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Keywords:

  • Chronic toxicity test;
  • Poisson regression mixture model;
  • Ceriodaphnia dubia;
  • Inhibition concentration

Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. CURRENT POTENCY ESTIMATORS
  5. MIXTURE MODEL
  6. CHRONIC TOXICITY TESTING WITH ADJUSTMENT OF MORTALITY EFFECTS
  7. RESULTS
  8. INHIBITION CONCENTRATIONS AND THE CONFIDENCE INTERVALS
  9. CONCLUSION
  10. References

Chronic toxicity tests, such as the Ceriodaphnia dubia 7-d test are typically analyzed using standard statistical methods such as analysis of variance or regression. Recent research has emphasized the use of Poisson regression or more generalized regression for the analysis of the fecundity data from these studies. A possible problem in using standard statistical techniques is that mortality may occur from toxicant effects as well as reduced fecundity. A mixture model that accounts for fecundity and mortality is proposed for the analysis of data arising from these studies. Inferences about key parameters in the model are discussed. A joint estimate of the inhibition concentration is proposed based on the model. Confidence interval estimation via the bootstrap method is discussed. An example is given for a study involving copper and mercury.


INTRODUCTION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. CURRENT POTENCY ESTIMATORS
  5. MIXTURE MODEL
  6. CHRONIC TOXICITY TESTING WITH ADJUSTMENT OF MORTALITY EFFECTS
  7. RESULTS
  8. INHIBITION CONCENTRATIONS AND THE CONFIDENCE INTERVALS
  9. CONCLUSION
  10. References

Protection of freshwater resources requires evaluation of potential toxicity from possible pollutants in fertilizers, pesticides, and industrial and municipal discharge [1]. While traditional approaches to the study of toxicity are focused on mortality, recent attention has shifted to chronic effects such as growth or fecundity. The Ceriodaphnia test is perhaps the most common of the chronic tests. The experimental design consists of placing organisms in beakers, treating the units with the chemical or dose of interest, and monitoring the number of young produced over time (usually 7 d). The traditional method of analyzing these data is to apply statistical tests based on analysis of variance followed by pairwise tests to detect treatment concentrations that differ from the control group. Contemporary methods also use regression models based on a generalized linear model framework [2] or a quasi-likelihood technique [3].

When mortality occurs, interpreting the chronic response is often complicated. The methods mentioned above are limited to separate analysis of mortality and fecundity. Since mortality often occurs during the test (especially for high doses), extra zeroes in the response result and the above methods may be inappropriate and produce biased estimates. Morgan [4] discussed the use of Abbott's formula [5] and other methods to adjust the natural or control mortality in mortality modeling. However, not much has been done to adjust for treatment mortality in chronic toxicity testing. A mixture approach is proposed to model and analyze the fecundity data with adjustment for mortality effects. In addition, a combined estimate of potency is described that incorporates both effects.

CURRENT POTENCY ESTIMATORS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. CURRENT POTENCY ESTIMATORS
  5. MIXTURE MODEL
  6. CHRONIC TOXICITY TESTING WITH ADJUSTMENT OF MORTALITY EFFECTS
  7. RESULTS
  8. INHIBITION CONCENTRATIONS AND THE CONFIDENCE INTERVALS
  9. CONCLUSION
  10. References

Currently, there are two commonly used approaches for analyzing and estimating critical concentrations from chronic Ceriodaphnia dubia tests. The approaches use different statistical models, the analysis of variance or means model, and the regression model. The analysis of variance approach results in quantities such as the chronic value (ChV), no observed effect concentration (NOEC), or the lowest observed effect concentration (LOEC). For example, the ChV procedure uses sequential statistical tests following analysis of variance to detect treatment concentrations that differ from the control group. Either parametric or nonparametric hypothesis tests are recommended depending on the distributional assumptions considered for the response variable [1]. The NOEC and the LOEC are obtained based on these statistical test results. The NOEC is the highest concentration level where the mean response does not differ statistically from the mean response of the control group (zero concentration group). The LOEC is the lowest concentration level where the mean response is observed to be statistically different from the mean response in the control group. A point estimate of an estimated safe concentration, the ChV, is then calculated as the geometric mean between these two concentrations. Since the ChV calculation is based on an hypothesis testing procedure, it is highly dependent on the sample sizes and the intratreatment variation [6,7]. The chronic value estimate also depends on the design of the experiment because the LOEC and the NOEC can only be tested concentration levels.

