Technical Basis for the Lowest 

Concentration Minimum Reporting Level 

(LCMRL) Calculator

 

 

Office of Water (MLK 140) EPA 815-R-11-001 December 2010 http://water.epa.gov/drink
LCMRL Technical Report December 2010

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Table of Contents

1.0 Introduction.......................................................................................................................... 1

1.1 Programmatic Background............................................................................................... 1

1.2 Scientific Background ...................................................................................................... 2

1.3 Improvements over Previous LCMRL Methodology ...................................................... 3

2.0 General Notation.................................................................................................................. 4

3.0 LCMRL Study Design ......................................................................................................... 5

4.0 Data Conditioning................................................................................................................ 6

5.0 Estimation of Replicate Variance ........................................................................................ 6

5.1 Introduction ...................................................................................................................... 6

5.2 Modified Hodges-Lehmann Location Estimator.............................................................. 7

5.3 Mean Absolute Deviation Scale Estimator ...................................................................... 7

5.4 Huber Location Estimator and Weights........................................................................... 7

5.5 Tukey’s Biweight Location Estimator and Weights........................................................ 8

5.6 Final Location-Scale Estimator and Weights................................................................... 9

6.0 Replicate Variance Model.................................................................................................. 10

6.1 Derivation....................................................................................................................... 10

7.0 Conditional Means Model.................................................................................................. 12

8.0 Conditional Mean Squared Error Model............................................................................ 14

8.1 Estimating Conditional Mean Squared Errors ............................................................... 14

8.2 Fitting Conditional Mean Squared Errors Model........................................................... 14

9.0 Estimating the LCMRL...................................................................................................... 15

9.1 Introduction .................................................................................................................... 15

9.2 Prediction Variance ........................................................................................................ 15

9.3 Conditional Distributions............................................................................................... 16

9.4 Search for LCMRL......................................................................................................... 17

10.0 Estimating the Modified Hubaux-Vos DL......................................................................... 20

10.1 Estimating the Critical Level...................................................................................... 20

10.2 Search for mHV-DL................................................................................................... 21

11.0 Summary, Conclusions and Recommendations................................................................. 22

12.0 References.......................................................................................................................... 23

Appendix A: Computation of MRL............................................................................................ A-1
LCMRL Technical Report December 2010

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1.0 Introduction

1.1 Programmatic Background

The Safe Drinking Water Act Amendments of 1996 require EPA to establish criteria for a 

monitoring program and to publish a list of not more than 30 unregulated contaminants for which 

public water systems (PWS) are to monitor. The monitoring program will provide a national 

assessment of the occurrence of these contaminants in public drinking water that will be used to 

help decide which contaminants may or may not require regulation in the future. In 1999, EPA 

revised the approach for unregulated contaminant monitoring in the Unregulated Contaminant 

Monitoring Regulation (UCMR) (64 FR 50556, USEPA, 1999) and subsequent revisions. 

PWSs will be required to monitor for a variety of contaminants under the UCMR. A Minimum 

Reporting Level (MRL) will be assigned to each contaminant. Under the UCMR, laboratories 

will be required to report all occurrences of listed contaminants at concentrations that are equal 

to or greater than the established MRL. 

MRLs represent an estimate of the lowest concentration of a compound that can be quantitatively 

measured by members of a group of experienced drinking water laboratories. It is based on a 

Measurement Quality Objective (MQO) of 50%-150% recovery of spiked concentrations. 

Informally, at or above the MRL, competent drinking water laboratories should be expected to 

obtain 50%-150% recovery or better. 

In both the Information Collection Rule (ICR) and the UCMR, the EPA Office of Ground Water 

and Drinking Water (OGWDW) specified MRLs and an accuracy requirement for recovery at 

the MRL. The MRL was introduced with the new analytes and new methods for implementing 

the new UCMR. 

The Method Detection Limit procedure (MDL; see 40 CFR part 136, Appendix B) was 

developed by EPA (Glaser et al., 1981) to establish a decision threshold for detection in lowlevel samples. It is based solely on the standard deviation of repeated measurement of low-level 

spikes. The MRL differs from the MDL by considering not only the standard deviation of low 

concentration analyses (precision) but also the accuracy of the measurements as they impact 

achievement of the MQO for spike recovery. 

The practical quantitation limit (PQL) was established by EPA’s drinking water program. The 

PQL is defined as "the lowest concentration of an analyte that can be reliably measured within 

specified limits of precision and accuracy during routine laboratory operation conditions" (52 FR 

25690, USEPA, 1987). The PQL is problematic as a practical matter since at least three different 

methods have been used to determine PQLs: 

1. analysis of performance evaluation (PE) sample data, 

2. a risk-based multiple of the MDL, and 

3. adoption of Contract Laboratory Program Contract Required Quantitation Limits 

(CRQLs) based on lowest nonzero standard in a linear calibration.
LCMRL Technical Report December 2010

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The MRL may be useful as an alternative to the PQL for setting future regulatory limits. 

A necessary element of determining the MRL is a procedure to estimate the lowest level at which 

a single laboratory can meet the specified MQOs. EPA has developed a statistical approach for 

determination of single-laboratory Lowest Concentration MRLs (LCMRLs). This approach uses

variance function modeling, regression and modeling instrument response distributions. 

The MRL will be determined using a Bayesian bootstrap (BB) (Rubin, 1981) of the LCMRL 

estimator using the LCMRL study data from each of several experienced drinking water 

laboratories. The BB replicates that were generated from each laboratory’s data, serve to

estimate the distribution of estimated LCMRL values that each laboratory might generate on 

repeated performance of the LCMRL study. The distribution of pooled BB replicates, generated 

from the LCMRL study data from a sample of experienced drinking water laboratories,

approximates the distribution of estimated LCMRL values which might be generated from the 

population of experienced drinking water laboratories. 

Reasonable values for programmatic required MRLs can be determined based on these simulated 

distributions. Drinking water laboratories will confirm that they are capable of meeting a 

required MRL during their Initial Demonstration of Capability (IDC) for an analytical method. A 

technical description of the MRL procedure is given in Appendix A.

1.2 Scientific Background

The Lowest Concentration Minimum Reporting Level (LCMRL) is defined as the lowest spiking

concentration such that the probability of spike recovery in the 50% to 150% range is at least 

99%. The algorithms and procedure presented in this report provide a means of reliably 

estimating the LCMRL.

The LCMRL calculator constructs conditional (on spiking concentration) mean and variance 

models for analytical measurements. For analytical methods which cannot give negative ‘raw’ 

measurements or negative results (except possibly at low level due to a negative intercept in the 

calibration function), the gamma distribution is used as the distribution of measurements. 

Reported negative measurements due to a calibration artifact are set to zero. Typically, these 

analytical methods are chromatographic methods in which the response is integrated valley to 

valley.

Other classes of methods can obtain ‘raw’ measurements at very low concentration that are 

negative. These methods may be spectroscopic or radiochemical methods and often involve 

subtraction of ‘dark current,’ reference cell readings or background. In these cases the normal 

distribution is used as the distribution of measurements. 

The conditional mean and variance models then specify the parameters for the conditional 

measurement distribution as a function of spiking level. It is this model of the distribution of 

repeated measurements at a given spiking level that is used to estimate the LCMRL.
LCMRL Technical Report December 2010

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The critical level, Lc is defined as an upper percentile of the distribution of repeated 

measurements at zero true concentration (see Section 10.1, and Currie, 1968). The modified 

Hubaux-Vos Detection Limit (mHV-DL) is defined as the spiking concentration that has a 95% 

probability of exceeding the 95% Lc. The conditional distributional model constructed for 

determining the LCMRL is also used to determine the mHV-DL.

