Procedures for Formation of Composite Samples from Segmented Populations

Procedures for Formation of Composite Samples from Segmented Populations M A R Y C . F A B R I Z I O , * A N T H O N Y M . F R A N K , A N D J A C Q U E L I N E F . S A V I N 0

National Biological Service, Great Lakes Science Center, 1451 Green Road, Ann Arbor, Michigan 48105

We used a simulation approach to investigate the implication of two methods of forming composite samples to characterize segmented populations. We illustrate the case where the weight of individual segments varies randomly, a situation common with fish samples. Composite samples from segments such as whole fish or muscle tissue should be formed by homogenizing each segment separately and combining equal- sized portions randomly drawn from each homogenate. This approach permits unbiased estimation of the mean concentration per fish. Estimates of mean contaminant concentration varied little with variation in the number of composite samples analyzed or with composite size (number of segments in a composite sample). However, for a fixed number of composite samples, the precision of the variance estimate increased as composite size increased. In addition, for a fixed number of composites, the estimate of the variance stabilized as more segments were included in the composite samples. Estimates of the variance among fish or other population segments can be recovered using appropriate compositing procedures and specially-designed studies.

Introduction One of the practices associated with environmental studies is the combining of samples into a composite that is analyzed to obtain a single ‘average’ measure. For example, five discrete samples of sediment or five fish are combined (homogenized), and a single analytical measurement of contaminant concentration is obtained from a subsample of the composite. The process of compositing is a physical averaging of the materials used to form the composite sample. Thus, for a well-formed composite, a single measuredvalue should be similar to the arithmetic average ofvalues measured from the components of the composite. The formation of a composite may be justified when the cost of analysis is high relative to the cost of collecting individual samples (1,2). There are also other advantages to composite sampling, such as confidentiality oftest results (3). Particularly for environmental studies, composite samples may be essential when the analytical technique requires more material than a single grab sample or individual specimen can provide for minimum detection ( 1 , 2).

In a recent review that attempts to develop consistent terminology, Lovison et al. (3) identified five stages of composite sampling: identification of the target population, statistical sampling, compositing (the actual formation of composite samples), subsampling, and measurement. Materials sampled for composites are drawn from either bulk or finite populations (3). With bulk populations, the representative unit is created by the sampling process (3); examples include wastewater and soils. Composite samples are widely used for contaminant analysis of bulk samples, and sampling, compositing, and subsampling methods are well described (I-@, as are statistical considerations (e.g., ref 1). Representative units that are identifiable before sampling occurs are said to belong to finite populations [following the terminology of Lovison et al. (311 or segmented populations; individual organisms and materials occurring in partitioned form such as bales, bags, or bins are population segments. Composite samples have been used to measure contaminant concentrations from segmented populations including Great Lakes fish (e.g., ref 7), marine fish (e.g., ref 81, and birds (e.g., ref 9). However, many studies fail to describe how composite samples were formed (e.g., refs 10-12), and others provide descriptions of sample preparation techniques that have the potential for yielding biased results. Composites formed from identifiable units have unique characteristics and statistical properties that have not been addressed as thoroughly as those for composites formed from bulk samples. In particular, there is the added complexity associated with random weights of the units (3).

Composite samples of whole organisms or specific tissues have other drawbacks. There may be difficulty in physically forming a homogeneous mixture (13), and information about variation among individual specimens may be lost (1, 2). An estimate of the among-segment variation may be desirable for many studies; e.g., models

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VOL. 29, NO. 5,1995 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 1 1137

describing the fate and transport of contaminants in various components of an ecosystem (such as water, plankton, and fish in large lakes) require variance estimates. Recovering an estimate of the variation among segments is possible, if composites are formed properly and appropriately designed studies are executed (see, e.g., refs 3 and 14).

In this paper, we focus on the properties of composite samples formed in various ways from segmented populations. We specifically address the issue of randomweights of segments and note that either the entire unit or a portion thereof can contribute to composite formation. Our statistical approach is based on composite sampling from segmented populations developed by Brown and Fisher (5) and Elder et al. (15).

