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SOIL PHYSICS

Optimum Allocation for Soil Contamination Investigations in Hsinchu, Taiwan, by Double Sampling

National Chiao Tung Univ., 75 Po-Ai St., Hsinchu, 300 Taiwan

^* Corresponding author (hdyeh{at}mail.nctu.edu.tw).

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
The double sampling (DS) scheme is a cost-effective sampling method that combines an expensive measurement procedure with an inexpensive but less accurate one. Double sampling works when the true correlation of determination (²) between two techniques is known in advance, but that is hardly ever the case. By assuming a ² < 0.9, a three-stage procedure (TSP) in DS selects a preliminary pool of samples and estimates ², from which the optimal allocation of samples is determined. There may be excessive and unnecessary sampling during a TSP when ² > 0.9. The main objective of this study was to extend the TSP and determine the optimum allocation of the samples under a fixed budget condition for the case ² > 0.9. A soil from Hsinchu, Taiwan, contaminated with heavy metals Zn, Cu, Pb, Ni, Cr, and Cd was sampled at 0.15, 0.3, 0.45, and 0.6 m deep at 36 sites (144 samples). All samples were analyzed with field-portable x-ray fluorescence and a subset of 40 samples was selected for analyses of Zn, Cu, and Pb with standard methods in the laboratory. Results from both measurements were linearly correlated with estimated ² values of 0.96, 0.95, and 0.97 for Zn, Cu, and Pb, respectively. Considering a ² = 0.99, the optimum subsample and sample sizes were 5 out of 167, 4 out of 173, and 5 out of 167 for Zn, Cu, and Pb, respectively. The extended TSP analyses reduced the number of superfluous samples to only two or three, which was less than obtained by TSP (9–16).

Abbreviations: DS, double sampling • FPXRF, field-portable x-ray fluorescence • HEPB, Hsinchu Environmental Protection Bureau • LAB, standard method in laboratory • NTD, Taiwan dollars • SDR, Studentized deleted residual • TSP, three-stage procedure

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
In a sampling program, two different techniques may be used to analyze samples. One would be an expensive procedure that records accurate measurements for the variable of interest, while the other would be an inexpensive procedure that is subject to large measurement error or poor accuracy recording an auxiliary variable. In addition, the budget for site characterization is usually fixed and the number of samples is crucial for obtaining accurate results under a fixed budget. In such a situation, the DS method is useful for estimating the mean and variance of a distribution. Double sampling is simple random sampling without replacement in which the variable of interest is available only for the subsample, and an auxiliary variable is available for the full sample (Cochran, 1977). The DS scheme has been studied for many years and is widely used in survey sampling (Bolfarine and Zacks, 1992). Buonaccorsi (1990) applied the DS scheme for multivariate measurement error problems and focused on inference for the regression parameters.

Tenenbein (1970) demonstrated that the gains of the DS scheme are quite substantial when the correlation coefficient and the relative cost of the two different measuring devices are high. The optimum sample size that minimizes the variance for the fixed budget in the DS scheme depends on the correlation coefficient () or the coefficient of determination (²), which, in many cases, is unknown before sampling. Thus, Tenenbein (1971) provided a TSP for determining the optimum sample size based on the DS scheme. In the first stage, ² is estimated from a preliminary sample of m = 30 accurate–inaccurate data pairs. An estimated sample size of is determined and ² is recalculated in the second stage. The third stage uses this correlation to determine the optimum sample size of n. Tenenbein (1974) further recommended a rule to select the value of m for the TSP in a DS scheme for estimating means. Gilbert (1987) used a DS scheme and the TSP to determine the optimal allocation of the samples under fixed-budget and fixed-variance conditions in an experiment with nuclear weapon material in an isolated region of Nevada. Cochran (1977) compared DS with a regression estimate to simple random sampling with no regression adjustment and demonstrated that, under some conditions, the linear regression DS will yield a more precise estimate of the population mean than would be achieved by simple random sampling.