The regression approach involves the fitting of a regression model and the estimation of an inhibition concentration (IC). The method results in an estimate, IC[100·q], corresponding to the concentration that causes a specified percent reduction (100·q%, where 0 < q < 1) in reproduction output [1,8,9]. The concentration of interest is the concentration where the mean response is equal to the proportion (1 − q) of the mean response of the control. This may be represented as the concentration x such that

  • equation image

and μC is the mean response of the control. This analysis usually assumes reproduction decreases monotonically as concentration of toxicant increases. There are a variety of approaches for estimating ICs, including inverse regression and interpolation. For the interpolation method, linear interpolation between the two nearest concentration values that produce inhibition levels above and below the specified percentage is used to calculate the point estimate endpoint. Hence, in order to obtain the IC50, the value μx = 0.5·μC is needed. It has been observed that the 50% inhibition point (IC50) roughly corresponds to calculated ChV (= geometric mean of LOEC and NOEC) values and that the 25% inhibition point (IC25) corresponds roughly to NOECs from hypothesis testing analyses [8,9]. As pointed out by Oris and Bailer [6], these relationships may be spurious and related in part to the design of the studies.

The linear interpolation used in the IC procedure often obtains unsatisfactory results because the linear assumption may not be appropriate. Also since the calculation uses the observed means, the estimate can be affected by outliers or atypical values. Quite often these represent zero fecundity due to mortality.

Several authors have suggested regression-type models based on a dose-response relationship to improve the interpolation-based IC procedure and discuss the advantages of using them in chronic toxicity testing [10,11]. Bailer and Oris [11] proposed a concentration-response model as an alternative to the ChV calculation and the linear interpolation method. They modeled the relationship between exposure concentration and total number of offspring and estimated concentrations that lead to some specified level of inhibition relative to the control group. Bailer and Oris [11] found the commonly used Poisson assumption to be adequate for their model involving count data. Other authors also proposed regression-type models based on the generalized linear model framework [2] and the quasi-likelihood technique [3]. Bailer and Oris [12] present a general approach for calculation of various endpoints using a regression approach and distributions other than the Poisson.

MIXTURE MODEL

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. CURRENT POTENCY ESTIMATORS
  5. MIXTURE MODEL
  6. CHRONIC TOXICITY TESTING WITH ADJUSTMENT OF MORTALITY EFFECTS
  7. RESULTS
  8. INHIBITION CONCENTRATIONS AND THE CONFIDENCE INTERVALS
  9. CONCLUSION
  10. References

Even though the proposed regression models [11] have many advantages over the ChV and inhibition IC methods in chronic toxicity testing, their use is limited to the modeling of fecundity apart from mortality or mortality apart from fecundity. Since mortality typically occurs for some organisms during the test, mortality often complicates the response of interest. Approaches based on ignoring animals that die or treating fecundity as zero are inappropriate and produce biased estimates. In order to incorporate these extra zeroes due to mortality, a mixture approach is proposed to model and to account for mortality and analyze the fecundity data. The mixture approach has been applied in other fields to combine zero and nonzero observations. Lachenbruch [13] proposed a mixture model to analyze cell transplantation data with zeroes due to transplant failure. Feuerverger [14] mixed zeroes with a gamma distribution to model rainfall data. Heilbron [15] and Lambert [16] separately developed regression models mixing zeroes with the Poisson distribution to analyze high-risk behavior data and quality control data, respectively.