We demonstrate a few properties of composite samples formed from segments using computer simulations based on actual contaminant concentrations measured from individual fish. Although our treatment of segmented populations relies on fish, it is applicable to composite sampling of other segmented populations. The objectives ofthis paperare (1) to idenUfypotentialproblemsinforming a suitable composite sample from a population consisting of segments having random weights and (2) to provide guidelines for determining necessary sample sizes when working with such composites. In particular, we examine the number of individual units to collect when a fured number of composite samples is analyzed and the number of composites to form when the number of individual units is fixed. We assume that a single subsample is drawn from each composite and a single analytical measurement is taken from that subsample. Thus, we do not address the variation among composite subsamples or the repeat- measurement variability. These components of variance are either due to the nature of the object or specimen sampled, the analytical procedures used, or are under partial control of the investigator. Often, additional studies must be performed to estimate these variance components. The interested reader is referred to Gilbert (1) and Lovison et al. (3).

Original Population Our original sample consisted of observations on individual Contaminant concentrations in 195 striped bass ranging between 55 and 116 cm fork length. These fish, collected off the coast of southern New England from 1982 to 1986, were from the Hudson River stock, one of the major East Coast spawning stocks of striped bass. A complete de- scription of the collection and analytical methods is given in Fabrizio et al. (16). Concentrations of total polychlo- rinated biphenyls (PCBs) in muscle tissue from these fish ranged between 0.10 and 40.70 mglkg wet weight or ppm; the mean concentration per fish was 3.57 ppm, and the variance was 24.105 (Figure 1). Although we examined total PCBs, our results apply equally well to any contaminant (organochlorine compound, heavy metal, etc.) that exhibits similar variability among individual organisms sampled randomly from a population.

Simulation Study Composite Formation Using the Batch and Individual Methods. To demonstrate the effect of two composite formation procedures on estimates of the mean contaminant concentration and its variance, we simulated the formation of two types of composite samples. We used

0 1 2 3 4 5 6 7 0 9 l O . . 2 0 3 0 4 0 Y

Concentration (pprn) FIGURE 1. Empirical distribution of total PCB concentrations (in ppm) for 195 striped bass collected from southern New England waters. The mean and variance ware 3.57 ppm and 24.105.

Monte Carlo methods to randomly select fish from the original population; we simulated sampling with replace- ment, so each fish had an equal probability ofbeing selected. Our simulated composite samples consisted of five fish.

Next, we calculated the contaminant concentration for each composite sample according to one of two methods. When forming a composite sample for fish contaminant analysis, whole fish are often combined in a large mixer or homogenizer (Figure 2). This procedure, which we call the batch method, produces a naturally-weighted composite sample with fish mass as the weighting factor. In a composite sample formed by the batch method, larger fish contribute proportionally more mass to the composite. Although fish of the same age class are sometimes used to form the composite, size variation within an age class may be significant. We simulated contaminant concentrations in the batch homogenate of muscle tissue by calculating a weighted average concentration for the fish comprising the homogenate; fish mass was the weighting factor:

k

C w F i i= 1

wi i= 1

where B a t c h is the concentration of the composite sample formed by the batch method, wi is the weight (mass) of the ith unit or portion, xi is the concentration in the ith unit or portion, and k is the total number of units (or portions) in each composite.

For the individual method, each fish is homogenized separately, an equal-sized aliquot is drawn from each homogenate, and the aliquots are combined to form the composite (Figure 2). We simulated the combining of equal portions of homogenate from each fish by calculating a simple average of individual contaminant concentrations and setting wi = l / k

q k k

We applied the Monte Carlo approach to simulate multiple samples of composites by repeating the sampling

1138 4 ENVIRONMENTAL SCIENCE &TECHNOLOGY / VOL. 29. NO. 5 , 1 9 9 5

Batch Method I ndivid ual M e t hod

take equal-sized portions 1 1 U - l from each homoaenate remove composite sample

1 remove composite sample

Q FIGURE 2. Two procedures used to form composite samples from whole fish. The batch method produces a weighted average concentration because whola fish ara homogenized together. The individualmethod combines equal portions of homogenate from each fish and produces a simple average of the concentrations in each fish.