1.3 Improvements over Previous LCMRL Methodology

The currently proposed LCMRL methodology offers several improvements over the previous 

LCMRL (USEPA, 2004). 

The first improvement is the fundamental use of a probability-based definition for the LCMRL in 

deriving the estimation algorithms. As described above, the LCMRL is defined as the lowest 

spiking concentration such that the probability of spike recovery in the 50% to 150% range is at 

least 99%. The previous LCMRL calculator used a graphical technique which did not 

completely agree with the probability- or MQO-based definition of the LCMRL. The 

probability-based definition leads to an approach based on characterizing the distribution of the 

analytical response as a function of the spiking level.

The second improvement is that the piecewise linear replicate variance function in the first 

LCMRL calculator has been replaced by a smooth parametric function. The piecewise linear 

model was based on linear interpolation of pointwise sample variances. The proposed 

methodology replaces the ordinary sample variances with robust estimates of variance based on 

modern statistical methods (M-estimation). This enables the potential impact of outliers to be 

effectively eliminated without requiring outlier testing and removal. 

The third improvement is that the robust replicate variance estimates are fit to a model based on 

an additive and multiplicative error structure for trace level analytical chemistry measurements. 

The fourth improvement is that the average response model (regression of measured onto spiked 

concentrations) is fit using iteratively reweighted least squares (IRLS) in which the weights are 

determined not only by the reciprocal of the robust variances but also by M-estimation weights 

determined as part of the robust variance estimation process. This again enables the potential 

impact of outliers to be effectively eliminated without requiring outlier testing and removal. 

The fifth improvement is in considering lack of fit for the average response model as a 

component of prediction variance. This is done by computing robust estimates of mean squared 

error (MSE) and fitting a parametric model for conditional MSE, similar to what is done for 

replicate variance. The prediction variance function is constructed using the pointwise 

maximum of the replicate variance and conditional MSE functions. This is multiplied by a 

function that includes the additional uncertainty inherent in the estimation of the average 

response model. This is a conservative approach based on the author’s perception that the 

greater danger arises from underestimating the prediction variance. 

The sixth improvement is that the gamma distribution is used as a response model for analytical 

methods which do not give negative results at low level. The normal distribution is used as the 
LCMRL Technical Report December 2010

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response model for analytical methods which can give negative results at low level. The degrees 

of freedom in the normal case are estimated from the approximate degrees of freedom for the 

variance predictions given by the replicate variance model.

The seventh improvement is using the conditional response distribution estimated to determine 

the LCMRL to estimate Lc and the detection limit using a modification of the Hubaux-Vos 

approach.

2.0 General Notation

Below are definitions of key variable notations used throughout this paper. Additional notations 

included in this document are described as they are introduced.

m is the number of spiking levels.

n is the total number of observations.

ni is the number of replicate observations at the i

th spiking level.

xi is the spiking concentration for the i

th spiking level.

yij is the measured concentration for the j

th measurement at the ith spiking level.

Yij is the j

th measurement at the ith spiking level represented as a random variable.

wij is the weight for the j

th measurement at the ith spiking level.

y is the nx1 vector of measured concentrations.

p is the number of model parameters.

β is a px1 vector of regression coefficients (model parameters).

E(.) is the expected value function for a random variable.

var(.) is the variance function for a random variable.

μi is the expected value of Yij.

2

σ i is the variance of Yij.

Z is a design matrix.

Zi is a design vector (row of the design matrix) for the measurements at the i

th spiking level.

CV is coefficient of variation (also known as relative standard deviation).

Lc is the critical level.

C γ is the coverage probability for the critical level, Lc.

D γ is the coverage probability for the mHV-DL.

Q γ is the coverage probability for the quality specification interval (50%-150% recovery) at 

the LCMRL.
LCMRL Technical Report December 2010

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3.0 LCMRL Study Design

Because the LCMRL methodology is a regression-based methodology, LCMRL study designs 

involve replicate spiking at multiple levels. The relevant design parameters are:

• the number of spiking levels,

• the spacing of spiking levels,

• the number of replicates at each spiking level (not necessarily equal), and

• inclusion of method blanks.

The relevant issues associated with the design of detection and quantitation studies include the 

impact of the study design on: 

• identification of nonlinearity and the correct model form for the average response 

function,

• parameter estimation in the replicate variance and conditional MSE model, and

• parameter estimation in the average response function model.

Because the LCMRL methodology is complex and multistage, definitive recommendations for 

study designs are not available at present. This is an issue for further development. 

Hubaux and Vos (1970) study a variety of standards spacings, which they refer to as repartitions 

of standards. They study linear spacing, parabolic spacing, two point spacings (that is, two 

spiking levels) and three point spacings. They recommend that the ratio of the maximum to 

minimum (nonzero) spiking level be about 10. They also recommend having all the replicates 

(save one) at the lowest and highest spiking levels. This is their three point design. This report 

rejects that particular recommendation for the following reasons.

Both the average response and replicate variance models are regression models. In design for 

regression, having more levels of the predictor (the spiking level in this case) is much more 

important than having more replication at each level. However, for the LCMRL, estimation of 

the replicate variance function is also vital. Therefore, some replication is also essential. Under 

Gaussian sampling, the coefficient of variation of the ordinary sample variance is 1 for three 

replicates, 3 0.82 2 ≈ for four replicates, 5 0.63 2 ≈ for six replicates, 7 0.53 2 ≈ for eight

replicates and 9 0.47 2 ≈ for ten replicates.

Nonlinearity of the response function at low concentration is a well known phenomenon. It is 

very important to characterize nonlinearity at low level in order to accurately estimate the 

LCMRL and DL because the mean response function at low level is a key component of the 

LCMRL/mHV-DL methodology. For this reason, we allow a polynomial model, up to a cubic, 

for the mean response. In order for the coefficients of a cubic polynomial to be identifiable 

(estimable), there must be at least four spiking levels.

It is also very important to estimate the variance function as accurately at low level as possible. 

For these reasons, this report recommends the parabolic design for spiking levels with as many 

levels as feasible and at least four replications per spiking level. 
LCMRL Technical Report December 2010

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The spiking levels may be computed as follows

( )( )

2

1

1 1 1 , for 1, , , i

i m m xx x x i m −

− =+ − =  (1)

where x1 and xm are the designed minimum and maximum spiking levels.

The proposed LCMRL calculator recommends seven spiking levels with four replicates per level 

for a total of 28 analyses. Fitting a cubic polynomial in the mean response model requires a 

minimum of four spiking levels. The minimum values that the LCMRL calculator will accept 

are four spiking levels and three replicates per level. Given that the size of the LCMRL/DL 

study must be limited by cost and resource availability, a tradeoff must be made between 

estimating replicate variance and MSE at individual spiking levels and estimating the regression 

models for mean response, replicate variance and MSE. The recommended numbers of spiking 

levels and replicates are a reasonable compromise. 

4.0 Data Conditioning

Because the LCMRL methodology is a complex, multistage, model-based design, care must be taken

in screening the input data for use in the model development. The model estimation procedure is 

sensitive to departures from a monotonic variance model. Having non-zero spiking levels with nonzero replicate variances is problematic for modeling. Therefore, data values that would contribute to

invalidating these assumptions are dropped from the estimation procedure. Data are dropped from 

the estimation procedure for non-zero spiking levels where 50% or greater of replicate responses are 

zero. If less than 50% of the replicate responses of a non-zero spiking levels are zero, then the zero 

responses are replaced by the minimum non-zero replicate response within the spiking level or within 

a lower spiking level. 