In this paper, the model allows zeroes to be produced from two different sources, mortality and nonreproductive organisms. The response Y (number of offspring) depends on two variables, X and Z via the model Y = X·Z, where X is a nonnegative variable representing the number of offspring and Z is a dichotomous variable representing mortality, with Z = 0 or 1. When mortality information is provided, Z is observed and Y is observed when the organism is alive. However, most of the time, mortality information is missing and only Y is observable. This mixture model accounts for a certain number of animals that die before any eggs are produced. However, the model does not account for the actual time of death. Time of death can be incorporated into the model and will be discussed in a later paper. The analysis of the mixture model is based on the likelihood function of the data. The likelihood function measures how likely the data are for a given model and set of parameters. By varying the parameters, optimal parameters may be obtained (referred to as maximum likelihood estimates) and the model likelihood evaluated. The model likelihood is usually calculated in terms of logarithms, and a common summary measure is the log of the likelihood given the parameter estimates. Models with small (close to zero) likelihoods and a small number of parameters are usually considered better models.

Table Table 1.. The means and variances for the zero inflated Poisson and Poisson distributions; λ is the mean number of offspring and p is the probability of death; X refers to the random variable representing the fecundity
Thumbnail image of

The Poisson and zero inflated Poisson distributions are represented in Table 1 along with their means and variances. The effect of mortality may be evaluated by comparing the mean and variance of the two distributions. As the probability of mortality increases, the mean of the zero inflated Poisson decreases relative to the Poisson mean. Thus, the ordinary average will underestimate the mean fecundity. The bias will thus be the greatest when there is high fecundity and also high mortality. The effect of additional zeroes is generally to increase the variance. The variance for the zero inflated Poisson will be greater than that of the Poisson unless (1 − p)λ is less than one. The formula also suggests that the variance should be a function of dose. At low doses, mortality will tend to increase the variance of the counts relative to the Poisson model, while at large doses, fecundity will be low and mortality high and the variance may decrease relative to the Poisson variance.

CHRONIC TOXICITY TESTING WITH ADJUSTMENT OF MORTALITY EFFECTS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. CURRENT POTENCY ESTIMATORS
  5. MIXTURE MODEL
  6. CHRONIC TOXICITY TESTING WITH ADJUSTMENT OF MORTALITY EFFECTS
  7. RESULTS
  8. INHIBITION CONCENTRATIONS AND THE CONFIDENCE INTERVALS
  9. CONCLUSION
  10. References

A typical chronic toxicity test uses less than 24-h-old juveniles (all released within the same 8-h period) and measures survival and reproductive output of 10 to 20 individuals for each treatment in a 7-day period. We refer to treatment as a mix of toxicants (effluent) or a single chemical. One organism is placed in each container and a dose applied. Each treatment is usually replicated 10 to 20 times, depending on the individual experiment. The number of young produced each day is counted and the experiment is run for 7 d. The response is the number of young produced by each organism during the 7-d period. The analysis is done on the total number of young over the 7-d period for each treatment. If the dose is sufficiently great, there is mortality and a high zero count may occur. A mixture model is used to adjust the mortality effects. Since the responses (Y's) are young counts, the model assumes that the counts are independent. Also, it is assumed that mortality occurs with certain probability pi (for the ith dose of the treatment) and the counts follow a Poisson distribution with mean λ. Hence, this mixture model accounts for a certain number of animals that die before any young are produced. This model is the same as the zero inflated Poisson (ZIP) regression model proposed by Lambert [16] and the added zero model proposed by Heilbron [15].

Table Table 2.. Summary of single regressor Poisson models for chromium (Cr), copper (Cu), mercury (Hg), and zinc (Zn); estimated model parameters for the intercept and slope are given with standard error in parentheses; SSR denotes the sum of squared residuals for the fitted model
 CrCuHgZn
β02.812.882.842.66
 (0.0151)(0.0143)(0.0123)(0.0171)
β1−0.208−0.096−0.120−0.0049
 (0.00967)(0.00364)(0.00350)(0.00063)
SSR1,6191,2391,0811,838
−log likelihood−13,471−13,854−13,923−13,256

The model

The observations from the study may be represented as Yij, where i indexes the dose and j the replicate (in most studies, there will be d doses and n replicates). These observations are assumed to be independent (i.e., there are separate beakers for each replicate [organism] and these are randomly located). Then

  • equation image

The relationship with dose may be modeled generally through the parameters λ and p using the linear model

  • equation image

where β0, β1, γ0, and γ1 are parameters.