procedure many times; this yielded an approximation of the bootstrap distribution of contaminant concentrations in composite samples (1 7). Because we drew our samples from the empirical distribution of the original population, our implementation is termed the nonparametric bootstrap (18,19). Thebootstrapis awell-describedstatisticalmethod that is widely used to calculate the bias and variance of an estimator (17, 18, 20). [Although the bootstrap variance estimate is biased downward (181, this bias should not affect our comparisons among composite formation methods.] We examined the results of 10 000 bootstrap replications of single composites to determine how the mean and variance were affected by the method used to form the composite. The ‘true’ (unweighted) mean and variance were calculated from contaminant concentrations in the fish that constituted the original population. We tested the null hypothesis that the difference between the true mean and estimated mean was zero using the t-test; this test provided an indication of bias.

Because more than one composite is usually formed to estimate contaminant concentrations in a population of interest, in our next example we formed sets of 20 composite samples of five fish each. We repeated these procedures 1000 times for each set of 20 composites. Again, we compared results from the simulation to those from the original sample. If we assume no measurement error and w, = l /k , then the expected value and variance of the jth composite sample is:

d d VarW.) = - + - ’ b k (4)

where is the variation among segments (fish), is the variation within segments (among aliquots within fish), b is the number of segments (fish or specific tissues) per composite, and k is the number of aliquots per composite (3). In our case, because there is one aliquot per segment, b = k. The composited mean, estimated from c composites is

and has the expected value:

(5)

With respect to the variance of the mean, Lovison et al. (3) gave the following formula (assuming no measurement error):

(7)

Necessary Sample Size. Researchers workunder several constraints in obtaining samples. In some cases, the number of samples that can be processed is limited, whereas in other cases, the number of organisms that can be collected is limited. We examined the tradeoffs in precision of the estimates of the mean and variance as we varied composite size (b, the number of fish in a composite), the number of composites (c) analyzed, and the number of

VOL. 29, NO. 5,1995 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 1139

TABLE 1

Mean PCB Concentration (ppm) and Associated Variance and Standard Error (SE) for Original Population as Well as for 10 000 Simulated Composite Samples Formed from Striped Bass 55-1 16 cm Fork Lengthe

method parameter original population individual batch

mean 3.57 3.55 3.64 variance 24.105 4.746 5.649 SE 0.352 0.022 0.024 a Each simulated composite contained five fish. The individual

method simulated the combining of equal portions of homogenate from each fish; the batch method simulated a homogenate formed from muscle tissue from five whole fish.

organisms collected (8, such that

F = cb (8)

We varied composite size (b) from 1 to 50, composite samples (c) from 2 to 50, and fish collected (8 from 2 to 2500. We repeated each combination 1000 times and examined the mean concentration and average variance among the bootstrap replicates. Because we performed the simulations 1000 times, we also examined the variance of the variance, a measure of the relative precision of the variance estimate. Comparisons with the true mean and variance, estimated directly from the original population, were used to interpret our results.

Using the individual method of forming composites and the same simulation approach described before, we first examined the effect of changing the composite size when the number of composites analyzed is fiied. Specifically, we addressed the question ‘If we want to obtain reliable estimates of the mean and variance, and we can process only a fixed number of analytical “samples”, how many fish should be collected to form those composites?’

We next examined the effect of changing the number of composites when the number of fish collected is fKed. The question we examined was ‘If we collect a fixed number of individual fish, how many composites can be formed without losing reliability, and how many fish should be combined to form each composite sample?’

Results Composite Formation. (1) Single Composites. The mean obtained by the individual method (Table 1) was similar to the true (unweighted) mean, 3.57 ppm, and was an unbiased estimate of the population mean (t= -0.92, P > 0.05). The same was not true for the batch method. The mean obta ined by t he batch m e t h o d was significantly greater than the true mean ( t = 2.90, P < 0.051, thus showing the bias in batch method computations. We believe the bias of the batch method depends, in part, on the distribution of fish weight in the population and in the composite sample. For this study, fish weight varied by a factor of about 2. In some special instances, the batch method may produce an unbiased estimate of the mean concentration; we found this to be true when the range of fish weight in the original population was restricted. The individual method provided an unbiased estimate of the population mean and is therefore recommended for composite formation. We used

Concentration (ppm)

FIGURE 3. Distribution of mean contaminant concentration (in ppm) for loo0 bootstrap replicates, each replicate consisting of 20 composites of five fish each. The mean and variance of this distribution are 3.58 ppm and 0.240.

the individual method to form composites in all subsequent simulations.