After data conditioning the procedure determines if the requirement of a minimum of four spiking 

levels and three replicates is achieved before estimation of the LCMRL.

5.0 Estimation of Replicate Variance

5.1 Introduction

Because of the potential for outliers, replicate variance at each spiking level is estimated using 

weighted variances using weights computed by M-estimation. As a byproduct of this process, 

robust weights are available for estimating the conditional mean (regression) model. Since at 

each spiking level the number of results may vary from 3 to perhaps 10 or more, this is a very 

challenging undertaking.

The objective is not to ‘identify’ outliers as such but merely to down-weight them sufficiently 

that they do not overly influence the means model, replicate variance model or conditional MSE 

model, without overly compromising efficiency. In contrast, testing and removal of outliers 

constitutes a data weighting scheme where the only possible weights are 0 or 1

n , where n is the 

number of retained observations. This is very inflexible compared to weighting based on M-
LCMRL Technical Report December 2010

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estimation. Not only is testing and removal of outliers awkward procedurally, it causes a loss of 

statistical efficiency. 

A modified Hodges-Lehmann estimator is used for the initial resistant location estimate. The 

scaled mean absolute deviation from the Hodges-Lehmann estimator is used as the resistant scale 

estimator. Using this initial location-scale estimation pair, the Huber M-estimate of location and 

its associated weights are computed. A robust scale estimator is computed based on the

weighted mean absolute deviation from the Huber location estimator and using the Huber 

weights. The Huber location-scale estimates are then used as an initial location and robust scale 

estimates for Tukey’s biweight procedure. 

Finally, the Tukey’s biweight weights are used to compute location and variance estimators at

each spiking level. The weights are also later used in computing the conditional mean response 

(regression) model. 

5.2 Modified Hodges-Lehmann Location Estimator

The Hodges-Lehmann location estimator (Randles and Wolfe, 1991) is the median of the set of 

means of all pairs of observations. The modified Hodges-Lehmann location estimator (modified 

for this application) at the i

th spiking concentration combines the median of the measurement 

data with the set of means of pairs of data and takes the median of the resulting set of numbers. 

For a set of ni observations {y y i in 1, ,  i} for the i

th spiking level, this is given by 

, 1 median ,median , , { } 2 i

ij ik

HL i i in j k

y y

m y y <

  + =       . (2)

5.3 Mean Absolute Deviation Scale Estimator

This estimator is based on the long known (Kelley, 1921) Absolute Deviation (AD) scale 

estimator. It uses the mean absolute deviation from the Hodges-Lehmann estimator (instead of 

from the sample mean). It is computed as 

1

, 1

1.4826 i n

i n ij HL i j

AD y m = =⋅ − ∑ . (3)

 

The factor 1.486 is a normalization constant, used so that the expected value of the AD in normal 

samples equals the population standard deviation.

5.4 Huber Location Estimator and Weights

The Huber estimator (Hoaglin et al., 1983) is calculated iteratively as a weighted estimator (westimator) using a relative convergence criterion of 10-6

. The process is started with a resistant 

initial location estimate (0) TH i, given by the modified Hodges-Lehmann estimator and a resistant 

auxiliary scale estimate given by the modified AD estimator, ADi . This is a departure from the 
LCMRL Technical Report December 2010

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usual practice, in which the median is used as the initial location estimate and the median 

absolute deviation (MAD) scale estimator is used as the auxiliary scale estimator for the Huber

location estimator. The modification is prompted by the very small sizes of the data sets in this 

application.

The Huber weights and estimator are determined by a tuning constant cH according to the 

equations below. The value of the parameter cH is taken to be 1, which is a standard value often 

used in practice. The index k tracks the iteration number. For a set of ni observations 

{y y i in 1, ,  i} at the i

th spiking level the iteration proceeds as follows

( ) ( )

( )

( )

( )

( ) ( )

1

,

,

, ,

, 1 1

, 1, ,

1, if 

, 1, , , if 

k

ij H i

i

i i

k y T

ij AD i

k

H k ij H

ij k i k

H ij ij H

k n n H k H k

H i ij ij ij j j

u jn

u c

w j n

c u u c

T wy w

− −

= =

= =



 ≤ =  =

 > 

= ∑ ∑



 (4)

until the estimate ( )

,

k TH i converges, at which time the weights are assumed to have converged

based on a relative convergence criteria of 10-4

. The Huber M-estimate of location is denoted by

TH i, and the associated weights by Hwij . 

The Huber scale estimator, 2

H i, s (equation 5), is computed as the weighted variance around the 

Huber location estimate using the Huber weights. This formula is derived from the variance of a 

weighted mean. It also follows from an approximation to the A-estimators of scale (Lax, 1985, 

Section 4.6.2) based on the asymptotic variance of M-estimators (Hoaglin et al., 1983). This 

completes the circle, since M-estimators can always be represented as weighted means (Westimators).

( )

( )

2

,

1

2 2

, , 1

,

1 i

i

n

H

w i i ij

j

i n H

H i ij ij H i j

w i

nn w

n s wyT

n

=

=

  = −    

= −

∑

∑

(5)

5.5 Tukey’s Biweight Location Estimator and Weights

The Tukey’s biweight M-estimator (Hoaglin et al., 1983; Horn, 1988) is calculated iteratively as 

a weighted estimator using a relative convergence criterion of 10-4. 

The biweight is more robust statistically in certain respects than the Huber estimator. It is very 

efficient and can down-weight extreme observations to zero. On the other hand, it is not as 

robust computationally as the Huber estimator. Without a good auxiliary scale estimate and a 
LCMRL Technical Report December 2010

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good starting location estimate, it can misbehave. To address this issue the process is started 

with a robust initial location estimate (0) TBW given by the Huber estimator and a robust auxiliary 

scale estimate given by Huber weighted AD scale estimator, H i, s . This is a departure from the 

usual practice, in which MAD is used as the auxiliary scale estimator. It is prompted by the very 

small sizes of the data sets in this application. 

The Tukey weights are determined by a tuning constant cBW according to the equations below. 

The value of the tuning parameter cBW is taken to be 9. The index k tracks the iteration number. 

For a set of ni observations {y y i in 1, ,  i} at the i

th spiking level the iteration proceeds as follows

( ) ( )

( )

( ( ) ) ( )

( )

( ) ( )

1

,

,

2 2

,

, ,

, 1 1

, 1, ,

1 , if 1

, 1, ,

0, if 1

k

ij BW i

BW H i

i i

k y T

ij c s i

k k

BW k ij ij

ij i

k

ij

k n n BW k BW k

BW i ij ij ij i i

u jn

u u

w j n

u

T wy w

− −

= =

= =



 − < =  =

 ≥ 

= ∑ ∑



 (6)

until the estimate ( )

,

k TBW i converges, at which time the weights are assumed to have converged. 

The location estimate is denoted by TBW i, and the weights by BW wij . Note that the definitions of 

(k )

ij u are similar but different in the two cases (Huber and biweight).

5.6 Final Location-Scale Estimator and Weights

The final robust weights are the Tukey biweights. The robust location and variance estimates for 

the i

th spiking level are computed as a weighted mean, mw i, , and weighted variance, 2

w i, s , of the 

data as indicated below where i nw is the degrees of freedom associated with the weighted 

observations. 