The problem with the TSP is that the preliminary sample pool recommended by the TSP assumes that the true ² is <0.9. Unnecessary sampling may occur when the true ² is >0.9. The objective of this study was to extend the TSP provided by Tenenbein (1974) to determine the optimum allocation of samples under a fixed-budget condition using the DS technique without the restriction that the true ² is <0.9. A set of field data of soil contamination in Hsinchu, Taiwan, was chosen to assess the performance of this extended TSP.

	THEORY

TOP NOTES ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Outlier Detection
Studentized deleted residuals (SDR), also known as jackknife residuals, are used to assess the assumption or goodness of fit of a model. Each residual is calculated from a model that includes all but the observation corresponding to the point in question. When there are np – 1 predictor variables X₁, ..., X_np–1, the regression model Y_i = ₀ + ₁X_i,1 + ₂X_i,2 + ... + _np–1X_i,np–1 + _i is termed a first-order model with ₀, ..., _np–1 being parameters and _i being the error terms for i = 1, ..., n.The fitted values are expressed as _i = ₀ + ₁X_i,1 + ₂X_i,2 + ... + _np–1X_i,np–1, with ₀, ..., _np–1 the least squares estimated regression coefficients, and the residual terms are expressed as e_i = Y_i –_i. The outliers can be identified if their SDRs are large in absolute value. The SDR t_i was denoted as (Neter et al., 1996, p. 368–375)

[1]

where SSE, the error sum of squares, = _i=1ⁿ(Y_i–_i)²=_i=1ⁿe_i² and h_ii, the diagonal element of the hat matrix, can be expressed as h_ii = X_i'(X'X)^–1X_i, with X being a n x np matrix:

[2]

Because n – 1 cases are used to predict the ith observation, each SDR t_i follows t distribution with (n – 1) – np = n – np – 1 degrees of freedom. In addition, one can conduct a formal test by means of a Bonferroni test procedure to determine whether the case with the largest absolute SDR is an outlier. The appropriate Bonferroni critical value therefore is t(1 – /2n; n – np – 1) (Neter et al., 1996, p. 368–375) with the significance level .

Double Sampling
The auxiliary variate x_i that is correlated with y_i has been used to make a regression estimate of the population mean = N^–1_Ny_i for the population of size N in some applications of DS. In the first (large) sample of size n', only x_i is measured; in the second, a random subsample of size n, both x_i and y_i are measured. The linear regression estimate of population mean , denoted by _lr is

[3]

where = n^–1_ny_i is the mean for the variable of interest for the subsample, ' = n'^–1_n'x_i and = n^–1_nx_i are the means of the x_i sample and subsample, respectively, and b=_n(x_i–)(y_i–)/_n(x_i–)² is the least squares regression coefficient of y_i on x_i, computed from the subsample. If b in _lr is replaced by the finite population regression coefficient B = S_yx/S_x², with S_x² and S_xy² being the population variance of x and covariance of x and y, then the error in the approximation is of the order 1/(n) relative to _lr (Cochran, 1977, Ch. 12). Therefore, one can examine the variance of the following approximation with the finite population regression coefficient B:

[4]

Let u_i = y_i – Bx_i. In the second phase, the large sample is considered a finite population. The small sample is drawn at random from the large sample. The expected value and the variance of _lr are, respectively,

and

[5]

where '= n'^–1_n'y_i is the mean for the variable of interest for the large sample and s_u'² is the variance of u within the large sample and is an unbiased estimate of S_y²(1 – ²). It follows that

[6]

where the subscript 1 in Eq. [6] and subscript 2 in Eq. [5] denote variations within the first and second phases of sampling, respectively. Thus, the linear regression variance estimator of a population _lr from design-based consideration is approximately (Cochran, 1977)

[7]

where S_y² = _N(y – _lr)²/(N – 1) is the population variance of y, and ² = S_xy²/(S_x²S_y²) is the coefficient of determination with the correlation coefficient . To find the optimal sample size that is economical and satisfies the minimum acceptable accuracy, based on the theory of DS, the Lagrange multiplier is used to obtain the general formula for n and n'. Suppose the total cost for survey sampling can be expressed as

[8]

where C is the total amount of money available for measuring collected samples, and c_A and c_F are the costs per unit of making a measurement from an accurate (laboratory) technique and a fallible (field) technique, respectively. Note that C does not include the cost of collecting samples. The variance shown in Eq. [7] is minimized and subject to the cost constraint when

[9]

and

[10]

where

[11]

and f₀ is set equal to 1 if Eq. [11] gives f₀ > 1. For a specified cost C and if both Eq. [9] and [10] hold, the variance is minimized as follows:

[12]

If all resources are devoted instead to a single sample with no regression adjustment, this sample has size C/c_A and the variance of its mean is

[13]

Hence, optimum use of DS gives a smaller variance if

[14]

If the relationship between the values from the laboratory and field techniques is linear and the coefficient of determination satisfies Eq. [14], then linear regression DS will, on average, yield a more precise estimate of the population mean than would be achieved by the accurate method on C/c_A units selected by simple random sampling (Gilbert, 1987). Tenenbein (1974) found that V^1/2 will not exceed about 15% of its minimal value if Eq. [9] and [10] are replaced by

[15]

and

[16]

when the true ² is <0.9 for the fixed-budget case. As noted above, since the true ² is never known in practice, it is necessary to conduct pilot studies or to use suitable data from prior studies to estimate ² and evaluate the linear regression hypothesis.

Three-Stage Double Sampling
If a pilot study is considered, the TSP recommended by Tenenbein (1974) may be used to estimate ² and then n and n'. For the fixed-budget case, the first stage consists of randomly selecting m units from the population and making both accurate and fallible measurements on each unit to estimate ². Using the definition of Tenenbein (1970), suppose a random sample of n' units is taken from the population and a subsample of n units is drawn from the main sample and each of these n units is classified by both measuring techniques. Then the remaining n' – n units are classified only by the fallible classifier. Let n_tf denote the number of units whose true classification is t and whose fallible classification is f (t,f = 0,1). Also let n_x and n_y denote the number of units whose fallible classifications are 1 and 0, respectively. The resulting data can be represented schematically as shown in Fig. 1 . For each unit in the sample, the random variable T_i (i = 1, 2, ..., n) is defined as

[17]

Let p = Pr(T_i = 1) and q = 1 – p. The maximum likelihood estimate of p and its asymptotic variance (V) can be expressed as (Tenenbein, 1971)

[18]

and

[19]

View larger version (12K): [in this window] [in a new window]   Fig. 1. Diagram for the classified units (Tenenbein, 1971) in which n' is a random sample taken from the polulation and n is the subsample drawn from n'. The classified units are: ntf, the number of units whose true classification is t and fallible classification is f(t,f = 0,1); and nx and ny are the number of units whose fallible classifications are 1 and 0, respectively.

[20]

where ₀ is the same expression as f₀ in Eq. [11], but with ² replaced by ². Table 1 shows the various values of 100_v for various values of R, ², and ². The variable r is the deviation from the true coefficient of determination ², that is,² = ² ± r where r = 0.05, 0.10, 0.20 and 0.25. The variance increases with the deviation from the true coefficient. The increase in variance is slight, however; even when |²–²|=0.20, the increase is <12% except when ²>0.9. Tenenbein (1974) found that a sample size of 21 satisfies the following equation for all values of ² and R:

[21]

The preliminary sample size m can be decided by the following rule for the fixed-budget case:

[22]

If Eq. [14] is satisfied, then and ' can be obtained by Eq. [9] and [10], respectively. If m < , then –m additional units should be randomly selected for measurement by both methods. This selection is the second stage of the sampling plan. At this point, ² can be estimated by using all data, and Eq. [14] can be rechecked before proceeding. The third stage consists of using for n in Eq. [10] to re-estimate n'. Once the data are in hand, V(_lr) is computed by Eq. [7]. The flowchart of the TSP recommended by Tenenbein (1974) is demonstrated in Fig. 2 .

