C H A P T E R 5 Tools for Dealing with Censored Data “As trace substances are increasingly investigated in soil, air, and water, observations with concentrations below the analytical detection limits are more frequently encountered. ‘Less-than’ values present a serious interpretation problem for data analysts.” ( Helsel, 1990a) Calibration and Analytical Chemistry All measurement methods (e.g., mass spectrometry) for determining chemical concentrations have statistically defined errors. Typically, these errors are defined as a part of developing the chemical analysis technique for the compound in question, which is termed “calibration” of the method. In its simplest form calibration consists of mixing a series of solutions that contain the compound of interest in varying concentrations. For example, if we were trying to measure compound A at concentrations of between zero and 50 ppm, we might prepare a solution of A at zero, 1, 10, 20, 40, and 80 ppm, and run these solutions through our analytical technique. Ideally we would run 3 or 4 replicate analyses at each concentration to provide us with a good idea of the precision of our measurements at each concentration. At the end of this exercise we would have a set of N measurements (if we ran 5 concentrations and 3 replicates per concentration, N would equal 15) consisting of a set of k analytic outputs, A i,j . for each known concentration, C i . Figure 5.1 shows a hypothetical set of calibration measurements, with a single A i for each C i , along with the regression line that best describes these data. Figure 5.1 A Hypothetical Calibration Curve, Units are Arbitrary steqm-5.fm Page 111 Friday, August 8, 2003 8:16 AM ©2004 CRC Press LLC Regression (see Chapter 4 for a discussion of regression) is the method that is used to predict the estimated measured concentration from the known standard concentration (because the standards were prepared to a known concentration). The result is a prediction equation of the form: M i = β 0 + β 1 • C i + ε i [5.1] Here M i , is the predicted mean of the measured values (the A i,j ’s) at known concentration C i , β 0 the estimated concentration at C i = 0, β 1 is the slope coefficient that predicts M i from C i , and ε i is the error associated with the prediction of M i . Unfortunately, Equation [5.1] is not quite what we want for our chemical analysis method because it allows us to predict a measurement from a known standard concentration. When analyses are actually being performed, we wish to use the observed measurement to predict the unknown true concentration. To do this, we must rearrange Equation [5.1] to give: [5.2] In Equation [5.2] β 0 and β 1 are the same as those in [5.1], but C i is the unknown concentration of the compound of interest, M i is the measurement from sample i, and ε ' i is the error associated with the “inverse” prediction of C i from M i . This procedure is termed inverse prediction because the original regression model was fit to predict M i from C i , but then is rearranged to predict C i from M i . Note also that the error terms in [5.1] and [5.2] are different because inverse prediction has larger errors than simple prediction of y from x in a regular regression model. Detection Limits The point of this discussion is that the reported concentration of any chemical in environmental media is an estimate with some degree of uncertainty. In the calibration process, chemists typically define some C n value that is not significantly different from zero, and term this quantity the “method detection limit.” That is, if we used the ε ' distribution from [5.2] to construct a confidence interval for C, C n would be the largest concentration whose 95% (or other interval width) confidence interval includes zero. Values below the limit of detection are said to be censored because we cannot measure the actual concentration and thus all values less than Cn are reported as “less than LOD,” “nondetect,” or simply “ND.” While this seems a rather simple concept the statistical process of defining exactly what the LOD is for a given analytical procedure is not (Gibbons, 1995). Quantification Limits Note that as might be expected from [5.2] all estimated C i values, , have an associated error distribution. That is: [5.3] C i M i β 0 – β 1 ε ’ i += c ˆ i c ˆ i κ i ε i += steqm-5.fm Page 112 Friday, August 8, 2003 8:16 AM ©2004 CRC Press LLC where κ i is the true but unknown concentration and ε i is a random error component. When is small, it can have a confidence interval that does not include zero (thus it is not an “ND”) but is still quite wide compared to the concentration being reported. For example, one might have a dioxin concentration reported as 500 ppb, but with