( )

( )

( )

2

, 1

, 1 1

2 2

, , 1

,

1 i

i

i

n BW

w i i ij j

n n BW BW

w i ij ij ij j j

n i BW

w i ij ij w i j

w i

nn w

m wy w

n

s w ym

n

=

= =

=

  = −    

=

= −

∑

∑ ∑

∑

(7)

As noted above, this is the formula for a weighted variance, derived from the variance of a 

weighted mean. It is also an approximation to the A-estimators of scale based on the asymptotic 

variance of Tukey’s biweight location estimator. 
LCMRL Technical Report December 2010

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Characteristics of the Huber and biweight estimators are combined by using the Huber estimates

as scale and starting location in computing the biweight. The reason for doing this is that for Mestimators with monotone ψ-functions, like the Huber, the equation defining the M-estimator is 

guaranteed to have a unique solution. For redescending estimators, like Tukey’s biweight, 

uniqueness is not guaranteed. This problem is particularly worrisome given the small data sets 

being used (results at each spiking level). Since the computer code for LCMRL/mHV-DL 

estimation must run unsupervised, the Huber estimate is a safe starting point for the biweight 

iterations.

6.0 Replicate Variance Model

6.1 Derivation

The replicate variance model, which models replicate variance as a function of spiking 

concentration, is based in part on the paper by Rocke and Lorenzato (1995). This paper 

identifies sources of both additive and multiplicative random error in trace level analytical 

chemistry measurements. Corresponding to this conceptual model, the LCMRL procedure fits 

constant plus power models to replicate variance and conditional MSE models. The constant 

term estimates the additive sources of variance and the power model term estimates the 

multiplicative sources. The power model term accommodates error variance behavior in higher 

concentration ranges that varies from near constant variance to shot noise to constant coefficient 

of variation (CV). 

The variance at extremely low concentrations may be distorted by processes such as 

thresholding, smoothing, rounding and truncation. Smoothing, which is often used in 

spectroscopic, chromatographic, and radiologic methods, reduces measurement variance, 

especially for low-level measurements. Rounding and truncation also reduce measurement 

variance. 

Response thresholding, which is used in chromatographic methods to manage data volume, can 

either reduce or increase measurement variance depending on the situation. Thresholding 

effectively replaces noise (except for the extreme upper tail of the noise distribution) and some 

low level signal with zeros. Thresholding then functions as a Type II censoring mechanism with 

an unknown censoring point (unless the laboratory reports the instrument settings).

Instrument thresholds have both direct and indirect impacts on estimating detection and 

quantitation limits. The main direct impact is that it is not possible to estimate the standard

deviation of measurements at zero. However, by definition, the standard deviation at zero is 

required to calculate Lc (Currie, 1968). The EPA MDL procedure (Glaser et al., 1981) was 

constructed to deal with this problem by providing a way to estimate a standard deviation at a 

low concentration, and included instructions for determination of a concentration as close to zero 

as is possible that will generate a measurement.

Theoretically, in the absence of such distortion, low-level measurement variance (even for 

method blanks) would always be positive. For this reason, method blank data are not used in 

estimating the replicate variance model. 
LCMRL Technical Report December 2010

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The replicate variance model takes the form

( ) 2 , v c

v v σ x a bx = + (8)

where x is the spiking concentration.

The replicate variance model parameters are estimated using a constrained iterated downhill 

Simplex Method minimization routine (Nelder and Mead, 1965) with starting values for b and c 

found by first fitting the robust variances using log-scale simple regression. Because of the 

potential for outliers, the ordinary sample variance is not used as the response in the fitting. 

Instead a robust estimate of variance is computed based on M-estimation, as described above. 

The starting values 0 b and 0 c , for b and c, are first found by fitting a reduced variance model of 

the form

( ) 2 ln[ ] ln( ) ln( ), v v σ x bcx = +

using ordinary regression using the robust variances 2

w i, s at each spiking level i x .

Following this, the full replicate variance model is fit using an iterated constrained downhill 

Simplex Method with starting values defined as

( ( ) )

2 2

, , 1

2

, 1

0

0

max ,1 8

min max 0, ,2

wi wi i

s

w i i

s

s

n s

a e

n

b b

c c

=

=

  =   −

 

=

=

∑

∑

and a constrained loss function defined at each iteration j of the optimization as

( )

( )

1

12

2 2 2

, 2

2 th

where

10 , if 0 or 0 or 0 or 2

, otherwise

, at the iteration of the optimization. j

n

i i

jjjj

i i i

w i

i

c

k kk

loss e

abcc

e s

n

x a bx k

σ

σ

σ

= =

 <<<>  =  − 



= +

∑

The optimization continues for estimates of a, b and c until the loss function is less than 10-16. 

Normal variances (that is, the statistic computed from data) have their sample variances 

proportional to their means. The robust variances used in the model behave similarly. It is this 
LCMRL Technical Report December 2010

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that motivates the choice of the functional form of the loss function. The large loss incurred 

when parameter estimates are out of valid ranges forces the Simplex Method to constrain the 

parameter values to valid ranges. 

7.0 Conditional Means Model

The conditional means model is estimated using a robust estimation procedure that implements a 

nested Iterative Reweighted Least Squares (IRLS) procedure. As part and as a consequence of 

the iterations, the conditional mean squared error model and the robust weights are also updated 

until convergence of the procedure. The IRLS procedure can handle non-standard regression 

assumptions such as non-homogenous variances and non-normal errors. This method is also 

resistant to outliers in the data set by using weights that are derived using Tukey’s bisquare 

method (See Section 4.5).

The general conditional means model is 

( ) ( )

( ) 2 3

E

1, , ,

T

i ij i

T

i ii i

xY Z

Z xx x

μ β = =

= (9)

To solve this in the IRLS framework, it is necessary to solve 

1 1

ˆ ( )( ) ( ) , n n

i i i i i ii i i w u y z z w u ez β = = ∑ ∑ − =

where ei is the residual at each point and w(ui) are the Tukey bisquare weights (estimated 

independently here for the means model). The Tukey weights, defined in Section 5.5, are in this 

case a function of the xi, the predicted values as an estimate location, the conditional mean 

squared error as an estimate of scale and a tuning constant of 9. The conditional mean square 

error is used instead of the traditional variance to allow for the model lack of fit.

It is first necessary to have initial starting values for ˆ

β and the conditional mean square error 

(see Section 8.0). Using the initial ˆ

β , say 0

ˆ

β , the initial predicated values, residuals and the 

conditional mean square error (see Section 8.0) are calculated. The initial 0

ˆ

β is found by 

performing weighted least squares regression using the robust weight matrix W (see Section 5.6).

The solution for 0 β

ˆ using weighted least squares regression is given below.
LCMRL Technical Report December 2010

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( )

{ }

( ) 1

1

0

1,1 ,

1,1 1,2 1, ,1 ,

1

1

1

ˆ

diag , ,

, ,, ,, ,,

 rows

 rows

m

m

T

m n

T

n m mn

T

T

T

m

m

T

m

y

w w

y yy y y y

Z

n

Z

Z

n

Z

β −=

=

=

        

  

  

=  

              

ZWZ ZW

W

Z











The nested IRLS is implemented for estimating ˆ

β as follows. Parameters epsilon and maxIter 

are 10-6 and 100, respectively.

1) From the initial 0

ˆ

β calculate the predicated values and residuals.

2) For the outer loop from the initial residuals compute the initial estimate of the conditional 

MSE function, mse0 .

3) For the inner loop compute the initial Tukey weights W0 using the predicted values and 

conditional MSE function mse0 and a tuning constant of 9.

4) Use weighted least squares to obtain the estimate ˆ

βi .

5) Use the parameter estimate ˆ

βi to obtain new predicated values and residuals.

6) Go back to step 3 until the estimate 1 ˆ ˆ | | β β i i − − converges pointwise such that the 

maximum pointwise difference is less than epsilon or the number of iterations exceeds 

maxIter. Set ˆ ˆ β β = i and go to step 7.