View this table: [in this window] [in a new window]   Table 1. Increase (Eq. [20]) in the variance resulting from sampling errors in estimating the coefficient of determination (2) with different ratios (R = cA/cF) of the cost per unit of making a measurement from an accurate technique (cA) to the cost per unit of making a measurement from a fallible technique (cF).

View larger version (28K): [in this window] [in a new window]   Fig. 2. Flowchart of the three-stage procedure, where m is the preliminary sample size, xi is the auxiliary variate; yi is the variate of interest; is the correlation coefficient between yi and xi; and ' are the estimated sample sizes of subsample n and main sample n', respectively; lr is the linear regression estimate of the population mean; and V(lr) is the variance of lr.

View this table: [in this window] [in a new window]   Table 2. Increase (Eq. [20]) in the variance if initial estimated coefficient of determination (2) is 0.99 with different ratios (R = cA/cF) of the cost per unit of making a measurement from an accurate technique (cA) to the cost per unit of making a measurement from a fallible technique (cF).

View larger version (20K): [in this window] [in a new window]   Fig. 3. Flowchart for the proposed approach with outlier detection and optimal allocation for soil sampling, where ti is the Studentized deleted residuals defined in Eq. [1], t(1 – /2; n – np –1) is the Bonferroni critical value, 2 is the coefficient of determination, is the significance level; np – 1 is the number of predictor variables, and n is the number of data.

	MATERIALS AND METHODS

TOP NOTES ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

View larger version (184K): [in this window] [in a new window]   Fig. 4. Location of sampling sites in the Hsiang-Shan industrial park.

In the LAB method, soil samples were digested using the National Institute of Environmental Analysis Method 321.60T. An approximately 3-g (dry weight) sample was weighed in the reactor. Concentrated HCl (21 mL) and 7 mL of concentrated HNO₃ were added and allowed to stand for 16 h at room temperature. The samples were then heated in a water bath to the degree of ebullition for 2 h. After quantifying the extracts, concentrations of these metals were analyzed by flame atomic absorption spectrometer.

	RESULTS AND DISCUSSION

TOP NOTES ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
The 40 pairs of LAB–FPXRF data showed a linear relationship, but for each metal one of the points clearly deviated from linearity (Fig. 5 ). The solid lines show the fitted regressions between the two techniques using all data points. The method of SDRs was used to test for outliers (Neter et al., 1996). The Bonferroni simultaneous test procedure with a family significance level of = 0.1 and t(0.99875; 37) was equal to 3.24. The points shown as outliers in Fig. 5 had absolute SDR values >3.24. The new ² calculated without the outliers were higher than those obtained considering the original data (Fig. 5).

View larger version (12K): [in this window] [in a new window]   Fig. 5. Plots of the 40 data points measured with both standard laboratory methods (LAB) and field-portable x-ray fluorescence (FPXRF) techniques for (a) Zn, (b) Cu, and (c) Pb.

[23]

Using the rule given by Eq. [22], we concluded that m = 21 samples should be collected and analyzed by both LAB and FPXRF methods in the first stage for these three heavy metals. Table 3 shows the statistics and Fig. 6 illustrates the regressions (solid lines) for 21 randomly selected LAB and FPXRF sample data for these three metals. Once ² was calculated in the first stage, Eq. [9–11] can be used in a second stage to obtain f₀, n and n', and ²(Table 4).

View this table: [in this window] [in a new window]   Table 3. Concentrations of Zn, Cu, and Pb determined by standard laboratory methods (LAB) and field-portable x-ray fluorescence (FPXRF) techniques and their statistics for 21 soil samples.

View larger version (13K): [in this window] [in a new window]   Fig. 6. Plots of 21 and 7 randomly selected data points measured with both standard laboratory methods (LAB) and field-portable x-ray fluorescence (FPXRF) techniques for (a) Zn, (b) Cu, and (c) Pb.