a 95% confidence interval of 200 to 1,250 ppb. This is quite imprecise and would likely be reported as below the “limit of quantification” or “less than LOQ.” However, the fact remains that a value reported as below the limit of quantification still provides evidence that the substance of interest has been identified. Moreover, if the measured concentrations are unbiased, it is true that the average error is zero. That is: [5.4] Thus if we have many values below the LOQ it is true that: [5.5] and for large samples, [5.6] That is, even if all values are less than LOQ, the sum is still expected to equal the sum of the unknown but true measurements and by extension, the mean of a group of values below the LOQ, but above the DL, would be expected to equal the true sample mean. It is worthwhile to consider the LOQ in the context of the calibration process. Sometimes an analytic method is calibrated across a rather narrow range of standard concentrations. If one fits a statistical model to such data, the precision of predictions can decline rapidly as one moves away from the range of the data used to fit the model. In this case, one may have artificially high LOQs (and Detection Limit or DLs as well) as a result of the calibration process itself. Moreover, if one moves to concentrations above the range of calibration one can also have unacceptably wide confidence intervals. This leads to the seeming paradox of values that are too large to be acceptably precise. This general problem is an issue of considerable discussion among statisticians engaged in the evaluation of chemical concentration data (see for example: Gilliom and Helsel, 1986; Helsel and Gilliom, 1986; Helsel, 1990a 1990b). The important point to take away from this discussion is that values less than LOQ do contain information and, for most purposes, a good course of action is to simply take the reported values as the actual values (which is our expectation given unbiased measurements). The measurements are not as precise as we would like, but are better than values reported as “<LOQ.” Another point is that sometimes a high LOQ does not reflect any actual limitation of the analytic method and is in fact due to calibration that was performed c ˆ i ε i∑ 0= c ˆ i ∑ κ i∑ ε i∑ += c ˆ i ∑ κ i∑ = steqm-5.fm Page 113 Friday, August 8, 2003 8:16 AM ©2004 CRC Press LLC over a limited range of standard concentrations. In this case it may be possible to improve our understanding of the true precision of the method being used by doing a new calibration study over a wider range of standard concentrations. This will not make our existing <LOQ observations any more precise, but may give us a better idea of how precise such measurements actually are. That is, if we originally had a calibration data set at 200, 400, and 800 ppm and discovered that many field measurements are less than LOQ at 50 ppm, we could ask the analytical chemist to run a new set of calibration standards at say 10, 20, 40, and 80 ppm and see how well the method actually works in the range of concentrations encountered in the environment. If the new calibration exercise suggests that concentrations above 15 ppm are measured with adequate precision and are thus “quantified,” we should have greater faith in the precision of our existing less than LOQ observations. Censored Data More often, one encounters data in the form of reports where the original raw analytical results are not available and no further laboratory work is possible. Here the data consist of the quantified data that are reported as actual concentrations, the less than LOQ observations that are reported as less than LOQ, together with the concentration defining the LOQ and values below the limit of detection, that are reported as ND, together with concentration defining the limit of detection (LOD). It is also common to have data reported as “not quantified” together with a “quantification limit.” Such a limit may reflect the actual LOQ, but may also represent the LOD, or some other cutoff value. In any case the general result is that we have only some of the data quantified, while the rest are defined only by a cutoff value(s). This situation is termed “left censoring” in statistics because observations below the censoring point are on the left side of the distribution. The first question that arises is: “How do we want to use the censored data set?” If our interest is in estimating the mean and standard deviation of the data, and the number of nonquantified observations (NDs and <LOQs) is low (say 10% of the sample or less), the easiest approach is to simply assume that nondetects are worth 1/2 the detection limit (DL), and that <LOQ values (LVs) are defined as: LV = DL + ½ (LOQ − DL) [5.7] This convention makes the tacit assumption that the distribution of nondetects is uniformly distributed between the detection limit and zero, and that <LOQ values are uniformly distributed between the DL and the LOQ. After assigning values to all nonquantified observations, we can simply calculate the mean and standard deviation using the usual formulae. This approach is consistent with EPA guidance regarding censored data (e.g., EPA, 1986). The situation is even easier if we are satisfied with the median and interquartile range as measures of central tendency and dispersion. The median is defined for any data set where more than half of the observations are quantified, while the interquartile range is defined for any data set where at least 75% of the observations are quantified. steqm-5.fm Page 114 Friday, August 8, 2003 8:16 AM ©2004 CRC Press LLC Estimating the Mean and Standard DevIation Using Linear Regression As shown in Chapter 2, observations from a normal distribution tend to fall on a straight line when plotted against their expected normal scores. This is true even if some of the data are below the limit of detection (see Example 5.1). If one calculates a linear regression of the form: C = A + B • Z-Score [5.8] where C is the measured concentration, A and B are fitted constants, and Z-Score is the expected normal score based on the rank order of the data, A is an estimate of the mean, µ, and B is an estimate of the standard deviation, σ (Gilbert, 1987; Helsel, 1990). Expected Normal Scores The first problem in obtaining expected normal scores is to convert the ranks of the data into cumulative percentiles. This is done as follows: 1. The largest value in a sample of N receives rank N, the second largest receives rank N − 1, the third largest receives rank N − 2 and so on until all measured values have received a rank. In the event that two or more values are tied (in practice this should happen very rarely; if you have many tied values you need to find out why), simply assign one rank K and one rank K − 1. For example if the five largest values in a sample are unique, and the next two are tied, assign one rank 6 and one rank 7. 2. Convert each assigned rank, r, to a cumulative percentile, P, using the formula: [5.9] We note that other authors (e.g., Gilliom and Helsel, 1986) have used different formulae such as P = r/(N + 1). We have found that using P values calculated using [5.8] provide better approximations to tabled Expected Normal Scores (Rohlf and Sokol, 1969) and thus will yield more accurate regression estimates of µ and σ . 3. Once P values have been calculated for all observations, one can obtain expected normal or Z scores using the relationship: Z(P) = ϕ (P) [5.10] Here Z(P) is the z-score associated with the cumulative probability P, and ϕ is the standard normal inverse cumulative distribution function. This function is shown graphically in Figure 5.2. 4. Once we have obtained Z values for each P, we are ready to perform a regression analysis to obtain estimates of µ and σ . P r3/8 –() N1/4 +() = steqm-5.fm Page 115 Friday, August 8, 2003 8:16 AM ©2004 CRC Press LLC Example 5.1 contains a sample data set with 20 random numbers, sorted smallest to largest, generated from a standard normal distribution (µ = 0 and σ = 1), cumulative percentiles calculated from Equation 5.8, and expected normal scores calculated from these P values. When we look at Example 5.1, we see that the estimates for µ and σ look quite close to the usual estimates of µ and σ except for the case where 75% of the data (15 observations) are censored. Note first that even when we have complete data we do not reproduce the parametric values, µ = 0 and σ = 1. This is because we started with a 20-observation random sample. For the case of 75% censoring the estimated value for µ is quite a bit lower than the sample value of − 0.3029 and the estimated value for σ is also a good bit higher than the sample value of 1.0601. However, it is worthwhile to consider that if we did not use the regression method for censored data, we would have to do something else. Let us assume that our detection limit is really 0.32, and assign half of this value, 0.16, to each of the 15 “nondetects” in this example and use the usual formulae to calculate µ and σ . The resulting estimates are µ = 0.3692 and σ = 0.4582. That is, our estimate for µ is much too large and our estimate for σ is much too small. The moral here is that regression estimates may not do terribly well if a majority of the data is censored, but other methods may do even worse. The sample regression table in Example 5.1 shows where the Statistics presented for the 4 models (20 observations, 15 observations, 10 observations, 