Approaches to modeling this mixture depend on whether or not p and λ are directly related. In the present example, we assume that p and λ are unrelated and also that the relationship is linear in a single-dose variable. The model is easily generalized to situations where the logit model depends on a different independent variable or sets of variables. For example, the Poisson part may be linear in dose while the logit model is quadratic in dose. Under the above assumptions, the mixture model requires twice as many parameters as the Poisson regression model. By assuming an extra random variable Zij to indicate the source of zero observations, where Zij = 0 if the observation Yij is from the Poisson part and Zij = 1 otherwise, the likelihood function of the model can easily be broken down into several parts. In terms of the variables, there is incomplete information to model the data. For example, if the organism dies, the fecundity is unknown. An approach to modeling this type of data is through the expectation-maximization, or EM, algorithm. The EM algorithm is used to obtain the maximum likelihood estimators for the coefficients of the ZIP model, treating the Zij's as the missing values. Details of the EM algorithm for ZIP regression are presented by Lambert [16]. The algorithm was implemented using the S-plus [17] programming language. The case where mortality is known is discussed by Perry [18].

RESULTS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. CURRENT POTENCY ESTIMATORS
  5. MIXTURE MODEL
  6. CHRONIC TOXICITY TESTING WITH ADJUSTMENT OF MORTALITY EFFECTS
  7. RESULTS
  8. INHIBITION CONCENTRATIONS AND THE CONFIDENCE INTERVALS
  9. CONCLUSION
  10. References

The model was applied to an experiment involving four chemicals (copper, mercury, zinc, and chromium). The total young counts for the 7-d period are used as the response. Thus, the same data are used throughout with only the independent variable changing. In order to keep the comparison of the inhibition concentration simple, only the single-toxicant models are compared in this paper. The Poisson regression approach was first evaluated, followed by the ZIP model. Table 2 presents the results of fitting Poisson models to each chemical.

This approach assumes that mean fecundity follows a Poisson distribution, but it does not consider the mortality effect on the responses. The copper model and the mercury model are the best Poisson models among the four models with the lowest negative log-likelihood (−13,854 for copper and − 13,923 for mercury). The mixture model approach based on the ZIP model [16] is summarized in Table 3. The copper model and the mercury model are the best among the four models with the lowest negative log-likelihood (−15,400 for copper and − 15,485 for mercury; Table 3). However, the mixture model has four parameters, two more than the Poisson model.

Different hypotheses can be tested based on the likelihood ratio test. One hypothesis is that the Poisson model is better than the ZIP model. The test statistic follows an asymptotic chi-square distribution and, for the copper (CU) model, 2(15,400 − 13,854) = 2·1,546 = 3,092 with (4 − 2) = 2 df. This is highly significant and indicates that the ZIP model is better than the regular Poisson model. In fact, based on the log likelihood, the zero inflated model provides a better fit for all four data sets.

The fitted models for copper and mercury are plotted in Figures 1 and 2, respectively. The range of the dose axis is chosen to allow comparison of the fits for copper and mercury and is based on the range of copper concentrations. Both the Poisson and mixture models are plotted. The mixture model involves two different parts; the overall prediction is based on the product of two parts equation image, where equation image is the Poisson mean (reproduction) and equation image is the logistic (survival) probability. The survival probabilities are plotted at the bottom of the graphs and they show mercury has a higher mortality effect than copper (the survival probability of mercury drops off faster than that of copper). The plots also show that the mixture model is similar to the Poisson model. It is apparent from the graphs that the data exhibit high variability. The variability is partially due to the use of a single-dose model when there are multiple chemicals. A more complete model will be explored in a later paper.

Table Table 3.. Summary of zero inflated Poisson regression models, each with a single regressor β0, β1 are parameters for the Poisson regression part of the model and γ0, γ1 are for the binary part of the model; Values in parentheses are standard errors; SSR denotes the sum of squared residuals for the fitted model
 CrCuHgZn
β02.902.942.962.90
 (0.0155)(0.0142)(0.0124)(0.0166)
β1−0.052−0.037−0.060−0.002
 (0.0102)(0.0036)(0.0035)(0.00060)
γ0−2.06−1.80−1.99−1.31
 (0.166)(0.123)(0.149)(0.153)
γ10.6050.1310.2220.009
 (0.0773)(0.0151)(0.0243)(0.0051)
SSR1,6041,3041,0981,861
−log likelyhood−15,330−15,401−15,485−15,293