These results show that unnecessary samples were taken after using the TSP with a high correlation coefficient of determination. The preliminary sample size m recommended by Tenenbein (1974) is based on the assumption that ² 0.9. As shown in Table 1, the increase in the variance could be very high when ² > 0.9. The TSP is not adequate in the case of the Hsiang-Shan industrial park. We suggest using the estimated ² of 0.99 to calculate the estimated value n when ² > 0.9.

Estimation of the Sample Size by the Extended Three-Stage Procedure
The sample size n = 7 if ²=0.99 is used to determine the optimum sample size in the second stage for Zn, Cu, and Pb. The regression lines in Fig. 6 are the dotted lines. The coefficients of determination obtained from the seven pairs of LAB–FPXRF measurement were 0.996, 0.998, and 0.983 for Zn, Cu, and Pb, respectively (Table 5). Since the ² is known, the optimum n for Zn, Cu, and Pb were 5, 4, and 5, respectively, and the optimum n' were 167, 173, and 167, respectively. This shows that only two to three superfluous LAB samples were collected for these three metals using extended TSP.

	CONCLUSIONS

TOP NOTES ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

This study assessed a complete DS program that was slightly different from the TSP provided by Tenenbein (1974) for conditions of a fixed budget. The optimum allocation of the samples was determined under fixed-budget conditions based on the theory of DS. Since field data are time consuming and costly to obtain, this study provided a useful and valuable case study for designing a site characterization work plan to monitor environmental pollution, considering both cost and accuracy. In addition, the results from this study may be more generally useful in determining the optimum allocation of samples for reducing the cost of sampling.

	ACKNOWLEDGMENTS

 
This study was partly supported by the Taiwan National Science Council under Grant NSC 94–2211–E–009–012.

	NOTES

TOP NOTES ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher.

Received for publication March 27, 2006.

	REFERENCES

TOP NOTES ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

• Bolfarine, H., and S. Zacks. 1992. Prediction theory for finite populations. Springer, New York.
• Buonaccorsi, J.P. 1990. Double sampling for exact values in some multivariate measurement error problems. J. Am. Stat. Assoc. 85:1075–1082.
• Cochran, W.G. 1977. Double sampling. p. 327–358. In W.G. Cochran (ed.) Sampling techniques. 3rd ed. John Wiley & Sons, New York.
• Gilbert, R.O. 1987. Double sampling. p. 106–118. In R.O. Gilbert (ed.) Statistical methods for environmental pollution monitoring. Van Nostrand Reinhold, New York.
• Neter, J., M.H. Kutner, C.J. Nachtsheim, and W. Wasserman. 1996. Identifying outlying Y observations: Studentized deleted residuals. p. 368–375. In R.D. Irwin (ed.) Applied linear regression models. 3rd ed. McGraw-Hill, Chicago, IL.
• Tenenbein, A. 1970. A double sampling scheme for estimating from binomial data with misclassifications. J. Am. Stat. Assoc. 65:1350–1361.
• Tenenbein, A. 1971. A double sampling scheme for estimating from misclassified binomial data: Sample size determination. Biometrics 27:935–944.
• Tenenbein, A. 1974. Sample size determination for the regression estimate of the mean. Biometrics 30:709–716.
• Thermo Electron Corporation. 2006. NITON analyzer products. Available at www.niton.com/NITON-Analyzers-Products/default.aspx (accessed 4 Aug. 2006; verified 17 June 2007). Thermo Electron Corp., Billerica, MA.

This Article Abstract Figures Only Full Text (PDF) Free A correction has been published Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via Google Scholar Google Scholar Articles by Chang, Y.-C. Articles by Yeh, H.-D. Search for Related Content PubMed Articles by Chang, Y.-C. Articles by Yeh, H.-D. GeoRef GeoRef Citation Agricola Articles by Chang, Y.-C. Articles by Yeh, H.-D. Related Collections Data acquisition and assimilation Soil Pollution Statistics