(2) Sets of Composites. Using the individual method to simulate the formation of 1000 sets of 20 composites, we found that most sets of composite samples had means near 3.57 ppm, the true mean (Figure 3). However, the means rangedfrom2.45 to 5.61 ppm (Figure3). Based on bootstrap results and assuming d is negligible, we calculated the 95% confidence intervals around the composite means and found that 9.9% of the confidence intervals did not include the true mean. Notice that we selected an a-level of 0.05, but the actual significance level was 0.099. The reason for this difference in a-level is that the average estimated variance among composite samples (4.900) underestimated the true populationvariance (24.105). Aconfidence interval constructed without regard to the effect of multiple sources of variation will be incorrect in that the width of the interval will be smaller than expected. The actual bias associated with the estimate of the variance among composites will depend in part on the magnitude of the variation within segments (d). This variation may be quite large de- pending on the nature of the tissue(s) and the efficiency of homogenization. We have shown that even under ideal conditions (Le., nonsignificant variation associated with the mixing process, subsampling, analytical replication, and negligible 41, the variance of the mean estimated from a set of composite samples underestimates the variance among fish. Fortunately, there is a way to recover the estimate of the varv) using composite samples. However, the varv) cannot be estimated from a single set of composite samples because, as Lovison et al. (3) point out, cf and 4 are not separately estimable. Therefore, a special approach to permit direct estimation of by the grouping of composites into more than one set has been recommended (3). An estimate of d from sets or groups of composites is

(9)

where several aliquots are randomly drawn from each segment and randomly assigned to g groups; within each group, d composites are formed (3). In this protocol, there is a total of c = gd composites. With this estimate of 4, the estimate of can be recovered from the formula for the variance of the mean from several sets of composites. This type of study involving grouped composite samples from each segment is necessarywhen investigators are interested

1140 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 29, NO. 5,1995

3.65 \ 5

= 3.60 ' 1 : 3.55

3.60

- 9

3.55

3.65 1 10 c o m ~

. * . I i

s 3.50 1. L L 0 1 0 2 0 3 0 4 0 5 0 0 1 0 2 0 3 0 4 0 5 0

CmrnC.site size Composite Size

E

3.65 1 50 Cornpitea A

S I 3.50 1-

0 1 0 2 0 3 0 4 0 5 0 composite size

B 261

2 2 1 221 0 1 0 2 0 3 0 4 0 5 0 0 1 0 2 0 3 0 4 0 5 0

Composite S i Composite &e

221 0 1 0 2 0 3 0 4 0 5 0

GJmpasite-

FIGURE 4. Mean contaminant concentration (A) and average variance (B) for 1008 bootstrap replicates of 5,lO. and 50 composite samples; the number of fish in each composite varied between 1 and 50. The refenmce lines are at the true mean, 3.57 ppm (A), and true variance, 24.105 (B). For clarity, only three cases (5, 10, and 50 composites) are plotted.

in a measure of the variability in the sampled population. The accuracy of the estimate of the population variance from sets of composites (var(j?) depends on the number of composite samples, the magnitude of the variance associated with subsampling the segments (which is done in the process of forming composites), and other factors which we have not examined here.

Necessary Sample Size. (1) Fixed Number of Com- posites. Whether we formed 2 or 50 composites, the true mean concentration was estimated equally well by all composite sizes, although a small number of composites formed from few fish appeared to be less accurate (Figure 4A). The accuracy decreased because the estimate of the composite mean depends on the total number of fish not only on the number of composites. The average variance appeared to be better estimated with more fish per composite (Figure 4B). The precision of the variance estimate increased as composite size increased (Figure 5). That is, 10 composite samples each formed from 50 fish will yield better estimates of the variance and hence higher precision than 10 composites formed from two or five fish. As composite size (b) increased, precision increased relatively rapidly; for composites of five or more organisms, the greatest gain in precision occurred between 2 and 10 composites. Based on Figures 4 and 5, we estimated that at least five fish per composite will give a reliable estimate of the variance for our striped bass example when 10 or more composites are analyzed.