INHIBITION CONCENTRATIONS AND THE CONFIDENCE INTERVALS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. CURRENT POTENCY ESTIMATORS
  5. MIXTURE MODEL
  6. CHRONIC TOXICITY TESTING WITH ADJUSTMENT OF MORTALITY EFFECTS
  7. RESULTS
  8. INHIBITION CONCENTRATIONS AND THE CONFIDENCE INTERVALS
  9. CONCLUSION
  10. References

Since the mixture model involves two different parts, the overall prediction is based on the product of these parts equation image. Both of these parameters were modeled as being linear in the toxicant (x) following transformation. The Poisson part of the mixture model is λ = emath image and the logistic part of the model is p = 1/(1 + emath image). As mentioned in the previous section, IC[100·q] is the concentration of the toxicant required to induce a 100q% level of reproduction inhibition, where 0 < q < 1. For the mixture approach, the inhibition concentration can still be calculated using the regular definition; however, the formula is based on the mixture model. The concentration of interest is where the mean response is a proportion (1 − q) of the mean response of the control group, which can be represented symbolically as the concentration x such that

  • equation image

and μC is the mean response of the control. Assuming the simplest single-covariate model, the overall model prediction is μx = λ(1 − p) = (emath image)/(1 + emath image). The control mean is also needed for calculation of the IC level. This may be estimated directly if sufficient data exist, or the model-based estimate μC = (emath image)/(1 − emath image) may be used. Hence, in order to obtain IC [100·q], the following equation is needed:

  • equation image

By solving this equation, the value of IC[100·q] can be obtained. However, because of the complexity of the equation, it has to be solved by iterated root search methods.

Confidence intervals for the IC concentration can be obtained by using the bootstrap procedure [19]. When using the bootstrap method, the data are treated as a population (rather than a sample from a population). Because we have the entire population, it may be sampled (with replacement) to obtain estimates of the quantities of interest. By repeatedly resampling the population (with replacement) and recalculating the quantity of interest, the distribution (and hence confidence limits) may be obtained. There are two different ways of bootstrapping a regression-type model. One is resampling the response and covariate pairs, and the other is resampling the residuals. Since bootstrapping the response and covariate pairs is less sensitive to the assumptions of the regression model than bootstrapping the residuals [19], the bootstrapping response and covariate pairs method is used.

thumbnail image

Figure Fig. 1.. Plot of reproduction data and fitted model using the Poisson model and the zero inflated Poisson (ZIP) model for copper. The estimated values from the ZIP model and the reproduction part of the model (i.e., excluding zeroes) are plotted. Also plotted are the estimated survival probabilities from the logistic model. Parameter values are presented for the two Poisson and ZIP models.

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thumbnail image

Figure Fig. 2.. Plot of reproduction data and fitted model using the Poisson model and the zero inflated Poisson (ZIP) model for mercury. The estimated values from the ZIP model and the reproduction part of the model (i.e., excluding zeroes) are plotted. Also plotted are the estimated survival probabilities from the logistic model. Parameter values are presented for the two Poisson and ZIP models.

