cost expressions given in this paper for any situation where a compositing/ retesting scheme requires substantially more effort for storing and managing samples.
The resulting closed-form relative cost expressions for the exhaustive and sequential retesting protocols are:

Cost analysis of composite sampling
RC exhaustive
ˆ
C s
‡ C a
† ‡ 1 k
C c
C s
‡ C t
‡ C t
1 ‡ k … 1 q k
†
††† and
101
21 †
RC sequential
ˆ
C s
‡ C a
† ‡ 1 k
C c
k 2 † p ‡ 1 † ‡ C t
2 k … k 3 † q q 2
C s
‡ C t
†
1 q k ‡ 1 p
††
:
22 †
Figure 3.
Relative cost versus prevalence when the ratio of compositing to sampling costs … Cc = Cs † equals 0.1. Various composite sample sizes … k † are presented, sorted by retesting protocol and relative testing cost.

102 Johnson, Patil
Figure 4.
Relative cost versus prevalence when the ratio of compositing to sampling costs … Cc = Cs † equals 0.5. Various composite sample sizes … k † are presented, sorted by retesting protocol and relative testing cost.
For the binary split retesting protocol, Equations (13) and (14) are applied recursively within Equation (2).
Results are seen in Figs. 3 to 5, where some key features include the following:
1. In almost all cases, compositing with selective retesting is cost effective for classifying all the individual sample units when the prevalence of the variable used for classi®cation is less than 10%. Often, the prevalence can be up to 30% and compositing is still cost effective. A consistent exception is when the cost of testing and the cost of forming a composite both equal the cost of obtaining a sample unit, such that C t
= C s
ˆ 1 and C c
= C s
ˆ 1, for k ˆ 2.

Cost analysis of composite sampling 103
Figure 5.
Relative cost versus prevalence when the ratio of compositing to sampling costs … Cc = Cs † equals 1. Various composite sample sizes … k † are presented, sorted by retesting protocol and relative testing cost.
2. The cost of compositing has a substantial effect when the cost of testing equals the cost of obtaining a sample unit, such that C t
= C s
3. Cost effectiveness of compositing improves noticeably as the cost of testing, relative to sampling cost, increases from C ˆ 1 to C t
ˆ 1; otherwise, the extra cost of compositing can essentially be ignored for the higher testing costs.
t
= C s
= C s
ˆ 10. As the cost of testing further increases, changes in cost effectiveness appear negligible.
4. Exhaustive retesting appears to be the least cost risky protocol as the true prevalence, p , increases. When combined with its simplicity, this makes exhaustive retesting quite appealing; however, as the composite sample size, k , increases, the expected relative cost associated with the other retesting protocols becomes more appealing.

104 Johnson, Patil
A very strong application for compositing with selective retesting for classi®cation is for post-remediation veri®cation sampling of soil at a hazardous waste site, since the probability of exceeding a cleanup criterion should be very low if a good cleanup was performed. One must consider, however, that when compositing very heterogeneous material like soil, measurement error should be absolutely minimized by thoroughly homogenizing samples, as shown in EPA guidance (USEPA, 1985b).
For initial site characterization, however, compositing should be considered only with great caution. If there is strong suspicion of a low prevalence of contamination, then compositing may be viable, but if the prevalence is thought to be high or if one has no reasonable idea of the prevalence rate, then compositing may be best avoided.
Compositing should also be avoided when testing for volatile compounds in soil since the act of mixing and homogenizing individual sample units will reduce the actual concentration of volatiles, thus inducing a strong sample preparation bias.
The equations presented in this paper should allow investigators to evaluate the feasibility of composite sampling to reduce overall costs of a study. When false positive and false negative rates are known and considered signi®cant, they can also be included in the evaluation. If these rates are known to vary in a predictable way, then a distribution of values for relative cost can be simulated, given ®xed cost components, using randomly generated false positive and false negative values. The Beta distribution, which de®nes the distribution of a random variable on the closed unit interval ‰ 0 ; 1 Š , would be a logical choice for de®ning an input distribution of false positive and false negative rates.
Further research may also address variance of the relative cost. It is fairly simple to ®nd the closed form expression of variance for the exhaustive retesting protocol; however, this becomes more complex with the other protocols.
Proof of Equation (10): First note at least one composite is formed with certainty. For the expected number of extra composites, let J ˆ the number of retests until the ®rst positive test is obtained, therefore implying that J is a pascal random variable. We can then
®nd the E ‰ Y k
Š by conditioning on J as follows.
1 ‡ E ‰ Y k
Š ˆ 1 ‡ E ‰ E ‰ Y k j J ŠŠ
ˆ 1 ‡ j ˆ 1
E ‰ Y k j J ˆ j Š Pr … J ˆ j †
ˆ 1 ‡
X 2
E ‰ Y k j
Š q j ˆ 1 j 1 p ; k ˆ 3 ; 4 ; . . .
:
Therefore,