(2) Fixed Number of Fish. Given a fixed number of fish, any number of composite samples will provide an unbiased estimate of the mean (Figure 6A). However, the average variance stabilized as the number of composite samples increased (Figure 6B). Indeed, as the number of composite samples approached the number of fish collected (Le., a composite was formed from a single fish), we would expect the average variance to approach the true variance. The precision of the variance estimate was clearly related to

0 10 20 30 40 50 Number of Composkes

FIGURE 5. Precision of the variance estimate for loo0 bootstrap replicates of 2-50 composite samples; lines for composite sizes (W of 1,2, 5,lO. and 50 are shown. In this simulation, a single aliquot was drawn from each fish, so composite size is the number of fish in a composite sample.

both the number of composites formed (Figure 7A) and composite size (Figure 7B). This indicated that given a collection of say, 100 individuals, the variance estimate stabilized when no more than 10 fish were included in each composite sample.

Discussion We recognize that the formation of composites is sometimes essential, so our goal in this study was to highlight the proper procedure of composite formation for segmented populations. Furthermore, we assumed segmentsvaried randomly in weight. Several studies have addressed the question of the desirability of forming composite samples for monitoring pollutant concentrations, but these studies have specifically dealt with sampling from a bulk population such as a fluid (e.g., refs 6, 21, and 22). Statistical limitations

VOL. 29, NO. 5. 1995 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 1141

3.65 I 40 Fish 3.65 1 1CQ Fmh 3.65 1 200 Fish A

I I 4 4

3.50 I 0 1 0 2 0 3 0 4 0 5 0

Number of Composites

3.551

3.50 0 1 0 2 0 3 0 4 0 5 0


221 0 I 0 2 0 3 0 4 0 5 0


221 0 I 0 2 0 3 0 4 0 5 0


3.50 1 0 1 0 2 0 3 0 4 0 5 0


261 B

2 2 1 0 1 0 2 0 3 0 4 0 5 0


FIGURE 6. Mean contaminant concentration (A) and average variance (B) for 1RW bootstrap replicates of 40,100, and 200 fisk formed into 2-50 composite samples. The reference lines are at the true mean, 3.57 ppm (A), and true variance, 24.105 (B). For clarity, only three cases (40, 100, and 200 fish) are plotted.

associated with using composites of water samples to monitor water quality have been reported, including the inability to detect the presence and severity of extreme concentrations (21, 22) and the presence of bias in load estimation associated with time-proportional and volume- proportional sampling (23). Guidelines for statistical analysis of composites formed from water samples are well described in Brown and Fisher (3, Rohde (241, and Elder et al. (13 as well as in environmental sampling textbooks such as Gilbert (1) . However, these procedures and guidelines have not been directly translated to those appropriate for the formation of composites from segmented populations, particularly for aquatic organisms, and we believe that composite samples from whole organisms or their tissues are generally formed in a potentially biased manner. Using empirical data from contaminant concentrations in striped bass, we demonstrated the proper method of composite formation (the individual method) from segmented populations and suggested an approach to determining the number of segments to include per composite sample.

Throughout this paper, we assumed that a composite sample was a perfect mixture of tissue from individual segments that are equally represented in the composite. In reality, ideal homogenates are rarely obtained because entire organisms or specific tissues are difficult to blend perfectly. Additionally, composites are often comprised of different sized fish, and the concentration of a contaminant and the total body or tissue burden in an organism may or may not be related to its size. To further complicate whole organism composites, a contaminant may not be evenly distributed throughout the body but may be concentrated in an organ or a specific tissue (25-281. The contaminant concentration in a composite will estimate the population mean when individual segments are equally represented in the composite. A weighted composite, for example, one created based on the weights of the segment or portion used, will not have a concentration representative of the

population mean unless all of the fish or tissues used happen to be the same weight. Based on our simulation results, we recommend homogenizing entire segments or portions (whole organisms or specific tissues) and combining equal- sized aliquots from each homogenate into a single composite sample. Clearly, our specific results depend on the population from which we simulated our sampling; however, general results serve to illustrate some of the com- plexities of forming composite samples from populations of segments of random weight. Indeed, our observations on the batch method-a weighted approach-point out sampling problems similar to those explored by Smith and Parnes (29).