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Table Table 4.. Estimates and confidence intervals for parameters and inhibition concentrations based on 2,000 bootstrap samples; parameter estimates are for the coefficients of the zero inflation Poisson model; variance estimates are given below the parameter estimates; mean, median, and variance are calculated from the bootstrap distribution; the standard deviation is denoted by SD; confidence intervals (CI) are based on the percentile method
 β0β1γ0γ1IC10IC50IC90
Mercury       
 Estimate2.96−0.060−1.980.2221.176.3114.80
 (0.0124)(0.0035)(0.1496)(0.0243)   
 Mean2.97−0.060−1.990.2231.176.3214.82
 Median2.96−0.060−1.980.2321.176.3114.78
 Variance1.96 × 10−41.29 × 10−52.27 × 10−24.63 × 10−41.59 × 10−34.70 × 10−26.05 × 10−1
 SD1.40 × 10−23.58 × 10−31.51 × 10−12.15 × 10−23.98 × 10−22.17 × 10−17.78 × 10−1
 95% CI(2.93, 2.99)(−0.067, −0.053)(−2.34, −1.74)(0.184, 0.268)(1.10, 1.25)(5.90, 6.71)(13.44, 16.29)
 90% CI(2.94, 2.98)(−0.066, −0.054)(−2.27, −1.77)(0.189, 0.260)(1.11, 1.24)(5.951, 6.67)(13.57, 16.16)
 80% CI(2.94, 2.98)(−0.065, −0.055)(−2.18, −1.80)(0.195, 0.251)(1.12, 1.22)(6.04, 6.60)(13.84, 15.83)
Copper       
 Estimated2.94−0.037−1.800.1311.839.9623.89
 value(0.0142)(0.0036)(0.1231)(0.0151)   
 Mean2.94−3.70 × 10−2−1.811.35 × 10−11.839.9223.76
 Median2.94−3.68 × 10−21.351.35 × 10−11.839.8923.76
 Variance3.74 × 10−42.58 × 10−51.77 × 10−24.86 × 10−42.48 × 10−21.00 × 10−16.36
 SD0.0190.0050.1330.0220.1570.8372.50
 95% CI(2.90, 2.984)(−0.047, −0.027)(−2.09, −1.58)(0.096, 0.186)(1.53, 2.18)(8.24, 11.58)(18.74, 28.63)
 90% CI(2.91, 2.98)(−0.046, −0.029)(−2.03, −1.61)(0.103, 0.174)(1.57, 2.11)(8.54, 11.23)(19.57, 27.70)
 80% CI(2.92, 2.97)(−0.043, −0.031)(−1.99, −1.64)(0.108, 0.165)(1.65, 2.03)(8.83, 10.96)(20.52, 26.94)

The model is refitted 2,000 times along with the IC concentration calculation using these bootstrapping techniques. Each bootstrap observations set is sampled from the original observations. In order to maintain the design structure of the original experiment, each treatment is resampled with the same number of replications as in the original experiment.

Three IC levels (IC10, IC50, IC90) for the two best mixture models (copper model and mercury model) are obtained (the algorithm used the uniroot function in S-plus that uses the Newton search method). The confidence intervals of the estimated IC levels are then obtained based on the percentile method using 2,000 samples. For example, for a 95% interval, the bootstrapped quantities are first sorted, then the upper and lower values are located such that 2.5% of the observations are above or below the values. The bootstrap results are presented in Table 4. The IC50 for the mercury model is 6.31 with 95% confidence interval (5.90, 6.71), and the IC50 for the copper model is 9.96 with 95% confidence interval (8.24, 11.58).

CONCLUSION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. CURRENT POTENCY ESTIMATORS
  5. MIXTURE MODEL
  6. CHRONIC TOXICITY TESTING WITH ADJUSTMENT OF MORTALITY EFFECTS
  7. RESULTS
  8. INHIBITION CONCENTRATIONS AND THE CONFIDENCE INTERVALS
  9. CONCLUSION
  10. References

The traditional potency estimation method for chronic bio-assays does not take into account mortality that may occur during the experiment. Standard estimation may be inappropriate and highly biased if many deaths occur during the experiment. The mixture model presented here is able to incorporate the extra zeroes due to mortality. For the data set considered, a likelihood ratio test indicated that the mixture model does describe the data better than the regular Poisson model, although neither model fits the data well. However, this paper only considered the simplest, single-regressor model. The model can be extended to more complicated models involving more than one regressor and even interactions. However, the calculation and interpretation of the IC level are more difficult. Also, this paper discusses a mixture model using zeroes and a discrete (Poisson) distribution. It is possible to extend the analysis approach to other studies that involve continuous data with an excess of zeroes.

References

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. CURRENT POTENCY ESTIMATORS
  5. MIXTURE MODEL
  6. CHRONIC TOXICITY TESTING WITH ADJUSTMENT OF MORTALITY EFFECTS
  7. RESULTS
  8. INHIBITION CONCENTRATIONS AND THE CONFIDENCE INTERVALS
  9. CONCLUSION
  10. References
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    Efron B, Tibshirani RJ. 1993. An Introduction to the Bootstrap. Chapman & Hall, London, UK.