Cost analysis of composite sampling
E ‰ Y k 1
Š ˆ 1 ‡ k X j ˆ 1
E ‰ Y k 1 j
Š q j 1 p
105
ˆ 1 ‡ j 0 ˆ 2
E ‰ Y k j 0
Š q j 0 2 p where j 0 ˆ j ‡ 1.
Now, note that qE ‰ Y
Therefore E ‰ Y k
Š qE ‰ Y k 1 k 1
Š ˆ q ‡
P j k 2
0 ˆ 2
E ‰ Y
Š ˆ 1 q ‡ E ‰ Y k j 0 k 1
Š q j 0 1 p .
Š p , which implies
E ‰ Y k
Š ˆ 1 q ‡ E ‰ Y k 1
Š… p ‡ q †
ˆ p ‡ E ‰ Y k 1
Š :
Therefore, by recursion,
E ‰ Y k
Š ˆ … k 2 † p ‡ E ‰ Y
2
Š :
Since E ‰ Y
2 proof.
Š ˆ 1, then E ‰ Y k
Š ˆ … k 2 † p ‡ 1 fo r k ˆ 3 ; 4 ; . . .
, thus completing the
&
Proof of Equation (14): Let the outcome of testing a composite sample of size k be represented by the variable S , where S ˆ 1 is the event that the composite tests positive and S ˆ 0 is the event that the composite tests negative. The expected number of composites to be formed, based on an initial composite sample size k , can be found by conditioning on S as follows:
E ‰ Y k
Š ˆ E ‰ E ‰ Y k j S ŠŠ
ˆ E ‰ Y k j S ˆ 1 Š Pr … S ˆ 1 † ‡ E ‰ Y k j S ˆ 0 Š Pr … S ˆ 0 †
ˆ … 1 ‡ … E ‰ Y k
1
Š ‡ E ‰ Y k
2
Šj S ˆ 1 †† Pr … S ˆ 1 † ‡ 1 ?
Pr … S ˆ 0 †
ˆ Pr … S ˆ 1 † ‡ E ‰ Y k
1
‡ Y k
2 j S ˆ 1 Š Pr … S ˆ 1 † ‡ Pr … S ˆ 0 †
ˆ 1 ‡ E ‰ Y k
1
‡ Y k
2 j S ˆ 1 Š Pr … S ˆ 1 † :
Now, in the same manner,
E ‰ Y k
1
‡ Y k
2
Š ˆ E ‰ Y k
1
‡ Y k
2 j S ˆ 1 Š Pr … S ˆ 1 †
‡ E ‰ Y k
1
‡ Y k
2 j S ˆ 0 Š Pr … S ˆ 0 †
ˆ E ‰ Y k
1
‡ Y k
2 j S ˆ 1 Š Pr … S ˆ 1 † ‡ 2 q k :
Therefore,
E ‰ Y k
Š ˆ 1 ‡ … E ‰ Y k
1
‡ Y k
2
Š 2 q k †
ˆ E ‰ Y k
1
Š ‡ E ‰ Y k
2
Š ‡ 1 2 q k ; k ˆ 4 ; 5 ; . . .
; thus completing the proof.
&

106 Johnson, Patil
We are grateful to the following people for their assistance in providing cost information:
Larry Reed and Angelo Carasea of the EPA's Hazardous Site Evaluation Division, Of®ce of Emergency and Remedial Response; Randy Roush of the Pennsylvania Department of
Environmental Ptoection's Hazardous Sites Cleanup Program, Bureau of Waste
Management; John Carson of OHM Remediation Co.; and Wilson Hershey of Lancaster
Laboratories. Ultimately, we are very grateful to our peer reviewers who helped substantially in improving the presentation of this work.
This research was partially supported by the Statistical Analysis and Computing Branch,
Environmental Statistics and Information Division, Of®ce of Policy, Planning, and
Evaluation, of the United States Environmental Protection Agency under a Cooperative
Agreement Number CR- 821531. The contents have not been subjected to Agency review and therefore do not necessarily re¯ect the views of the Agency and no of®cial endorsement should be inferred.
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Cost analysis of composite sampling 107
Glen Johnson has developed a multidisciplinary career in environmental science, whose focus has ranged from small scale issues of chemical toxicology and microbiology to large scale issues of geography, landscape ecology and remote sensing. Glen has a B.S. in
Biology from the State University of New York College of Environmental Science and
Forestry, along with a M.S. in Ecology, a M.A. in Statistics and a Ph.D. (1999) in
Quantitative Ecology, all from Penn State University. His applied endeavors include six years with the Pennsylvania Department of Environmental Protection, and private consulting in Philadelphia. Currently, Glen works for the New York State Department of
Health where he specializes in geographic and environmental epidemiology.
G.P. Patil is Distinguished Professor of Mathematical Statistics and Director of the
Center for Statistical Ecology and Environmental Statistics in the Department of Statistics at the Pennsylvania State University.