7) Go to step 2 and compute a new conditional mean square error function msej using ˆ

β

and the residuals from step 5. 

8) If the maximum pointwise difference of 0 ˆ ˆ | | β β − (over the outer loop) is greater than the 

convergence tolerance and the maximum number of outer iterations has not been 

exceeded, set 0 β β

ˆ ˆ = and mse mse 0 = j and go back to step 1. Otherwise, stop and return 

the current values ˆ

β and msej .

Final model selection (linear, quadratic or cubic) is accomplished using Mallow’s Cp (Draper and 

Smith, 1981). Since the constant term in the model may be negative, it may be possible for mean 

concentrations near zero to have a negative mean estimate. Therefore, the mean response 

function at a spiking concentration, x, is defined as 
LCMRL Technical Report December 2010

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ˆ 0, if 0

( ) ˆ ˆ , if >0 

z

x

z z

β

μ

β β

 ≤  = 



8.0 Conditional Mean Squared Error Model

8.1 Estimating Conditional Mean Squared Errors

Since the sampling distribution for repeated measurements at a spiking level is based on the 

conditional means (regression) model and variance about the conditional mean, the contribution 

to variance by lack of fit must be accounted for. To account for the effect of lack of fit for the 

regression model, a model is fit for conditional mean squared error (cMSE) as a function of 

spiking level. 

First, robust estimates are made of MSE at each spiking level. This process is analogous to the 

estimation of replicate variance except that the raw residuals from the conditional means model 

at each spiking level are used as the data. The robust cMSE for the i

th spiking level, xi, is 

calculated as 

2 2

i wi wi , , cMSE m s = + (10)

with the quantities calculated as described in Section 5.6 but using the regression residuals. 

Clearly, the cMSE should be larger than the replicate variance at each spiking level.

8.2 Fitting Conditional Mean Squared Errors Model

The conditional mean squared error model, which models mean squared error as a function of 

spiking concentration, is assumed to be in the class of constant + power models 

2 , e c

i e ei τ = + a bx (11)

where i = 1, 2, ..., m indexes the spiking levels, xi. 

The cMSE model is fit in exactly the same manner as the replicate variance model. Because of 

the model fitting process however, there is no guarantee that the expected value from the cMSE 

model is larger than the expected value of the replicate variance model at every spiking level. 
LCMRL Technical Report December 2010

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9.0 Estimating the LCMRL

9.1 Introduction

To construct an estimated sampling distribution for repeated measurements at an arbitrary 

spiking concentration, a mean function, a variance function and a parametric distributional 

family are required. The mean function is estimated as described in Section 7.0.

As described in Section 1.2, the gamma distribution is used for analytical methods that give only 

non-negative measurements, and the normal distribution is used for methods that can give both 

positive and negative measurements at low concentration. The gamma distribution is appropriate 

in the first case since it is the maximum entropy distribution (Kagan et al., 1973) in the class of 

continuous distributions which support only non-negative values and have a specified mean and 

variance (or specified mean and geometric mean). The normal distribution is the continuous 

maximum entropy distribution with specified mean and variance and support on the real line.

9.2 Prediction Variance

As described in Sections 6.0 and 8.0, models are constructed for both replicate variance and 

conditional MSE (of residuals from the means model) as functions of spiking concentration. For 

estimation of the LCMRL and HV-DL, what is really needed is a model for prediction variance 

as a function of spiking concentration. Three additional considerations allow construction of a 

robust yet conservative model for prediction variance. 

The first is that, as discussed in Section 6.1, the variance at extremely low concentrations may be 

distorted by processes such as thresholding, smoothing, rounding and truncation. Therefore, a 

conservative prediction variance function should never give a zero prediction variance, not even 

at zero spiking concentration. 

Because the constant in either the replicate variance (equation 8) or cMSE (equation 11) models 

may be zero, a parameter called minVar has been created. The minVar parameter is set to the 

constant a if it is nonzero. Otherwise it is set to the mean of the robust variances at the two 

lowest nonzero spiking levels. The minVar parameter is denoted by 2 σ min and 2

min τ for the 

replicate variance and cMSE models, respectively.

The second consideration is that as noted in Section 8.1, although the replicate variance at each 

spiking level should always be smaller than the cMSE, it is possible for the replicate variance 

model estimate to be larger at some spiking concentrations than the cMSE model estimate. A 

conservative approach is to use the point-wise maximum of these functions. 

The third consideration is that the prediction variance is associated in a sense to the regression 

(conditional means model). The prediction variance should be augmented with regression model 

uncertainty just as it is in ordinary least squares (OLS) regression (Draper and Smith, 1981). 

This makes intervals calculated using the prediction variance comparable to tolerance intervals.

These considerations guide the construction of the prediction variance function:
LCMRL Technical Report December 2010

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( ) ( )

( )

( )

( )

( )

( ) ( ( ) ( )) ( ) ( )

( ( ) ( )) ( )

1

1

2

2

1

2

2

2 2 2

min min

2 2

2

22 2

1 1

max ,

max , , if max , 0

max , , , otherwise

v

e

m

n j j j

m

j j j

c

v v

c

e e

mm

v e

pred

mm

x nx

x x

f x

n nx x

x a bx

x a bx

x x fx a a

x

x x fx

σ

τ

σ στ

σ τ

σ

σσ τ

=

=

=

− =+ +

−

= +

= +

=

 ⋅ >  = 

 ⋅ 

∑

∑

(12)

Prediction variance estimates from this function are assumed to have p d nmn =− − degrees of 

freedom, where np is the maximum of the number of parameters fit in the variance model and the 

cMSE model. 

9.3 Conditional Distributions

Maximum entropy is one method of characterizing probability distributions (Kagan et al., 1973). 

Often there is a unique distribution that maximizes entropy while satisfying conditions of 

allowable values for a random variable and constraints on specified moments (or on other 

functionals). This unique distribution satisfies the given conditions while implying absolutely no 

additional information. Maximum entropy distributions are very important when working with 

scientific problems that have physical constraints. They can honor those constraints without 

adding any additional unwarranted information. 

The gamma distribution is the maximum entropy distribution in the class of continuous 

distributions with support (that is, allowed values) on the nonnegative real line and specified 

mean and geometric mean (Kagan et al., 1973). This meshes ideally with the model for lowlevel analytical error variance described by Rocke and Lorenzato (1995) and discussed in 

Section 6.1. It has both additive and multiplicative error components, corresponding to the linear 

and log-scale moment constraints in the gamma distribution. 

It can easily be shown (Kagan et al., 1973) that in this case, the constraint on mean and 

geometric mean is equivalent to a constraint on mean and variance. Thus, for specified 

conditional mean and variance functions with positive range the principle of Maximum Entropy 

generates a family of gamma distributions. Therefore, the gamma distribution is used as the

distributional model for response for methods which do not give negative results.

The normal distribution is the continuous maximum entropy distribution with specified mean and 

variance and support on the real line (that is, negative values are allowed). The normal 

distribution is therefore used as the distributional model for response for methods which can give 

negative results. In the case of the normal model for repeated measurements, the conditional 
LCMRL Technical Report December 2010

Page 17 of 24

mean and prediction variance functions directly give the parameters for the repeated 

measurement model. Since the variance function estimates have an associated degrees of 

freedom d that will typically be less than 30, probability calculations are made using the tdistribution with d degrees of freedom. 