5 observations) come from. The CONSTANT term is the intercept for the regression equation and provides our estimate of µ, while the ZSCORE term is the slope of the regression line and provides our estimate of σ . The ANOVA table is included because the regression procedure in many statistical software packages provides this as part of the output. Note that the information required to estimate µ and σ is found Figure 5.2 The Inverse Normal Cumulative Distribution Function steqm-5.fm Page 116 Friday, August 8, 2003 8:16 AM ©2004 CRC Press LLC in the regression equation itself, not in the ANOVA table. The plot of the data with the regression curve includes both the “detects” and the “nondetects.” However, only the former were used to fit the curve. With real data we would have only the detect values, but this plot is meant to show why regression on normal scores works with censored data. That is, if the data are really log-normal, regression on those data points that we can quantify will really describe all of the data. An important point concerning using regression to estimate µ and σ is that all of the tools discussed in our general treatment of regression apply. Thus we can see if factors like influential observations or nonlinearity are affecting our regression model and thus have a better idea of how good our estimates of µ and σ really are. Maximum Likelihood There is another way of estimating µ and σ from censored data that also does relatively well when there is considerable left-censoring of the data. This is the method of maximum likelihood. There are some similarities between this method and the regression method just discussed. When using regression we use the ranks of the detected observations to calculate cumulative percentiles and use the standard normal distribution to calculate expected normal scores for the percentiles. We then use the normal scores together with the observed data in a regression model that provides us with estimates of µ and σ . In the maximum likelihood approach we start by assuming a normal distribution for the log-transformed concentration. We then make a guess as to the correct values for µ and σ . Once we have made this guess we can calculate a likelihood for each observed data point, using the guess about µ and σ and the known percentage, ψ , of the data that is censored. We write this result as L(x i *µ, σ , ψ ). Once we have calculated an L for each uncensored observation, we can calculate the overall likelihood of the data, L(X *µ, σ , ψ ) as: [5.11] That is the overall likelihood of the data given µ, σ , and ψ , L(X *µ, σ , ψ ), is the product of the likelihoods of the individual data points. Such calculations are usually carried out under logarithmic transformation. Thus most discussions are in terms of log-likelihood, and the overall log-likelihood is the sum of the log- likelihoods of the individual observations. Once L(X *µ, σ , ψ ) is calculated there are methods for generating another guess at the values for µ and σ , that yields an even higher log-likelihood. This process continues until we reach values of µ and σ that result in a maximum value for L(X *µ, σ , ψ ). Those who want a technical discussion of a representative approach to the likelihood maximization problem in the context of censored data should consult Shumway et al. (1989). The first point about this procedure is that it is complex compared to the regression method just discussed, and is not easy to implement without special software (e.g., Millard, 1997). The second point is that if there is only one censoring value (e.g., detection limit) maximum likelihood and regression almost always give LXµσψ,,()Lx i µσψ,,() i1= N ∏ = steqm-5.fm Page 117 Friday, August 8, 2003 8:16 AM ©2004 CRC Press LLC essentially identical estimates for µ and σ , and when the answers differ somewhat there is no clear basis for preferring one method over the other. Thus for reasons of simplicity we recommend the regression approach. Multiply Censored Data There is one situation where maximum likelihood methods offer a distinct advantage over regression. In some situations we may have multiple “batches” of data that all have values at which the data is censored. For example, we might have a very large environmental survey where the samples were split among several labs that had somewhat different instrumentation and thus different detection and quantification limits. Alternatively, we might have samples with differing levels of “interference” for the compound of interest by other compounds and thus differing limits for detection and quantification. We might even have replicate analyses over time with declining limits of detection caused by improved analytic techniques. The cause