One of our observations indicated that under the restriction of similar mass of individual fish, the batch method of composite formation may provide unbiased estimates of the mean Concentration. Clearly, the acceptable range of fish mass would have to be determined for each case under investigation (e.g., species, season, and specific Contaminant), so we cannot provide generalized guidelines from these data on PCBs in striped bass. If the homogenization of individual organisms is costly and the composite sampling design has been deemed appropriate, then it may be prudent to determine the acceptable range

. of fish mass for composites and use the batch method. However, we note that the variability in the efficiency of the homogenization process may have a greater (and perhaps, unknown) effect on the mean contaminant concentration than the effect of bias due to fish weight. This question requires further research. In general, however, we agree with Lovison et al. (3), who caution against failure to account for randomness of weights because the estimate of the variance among segments from such a study will not be smaller than that from a study based on futed weights.

As we demonstrated in our experiment on sample size tradeoffs, ifthe purpose is to estimate ameanwithout regard to its precision, then a few composites of a few fish each

1142 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 29, NO. 5,1995

“Ol A

01 0 50 loo 150 200 250

Number of F i h Collected

-1 B

” 0 loo 200 300 400 500

FIGURE 7. Precision of the variance estimate for loo0 bootstrap replicates: (A) composite samples formed from 2-250 fish, where c (the number of composites) was 5,10, and 50; and (8) composite samples formed from 2-50 fish, where b (composite size) was 1, 5,lO. 20, and 50. Composite size is the number of fish in a composite sample. The total number of fish collected is cb.

Number of Fkh Collected

could be formed. In fact, because the mean depends only on the number of fish, a single composite would be adequate. Our empirical observations support the theo- retical work presented in Elder et al. (1.9, namely, that the average concentration from a set of composite samples is an unbiased estimate of the population mean. When a reliable estimate of the variance is also required, additional measures of variation associated with composite formation and subsampling are needed. Because our simulation produced ‘perfectly blended’ composites, we caution that our results are based on the assumption that the physical mixing of composite material is complete. Thus, although we do not explicitly consider this component of variance, we note that it may indeed be significant. Parrish et al. (14) present a model to estimate this variance component and an example illustrating the relation between the magnitude of the component and the mixing process used to form the composite.

In addition, when information on among-segment variation is required, our simulation study showed that the ‘best’ composite sample was formed from a single individual-a case that is no different from sampling the population and analyzing each sample separately. We

realize this may not be practical when costs are considered. Therefore, we recommend (1) determining the number of individuals that can be sampled for a fixed cost and (2) using the approach presented here, estimating the precision of the variance for composites formed of varying numbers of fish. Properly formed composites from whole organisms or specific tissues will still have limitations (such as the inability to detect extremes in a population), but the mean is unbiased and the variance among segments can be estimated from specially-designed studies of composite sampling.

Acknowledgments We thank Jon G. Stanley (NBS, Great Lakes Science Center, Ann Arbor, MI) for initiating our interest in composite sampling and Ronald J. Sloan (NY Department of Envi- ronmental Conservation, Albany, NYI for providing information on striped bass contaminants. George E. Noguchi, Michael J. Hansen, and Ray L. Argyle (NBS, Great Lakes Science Center, Ann Arbor, MI) provided comments on an earlier manuscript; recommendations from three anony- mous reviewers and discussions with Frank J. Ossiander (NOM, retired) greatly improved our presentation. Roann Ogawa (NBS, Great Lakes Science Center, Ann Arbor, MI) illustrated Figure 2. This article is Contribution 882 of the National Biological Service, Great Lakes Science Center, 1451 Green Road, Ann Arbor, MI 48105.

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Received fo r review January 11, 1994. Revised manuscript received October 19, 1994. Accepted January 31, 1995.@

ES9400300

@Abstract published in Advance ACS Abstracts, March 15, 1995.

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Procedures for Formation of Composite Samples from Segmented Populations