In the case of the gamma distribution (used for methods which cannot give negative results), the 

conditional gamma distribution parameters are computed as 

( ) ( ) ( )

( ) ( ) ( )

2 2

2 .

pred

pred

xx x

x xx

α μσ

βσ μ

=

= (13)

9.4 Search for LCMRL

An iterative search is conducted for the spiking level which satisfies the definition of the 

LCMRL. In the case of the normal distribution model, the LCMRL is the x which satisfies

( ) ( ) ( ) ( )

( )

( )

( )

2 1.5 0.5 Pr 0.5 1.5 , pred d d Q

pred pred

p

xx xx xY x x x T T

x x

d nmn

μ μ μ σ γ

σ σ

 −  −  ≤ < = − =      

=− −

(14)

where ( ) Td ⋅ is the cumulative distribution function for the t-distribution with d degrees of

freedom and 0.99 Q γ = . 

In the case of the gamma distribution model, the LCMRL is the x which satisfies

( ( ) ( ))

( ( ) ( )) ( ( ) ( ))

Pr 0.5 1.5 ,

1.5 , 0.5 , Q

xY x x x

F xx x F xx x

α β

αβ αβ γ

≤ < =

− = (15)

where F (⋅ α β, ) is the gamma cumulative distribution function with parameters α and β.

Figure 9-1 below shows an example of the coverage probability for the MQO recovery interval 

(50%-150%) as a function of spiking level. Figure 9-2 below is an example of an LCMRL plot 

showing the raw data, the estimated average response curve, the recovery limits, a 99% 

prediction envelope for the response and the LCMRL. The text on the figures also gives the 

LCMRL. These figures were created using the LCMRL calculator software application 

produced for laboratory use.

In this example, the spiking range is relatively narrow and the replicate variance and conditional 

mean squared error are relatively constant across the range. Figure 9-3 below shows an example 

of an LCMRL plot for an analyte with error (replicate variance and conditional MSE) increasing 

with spiking concentration.
LCMRL Technical Report December 2010

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Figure 9-1: Example Coverage Probability for MQO Recovery Limits

Figure 9-2: Example LCMRL Plot with Constant Error Variance

0 0.05 0.1 0.15 0.2 0.25

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1,4-dioxane--LCMRL Plot

Spike Concentration ug/L

Measured Concentration ug/L

Data 

LCMRL = 0.042 ug/L 

Y=X 

Regression 

50-150% Recovery 

Lower/Upper Prediction Limits

Spiked Concentration ug/L

0 0.1 0.2

0.92

0.94

0.96

0.98

1

LCMRL = 0.042 ug/L 

Qual Lim: 50-150% 

Coverage Prob: 0.99

1,4-dioxane--QC Interval Coverage Plot

Spike Concentration ug/L

Probability of QC Interval Coverage

Spiked Concentration ug/L

Probability of QC Interval Coverage
LCMRL Technical Report December 2010

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Figure 9-3: Example LCMRL Plot with Increasing Error Variance

Figure 9-4 below shows the increasing MSE as a function of spiking level. The points are the 

robust estimates of MSE. The line is an approximation to the constant + power model fitted to 

the cMSE. 

0 100 200 300 400 500 600 700

0

200

400

600

800

1000

Trichloroethene--LCMRL Plot

Spike Concentration ug/L

Measured Concentration ug/L

Data 

LCMRL = 57 ug/L 

Y=X 

Regression 

50-150% Recovery 

Lower/Upper Prediction Limits

Spiked Concentration ng/L

Trichloroethylene--LCMRL Plot

ng/L

Measured Concentration ng/L

mHV-DL = 18 ng/L

Lc = 19 ng/L

Equation: y = 8.9221 + 1.0564 * x
LCMRL Technical Report December 2010

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Spiked Concentration ng/L

Figure 9-4: Conditional MSE and fitted model showing increasing error with concentration

10.0 Estimating the Modified Hubaux-Vos DL

The proposed methodology differs from and improves on the original Hubaux-Vos methodology 

(Hubaux and Vos, 1970). First, the proposed method accommodates nonconstant variance in a 

very robust manner. Hubaux and Vos were aware of this consideration but did not include it in 

their methodology. Secondly, in cases where the instrument response cannot be negative, the 

proposed methodology uses the gamma distribution (which is the Maximum Entropy distribution 

in this situation) to model the response distribution. 

10.1 Estimating the Critical Level

Lc was defined by Currie (1968) as an upper percentile of the distribution of repeated 

measurements at zero concentration. For Currie, Lc represents an a posteriori decision limit for 

detection. As noted in Section 6.1, although in many situations it may be impossible to sample 

this distribution directly in an accurate way, it is possible to estimate using regression methods. 

This was the general approach of Hubaux and Vos (1970). 
LCMRL Technical Report December 2010

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As is the case for the LCMRL (Section 9.1), the methodology used to estimate Lc in the mHVDL method distinguishes two cases. In the first case, in which blank and low-level 

measurements can give negative instrument responses, the normal distribution is used to model 

the response distribution. Then Lc is estimated as 

( ) ( ) C 0 0 pred d C ( )

p

L t

d nmn

=+ ⋅ μσ γ

=− − (16)

where td C (γ ) is the C γ -quantile of the t-distribution with d degrees of freedom (see Section 

9.2), and 0.95 C γ = . 

In the second case, in which blank and low-level measurements cannot give negative instrument 

responses, the gamma distribution is used to model the response distribution.

Then, assuming that μ (0 0 ) > , Lc is estimated as 

( ( ) ( )) 1 0, 0 L F C C γα β − = (17)

where ( ) 1 F α β, − ⋅ is the quantile function of the gamma distribution with parameters α and β.

However, in the case that μ (0 0 ) = , the gamma distribution is not tenable. In this case, for want 

of a better alternative, a half-t distribution is used. Then Lc is estimated as

( ) ( ) 1

2 1 0 C L t C d pred

γ σ − =−⋅ (18)

where td (⋅) is the quantile function of the t distribution with d degrees of freedom.

10.2 Search for mHV-DL

An iterative search is conducted for the spiking level which satisfies the definition of the mHVDL. In the case of the normal distribution model, the mHV-DL is the x which satisfies

( ) ( ) ( ) ( )

( )

2 Pr , C

C pred d D

pred

L x YL x x T

x

μ μ σ γ

σ

−  ≥ ==     

(19)

where ( ) Td ⋅ is the cumulative distribution function for the t-distribution with d degrees of 

freedom (see Sections 7.2 and 7.3), and 0.95 D γ = .

In the case of the gamma distribution model, the mHV-DL is the x which satisfies
LCMRL Technical Report December 2010

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Pr( ( ), 1 ( )) ( ( ), ( )) Y L x x FL x x ≥ = C CD αβ αβ γ − = (20)

where α( x) and β ( x) are defined in equation 19, and F (⋅ α β, ) is the gamma cumulative 

distribution function with parameters α and β.

For the data shown in Figure 9-2, the mHV-DL and Lc values are 0.026 ug/L and 0.016 ug/L, 

respectively. In this case, which is typical, the mHV-DL is higher than Lc, as one would 

intuitively expect. In the case of the TCE data used for Figure 9-3, the reverse is true. The 

mHV-DL and Lc values are 0.018 ug/L and 0.019 ug/L (both are converted from ng/L, as 

presented in Figure 9-3), respectively.

Although this seems odd at first glance, in fact, the mHV-DL is in the domain of spiking 

concentrations while Lc is in the measurement domain. Systematic low level bias in the 

measurements can cause this apparent anomaly. Even if a calibration does not show a statistical 

significant intercept term, because the LCMRL study involves much more data and because it 

includes preparation steps, it tends to show low level bias and nonlinearity not evident in the 

calibration data. 