does not really matter, but the result is always a set of measurements consisting of several groups, each of which has its own censoring level. One simple approach to this problem is to declare all values below the highest censoring point (the largest value reported as not quantified across all groups) as censored and then apply the regression methods discussed earlier. If this results in minimal data loss (say, 5% to 10% of quantified observations), it is arguably the correct course. However, in some cases, especially if one group has a high censoring level, the loss of quantified data points may be much higher (we have seen situations where this can exceed 50%). In such a case, one can use maximum likelihood methods for multiply censored data such as those contained in Millard (1997) to obtain estimates for µ and σ that utilize all of the available data. However, we caution that estimation in the case of multiple censoring is a complex issue. For example, the pattern of censoring can affect how one decides to deal with the data. When dealing with such complex issues, we strongly recommend that a professional statistician, one who is familiar with this problem area, be consulted. Example 5.1 The Data for Regression Y Data (Random Normal) Sorted Smallest to Largest Cumulative Proportion from Equation 5.8 Z-Scores from Cumulative Proportions − 2.012903 − 1.920049 − 1.878268 − 1.355415 − 0.986497 − 0.955287 − 0.854412 − 0.728491 − 0.508235 − 0.388784 0.030864 0.080247 0.129630 0.179012 0.228395 0.277778 0.327161 0.376543 0.425926 0.475307 − 1.868241 − 1.403411 − 1.128143 − 0.919135 − 0.744142 − 0.589455 − 0.447767 − 0.314572 − 0.186756 − 0.061931 steqm-5.fm Page 118 Friday, August 8, 2003 8:16 AM ©2004 CRC Press LLC Statistics • Summary Statistics for the Complete y Data, using the usual estimators: Mean = − 0.3029 SD = 1.0601 • Summary Statistics for the Complete Data, using regression of the complete data on Z- Scores: Mean = − 0.3030 SD = 1.0902 R 2 = 0.982 • Summary Statistics for the 15 largest y observations (y = -0.955287 and larger), using regression of the data on Z- Scores: Mean = − 0.3088 SD = 1.1094 R 2 = 0.984 • Summary Statistics for the 10 largest y observations (y = − 0.168521 and larger), using regression of the data on Z- Scores: Mean = − 0.2641 SD = 1.0661 R 2 = 0.964 • Summary Statistics for the 5 largest y observations (y = 0.440684 and larger), using regression of the data on Z- Scores: Mean = − 0.5754. SD = 1.2966 R 2 = 0.961 The Regression Table and Plot for the 10 Largest Observations − 0.168521 0.071745 0.084101 0.256237 0.301572 0.440684 0.652699 0.694994 1.352276 1.843618 0.524691 0.574074 0.623457 0.672840 0.722222 0.771605 0.820988 0.870370 0.919753 0.969136 0.061932 0.186756 0.314572 0.447768 0.589456 0.744143 0.919135 1.128143 1.403412 1.868242 Unweighted Least-Squares Linear Regression of Y Predictor Variables Coefficient Std Error Student’s t P Constant − 0.264 0.068 − 3.87 0.0048 Z-score 1.066 0.074 14.66 0.0000 The Data for Regression (Cont’d) Y Data (Random Normal) Sorted Smallest to Largest Cumulative Proportion from Equation 5.8 Z-Scores from Cumulative Proportions steqm-5.fm Page 119 Friday, August 8, 2003 8:16 AM ©2004 CRC Press LLC R-SQUARED 0.9641 Estimating the Arithmetic Mean and Upper Bounds on the Arithmetic Mean In Chapter 2, we discussed how one can estimate the arithmetic mean concentration of a compound in environmental media, and how one might calculate an upper bound on this arithmetic mean. Our general recommendation was to use the usual statistical estimator for the arithmetic mean and to use bootstrap methodology (Chapter 6) to calculate an upper bound on this mean. The question at hand is how do we develop estimates for the arithmetic mean, and upper bounds for this mean, when the data are censored? One approach that is appealing in its simplicity is to use the values of µ and σ , estimated by regression on expected normal scores, to assign values to the censored observations. That is, if we have N observations, k of which are censored, we can assume that there are no tied values and that the ranks of the censored observations are 1 through k. We can then use these ranks to calculate P values using Equation [5.9], and use the estimates P values to calculate expected normal scores ANOVA Table Source DF SS MS F P Regression 1 3.34601 3.34601 214.85 0.0000 Residual 8 0.12459 0.01557 Total 9 3.47060 Figure 5.3 A Regression Plot of the Data Used in Example 5.1 steqm-5.fm Page 120 Friday, August 8, 2003 8:16 AM ©2004 CRC Press LLC [...]... Z-Scores from Cumulative Proportions − 0.16 852 1 0.0717 45 0.084101 0. 256 237 0.30 157 2 0.440684 0. 652 699 0.694994 1. 