Let xc denote the spiking concentration at which the expected response equals Lc. Then it is 

always true that xc < mHV-DL. In fact, xc is a very important quantity, since it is the natural 

censoring point to be used in data analysis for nondetects (measurements which are less than or 

equal to Lc). In the case of the TCE data used for Figure 9-3, the intercept for the mean response 

model, at about 0.009 ug/L, is about half Lc (0.019 ug/L), which is a significant fraction of Lc. 

Remember that Lc is the estimated 95th percentile of the response distribution at zero spiking 

concentration. To get a response greater than 0.019 ug/L 95% of the time, we estimate that we 

must spike at 0.018 ug/L (the mHV-DL). Although it may seem odd, this situation can be 

explained by the positive intercept of the mean response model being “not small” relative to Lc.

11.0 Summary, Conclusions and Recommendations

The proposed LCMRL procedure described in this report provides a statistically robust method 

of estimating the lowest concentration at which a laboratory can reliably achieve the MQO of 

50%-150% recovery. Although the procedure is complex and computationally intensive, 

implementing it in user-friendly software in the public domain and freely available over the 

internet makes it easy for laboratories to use. 

Furthermore, a robust method of estimating the LCMRL makes it possible, through Bayesian 

bootstrap resampling, to estimate by simulation the statistical distribution of LCMRL, mHV-DL 

and Lc values that would be generated by a randomly selected ‘experienced drinking water 

laboratory’ upon repeated execution of the LCMRL study design. This will enable the 

development of scientifically defensible MRL values for guidance and regulatory use.

Coupling the mHV-DL procedure with the modeling used for the LCMRL computation improves 

the estimation of Lc and the detection limit. Improvement of the estimation of Lc will improve 

statistical inference regarding low-level data sets. Improvement of the estimation of the 
LCMRL Technical Report December 2010

Page 23 of 24

detection limit will improve the accuracy of assessment of risk modification due to regulatory 

actions. 

These developments have great potential value for regulators, the regulated community, risk 

assessors and data analysts, but especially for laboratories. Use of these methods in the future 

should result in much better understanding and control of measurement data quality. In turn, this 

should lead ultimately to better data quality and decision-making.

The author has several recommendations for future work related to the LCMRL/Hubaux-Vos 

methodology. The first is the further characterization and potential improvement of the 

statistical methodologies proposed here, particularly viewing them as a combined procedure 

rather than isolated methodologies. Secondly is development of optimal design methodology 

(based on the statistical characteristics of the methodology) and development of more definitive 

recommendations for the design of LCMRL/DL studies.

The third recommendation is development or improvement of procedures for verification of 

MRL capability and ongoing monitoring of detection and quantitation capabilities in the context 

of a routine laboratory quality control program. The fourth recommendation is the development 

and release in the public domain of an R library (R Development Core Team, 2007) for the 

LCMRL/Hubaux-Vos methodology, which will facilitate the use and further development of this 

methodology in the research communities of statistics, chemometrics, environmental analytical 

chemistry and forensic and clinical chemistry. 

12.0 References

Currie, L.A. (1968). “Limits for Qualitative Detection and Quantitative Determination.” 

Analytical Chemistry, Vol. 40, pp. 586-593.

Dobson, A.J. (1983). Introduction to Statistical Modeling. Chapman and Hall, New York. 

Draper, N. and H. Smith (1981). Applied Regression Analysis: 2nd Edition. John Wiley and Sons, 

New York.

Glaser, J.A., D.L. Foerst, G.D. McKee, S.A. Quane, and W.L. Budde (1981). “Trace Analyses 

for Wastewaters.” Environmental Science and Technology. Vol. 15, pp. 1426-1435.

Hoaglin, D.C., F. Mosteller, and J.W. Tukey (1983). Understanding Robust and Exploratory 

Data Analysis. John Wiley, New York.

Horn, P.S. (1988). “A Biweight Prediction Interval for Random Samples.” Journal of the 

American Statistical Association. Vol. 83, No. 401. (Mar., 1988), pp. 249-256.

Hubaux, A. and G. Vos (1970). “Decision and Detection Limits for Linear Calibration Curves.” 

Analytical Chemistry. Vol. 42, No. 8, pp. 849-855.
LCMRL Technical Report December 2010

Page 24 of 24

Jaynes, E. (1983). Papers on Probability, Statistics and Statistical Physics. A reprint collection. 

R. D. Rosenkrantz (Ed.). Reidel. Dordrecht, Germany.

Kagan, A.M.; Y.V. Linnik, and C.R. Rao (1973). Characterization Problems in Mathematical 

Statistics. John Wiley. New York. 499 pp.

Kelley, T.L. (1921). “A New Measure of Dispersion.” Quarterly Publications of the American 

Statistical Association. Vol. 17, No. 134. pp. 743-749.

Lax, D.A. (1985). “Robust estimators of scale: Finite-sample performance in long-tailed 

symmetric distributions,” Journal of the American Statistical Association. Vol. 80, pp. 736-741.

McCullagh, P. and J.A. Nelder (1989). Generalized Linear Models: 2nd Edition. Chapman & 

Hall, London. 

Nelder, J.A. and R. Mead (1965). “A Simplex Method for Function Minimization”, Computer 

Journal. Vol. 7, pp. 308-313.

R Development Core Team (2007). R: A language and environment for statistical computing. R 

Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL 

http://www.R-project.org.

Randles, R.H. and D.A. Wolfe (1991). Introduction to the Theory of Nonparametric Statistics. 

Krieger Publishing, Malabar, Florida.

Rocke, D.M. and S. Lorenzato (1995). “A Two-Component Model for Measurement Error in 

Analytical Chemistry.” Technometrics. Vol. 37, No. 2, pp. 176-184.

Rubin, D.B. (1981). “The Bayesian Bootstrap.” Annals of Statistics, 9, pp. 130-134.

Thiel, H. and D.G. Fiebig (1984). Exploiting Continuity: Maximum Entropy Estimation of 

Continuous Distributions. Ballinger Publishing Co. Cambridge, MA. 246 pp.

USEPA (1987). “National Primary Drinking Water Regulations-Synthetic Organic Chemicals; 

Monitoring for Unregulated Contaminants; Final Rule.” Federal Register. Vol. 52, No. 130. p. 

25690, July 8, 1987. 

USEPA (1999). “Revisions to the Unregulated Contaminant Monitoring Regulation for Public 

Water Systems; Final Rule.” Federal Register. Vol. 64, No. 180. p. 50556, September 17, 1999.

USEPA (2004). Statistical Protocol for the Determination of the Single-Laboratory Lowest 

Concentration Minimum Reporting Level (LCMRL) and Validation of Laboratory Performance 

at or Below the Minimum Reporting Level (MRL). USEPA, Office of Ground Water and 

Drinking Water, Standards and Risk Management Division, Technical Support Center. EPA 815-

R-05-006. November 2004.
LCMRL Technical Report - Appendix A December 2010

Page A-1

Appendix A: Computation of MRL

The computations carried out for establishing the Minimum Reporting Level (MRL) are 

programmed in the R language. The Matlab code for the LCMRL calculator was first translated 

into R and then amended to also produce estimates of the MRL estimate for all analytes 

appropriate for use in the LCMRL calculator. There are two reasons for programming the MRL 

in R; the first being that the recoding of the Matlab code provided a validation check of the 

LCMRL code, and secondly it provided an interactive, non-compiled program that could be used 

internally by EPA to calculate MRL estimates and associated graphics.

The MRL is determined using a Bayesian bootstrap (BB) (Rubin, 1981) of the LCMRL estimator 

using the LCMRL study data from each of several experienced drinking water laboratories. The 

BB replicates that were generated from each laboratory’s data serve to estimate the distribution 

of estimated LCMRL values that each laboratory might generate on repeated performance of the 

LCMRL study. The distribution of pooled BB replicates, generated from LCMRL study data, 

can be used to approximate the distribution of estimated LCMRL values which might be 

generated from the population of experienced drinking water laboratories.

The MRL is calculated in three steps whenever there are three or more laboratories providing 

data with valid LCMRLs or calculated LCMRLs that are below the lowest non-zero spiking 

level. In the first step, 200 BB LCMRL replicates are calculated for each laboratory data set. In 

the second step a predicted distribution of some unknown and yet to be observed laboratory is 

built from the population of replicate laboratory LCMRLs using a random effects model. In the 

third and last step the MRL is taken to be the upper 95% one-sided confidence interval on the 

75th percentile of the predicted distribution referred to as the 95-75 upper tolerance limit (95-75 

UTL).

A description of the procedure is given in the following three steps.

Step 1: Bayesian Bootstrap of the Laboratory data

When three or more laboratories have either a valid LCMRL or an estimated LCMRL that is 

reported to be less than the lowest reported non-zero spiking level a BB is carried out on each 

laboratory’s data to produce 200 replicate LCMRL estimates, lkr . Here k is the number of 

laboratories ( k ≥ 3), and r is the number of replicates (200),. The BB is implemented by selecting 

random Dirichlet weights from the uniform distribution on the ni – 1 dimensional simplex, where 

ni is the number of measurements at the i

th spiking level. These weights are associated with the 

original replicate measurements within each of the i spiking levels of the laboratory data. The 

LCMRL procedure is then rerun using the Dirichlet weights to compute a weighted LCMRL. 

This is done repeatedly to generate the sample of BB replicate LCMRL estimates. 

Using Efron’s bootstrap on a small data set, like the set of replicates at each spiking level, is not 

appropriate. Statistics computed using Efron’s bootstrap are essentially calculated as randomly 

weighted statistics where the set of possible weights for each observation is { } 1 2 3 1 0, , , , , ,1 n

nnn n  −

with the additional constraint that all of the weights sum to 1. When n is small, the number of 

possible weights and therefore the number of values of any statistic that could be computed from 

the data given those weights, is small. Therefore our approach using the Bayesian Bootstrap 

with Dirichlet weights allows an extremely large number of weighted LCMRL values to be 

computed even from a relatively small data set.
LCMRL Technical Report - Appendix A December 2010

Page A-2

The portions of the LCMRL calculation that use robust weights are modified to use a product 

weight of the form

*

* *

1

* , 1, ,

/ , 1, , i

D RW

ij ij ij i

n

ij ij ij i j

www j n

w w wj n =

= =

= ∑ =





where D

wij is the Dirichlet weight and RW wij is either the robust Huber or Tukey bisquare weight. 

The weights are then normalized to produce the final weights wij.

Step 2: Estimating the Predictive Distribution 

The predictive distribution of an unknown and unobserved laboratory from the random sample of 

laboratories reporting data is estimated in three steps as follows

1. Power Transform of BB replicate Laboratory Data

The BB replicate results are transformed to approximate normality using the power 

transform (Box and Cox, 1964). The transformed replicate LCMRL values, *

kr l , are 

calculated as follows where λ is the transformation parameter

* , 1, , 1, ,200 if 0

ln( ), 1, , 1, ,200 if 0

kr

kr

kr

l k Kr

l

l k Kr

λ

λ

λ

λ

 =... =... > = 

 =... =... =

where K is the number of laboratories. The transformation parameter λ is constrained to 

be non-negative.

2. The values of the predictive distribution are calculated according to the following 

steps:

a. Calculate Initial Resistant Location and Scale Estimates

Location and scale parameters are estimated for the pooled set of transformed 

replicate values. These values are used as initial estimates in generating onesided Tukey bisquare weights to down-weight large estimates in the pooled 

transformed replicate LCMRL values *

kr l . The median of the *kr l is used for the 

initial resistant location estimate and the median absolute deviation from the 

median (MAD) of the *

kr l is used as the initial resistant scale estimate.

b. Calculate One-Sided Tukey Bisquare Weights for the pooled LCMRL 

replicate values

The one sided Tukey bisquare weights are calculated using a tuning constant, cBW, 

of 6. The initial starting values used are the median, (0) TBW , and MAD from “Step 

2a” above. If any of the weights are zero, the weights are recalculated with a 

tuning constant of 9. The weights are iteratively calculated at the t

th stage 

according to the following method
LCMRL Technical Report - Appendix A December 2010

Page A-3

( )

( ) ( )

( )

( )

( ( ) ) ( )

( )

( ) ( )

* 1 * 1

* 1

2 2

( )

( ) ( )

, 1, , , 1, ,200 , if 

0, if 

1 , if 1 , 1, , , 1, ,200

0, if 1

t

kr BW

BW

l T t

t c AD kr BW

kr t

kr BW

t t

BW t kr kr

kr

t

kr

t BW t BW t

BW kr kr kr kr kr

k Kr l T

u

l T

u u

w k Kr

u

T wl w

− − −

−



 = = > = 

 ≤ 



 − < =  = =



 ≥

= ∑ ∑

 

 

until a relative convergence of 1E-6 is reached for (t) TBW . After convergence the 

final one-sided robust weights are calculated as the normalized weights

( )

( )

BW t

BW kr

kr BW t

kr kr

w

w

w =

∑

c. Calculate Robust means and Variances for each of the K laboratory replicate 

LCMRL sets

Robust location and scale estimates are calculated by laboratory for the 

transformed replicate LCMRL values, *

kr l . These values are used in estimating the 

predicted laboratories variance on the transformed scale. For the k

th laboratory, an 

estimate of the mean, * mk , and variance , *2

k s , are calculated using the blended 

Huber and Tukey biweight methodology used in the LCRML routine except that a 

tuning constant of 6 is used instead of 9.

d. Estimate the variance of the predicted distribution

Robust location and scale estimates are estimated by laboratory for the 

transformed replicate LCMRL values. The predicted variance, 2

pred s , is then 

estimated as

( )

( )

( )( )

( )

( )

200 *

1 1

200 2 *2

1 1

200 2 * *

2 1 1

2 200

1 1

2 2 2

*

1

1 1/

w

b

b w

K

kr k k r

K

kr k k r

K

kr k k r

K

kr k r

pred

m w m

s w s

w mm

s

w

s Ks s

= =

= =

= =

= =

=

=

− =

−

=+ +

∑ ∑

∑ ∑

∑ ∑

∑ ∑
LCMRL Technical Report - Appendix A December 2010

Page A-4

e. Estimate the values of the predicted distribution

The values of the predicted distribution are generated as follows. First the 

transformed values are generated as

* * **

, * ( ) / , 1, , , 1, ,200 pred kr pred kr k k l m s l m s k Kr =+ − = =  

These values are then transformed back to the original scale as

*

,

, *

,

exp(ln( ) / ), if >0

exp( ), 0

pred kr

pred kr

pred kr

l

l

l if

λ λ

λ

 = 

 = 

Step 3: Estimate the MRL as the 95-75 UTL of the Predictive Distribution

The MRL is then estimated as the 95-75 UTL from the values of the predicted distribution using 

a weighted version of the Guttman non-parametric procedure (Guttman, 1970) using the weights 

wkr .

References

Box, G. E. P. and D.R. Cox (1964). “An Analysis of Transformations,” Journal of the Royal

Statistical Society, Series B. Vol 26, pp. 211-46.

Guttman, I. (1970). Statistical Tolerance Regions: Classical and Bayesian. Hafner Press, Darien, 

CT.

Rubin, D. B. (1981). “The Bayesian Bootstrap.” Annals of Statistics, 9, pp. 130-134.