352 276 1.843618 0.061932 0.186 756 0.31 457 2 0.447768 0 .58 9 456 0.744143 0.9191 35 1.128143 1.403412 1.868242 Data Calculated from Estimates of µ and σ Observed Exponential Transform of Calculated for Censored and Observed for Uncensored 0.84491 35 1.0743813 1.0877387 1.292 058 9 1. 351 98 25 1 .55 37696... Observed for Uncensored − 1.868240 − 1.403411 − 1.128143 − 0.9191 35 − 0.744142 − 0 .58 9 455 − 0.447767 − 0.31 457 2 − 0.186 756 − 0.061931 − 2. 255 831 − 1.760276 − 1.466813 − 1.243989 − 1. 057 429 − 0.89 251 8 − 0.741464 − 0 .59 94 65 − 0.463200 − 0.330124 0.1047864 0.1719973 0.230 659 4 0.2882319 0.3473474 0.4096230 0.4764 157 0 .54 91 052 0.6292664 0.7188341 steqm -5 . fm Page 122 Friday, August 8, 2003 8:16 AM Calculating the... Chemical Mixtures, 51 FR 3401 4-3 40 25 Gibbons, R D., 19 95, “Some Statistical and Conceptual Issues in the Detection of Low Level Environmental Pollutants,” Environmental and Ecological Statistics 2: 1 25 144 Gilbert, R O., 1987, Statistical Methods for Environmental Pollution Monitoring, Van Nostrand Reinhold, New York Gilliom, R J and Helsel, D R., 1986, “Estimation of Distributional Parameters for Censored... relationship: P ( max = 0 .5 1 / 200 ) Thus, P(max) = 0.99 654 3 We now determine from [5. 15] that SP = ZI (0.99 654 ) = 2.7007 4 The estimate for the logarithmic mean, µ, is given by [5. 16] and is: µ = Ln ( LOD – S P • σ ) µ – Ln ( 1 – ( 2.7007 • 1 ) ) µ = – 2.7007 5 Using [5. 17] the estimate for the geometric mean, GM, is: GM = e µ ©2004 CRC Press LLC steqm -5 . fm Page 126 Friday, August 8, 2003 8:16 AM GM = e... 3.8662 150 6.3193604 Statistics • Summary statistics for the complete exponentially transformed Y data from Example 5. 1 (column 1), using the usual estimators: Mean = 1.24 75 SD = 1.4881 • Summary statistics for the exponentially transformed Y data from column 4 above: Mean = 1.2621 SD = 1.4797 • Bootstrap percentiles (2,000 replications) for the exponentially transformed complete data from Example 5. 1... S-Plus Probability, Statistics and Information, Seattle, WA Rohlf, F J and Sokol, R R., 1969, Statistical Tables, Table AA, W H Freeman, San Francisco Shumway, R H., Azari, A S., and Johnson, P., 1989, “Estimating Mean Concentrations Under Transformation for Environmental Data with Detection Limits,” Technometrics, 31: 347– 356 USEPA, 1994, Guidance for the Data Quality Objectives Process, EPA QA/G-4... values for p will be the value that will produce the evidence with a probability of 0. 05 In other words we are 95 percent confident that p is less than this value Using Equation [2.23], this is formalized as follows: 0 ) f ( x=0 = 28 P u ( 1 – p u ) 0 28 ≥ 0. 05 Solving for pu, [5. 13] ) p u = 1.0 – ( 0. 05 1 / 28 = 0.10 The interval 0.0 ≤ p ≤ 0.10 not only contains the “true” value of p with 95. .. the LODs for the chemicals are high, one can have a situation where the HI is above 1 for a site where no hazardous chemicals have been detected! The solution to this dilemma is to remember (Chapter 2) that if we have N observations, we can calculate the median cumulative probability for the largest sample observation, P(max), as: P ( max = ( 0 .5 ) ) 1/N [5. 14] For the specific case of a log-normal... [5. 14] as follows, we may use the identity connecting the beta distribution and the binomial distribution to obtain values for pL and pU (see Guttman, 1970): Prob ( x < X p L =α ) ⁄ 2 Prob ( x ≤ X p U =α ) ⁄ 2 [5. 15] In the hypothetical case of one out of 28 observations reported as above the MDL, pL = 0.0087 and pU = 0.18 35 Therefore the 95 percent fiducial, or confidence, interval (0.0087, 0.18 35) ... II error is easily determined via Equation [5. 17]: K–1 ∑ Prob ( Type II Error K = 1.0 – ) k=0 where ( 0.1 < p ) ©2004 CRC Press LLC 7 k ) p ( 1.0 – p k 7–k [5. 17] steqm -5 . fm Page 130 Friday, August 8, 2003 8:16 AM Figure 5. 5 Probability of Type II Errors for Various Critical Counts Note that the risk of making a Type II error is near 90 percent for a critical count of 3, Prob(Type II Error|K . 1.868242 Observed 0.84491 35 1.0743813 1.0877387 1.292 058 9 1. 351 98 25 1 .55 37696 1.9207179 2.0036971 3.8662 150 6.3193604 50 % 75% 90% 95% Example 5. 1 1.2283 1. 450 1 1.6673 1.8217 Example 5. 2 1.2446 1.4 757 1.7019. 1.128143 − 0.9191 35 − 0.744142 − 0 .58 9 455 − 0.447767 − 0.31 457 2 − 0.186 756 − 0.061931 − 2. 255 831 − 1.760276 − 1.466813 − 1.243989 − 1. 057 429 − 0.89 251 8 − 0.741464 − 0 .59 94 65 − 0.463200 − 0.330124 0.1047864 0.1719973 0.230 659 4 0.2882319 0.3473474 0.4096230 0.4764 157 0 .54 91 052 0.6292664 0.7188341 x x x x x steqm -5 . fm. groundwater for contamination, − 0.16 852 1 0.0717 45 0.084101 0. 256 237 0.30 157 2 0.440684 0. 652 699 0.694994 1. 352 276 1.843618 0.061932 0.186 756 0.31 457 2 0.447768 0 .58 9 456 0.744143 0.9191 35