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A World of Solutions
LCMRL: Improved Estimation of Quantitation Limits
@ Pittcon 2015
John H Carson Jr., PhD CB&I Federal Services LLC
Robert O’Brien
Steve Winslow CB&I Federal Services LLC
Steve Wendelken USEPA, OGWDW
David Munch USEPA, OGWDW (retired)
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LCMRL stands for Lowest Concentration Minimum Reporting Limit
– Reporting Limit based on a defined accuracy of measurement objective
– Measurement Quality Objective (MQO)
– EPA is only organization using this term and concept
– Lowest true sample concentration such that individual measurements meet a specified MQO with high probability
Statistical estimate of the LCMRL
What is the LCMRL?
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Currie (1968) – Applied stat decision theory to detection limits (LC, LD)
– Determination Limit (LQ) -- S/N = 10
Hubaux & Vos (1970) – Applied regression analysis to Currie’s approach
Horwitz et al. (1980) – Power law for std dev of repeated measurement
Glaser et al. (1981) – EPA Lab in Cincinnati developed MDL, practical procedure for
determining Currie’s LC and LQ
Rocke and Lorenzato (1995) – Analytical error is combination of additive and multiplicative errors
– Error variance does not 0 as concentration 0
Summary of some prior developments
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MQO: the measured concentration is within 50% of true concentration (50-150% recovery)
LCMRL is lowest concentration that meets the MQO criterion with 99% probability
To use both precision and bias in the analysis
– non-constant error variance
– imperfect calibration
– possible nonlinearity of response
To make estimates robust against outliers in data
To develop a robust algorithm and computational code that can handle “bad data” without crashing
To develop an LCMRL calculator for end users
USEPA OGWDW Objectives
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Start with an experiment and a picture
0 100 200 300 400 500 600 7000
200
400
600
800
1000
Trichloroethene--LCMRL Plot
Spike Concentration ug/L
Mea
sure
d C
once
ntr
ati
on
ug/L
Data LCMRL = 57 ug/L Y=X Regression 50-150% Recovery Lower/Upper Prediction Limits
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The coverage probability for the MQO interval is a function of the true sample concentration
𝑄 𝜇 = Pr 0.5𝜇 < 𝑋 ≤ 1.5𝜇 𝜇 , for a future measurement
Estimating Q requires
– assumed predictive distribution family for replicate measurements
– estimates of mean and variance of replicate measurements as function of 𝜇
How to estimate probabilities?
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Estimation of conditional mean response function is a regression problem
– measured regressed onto “true”
– Also estimates conditional bias
Estimation of conditional replicate variance function is also a regression problem
Choice of measurement distribution family
– Prefer maximum entropy distributions with specified mean and variance of response distribution
– Normal when measurements can be negative
– Gamma, when measurements cannot be negative
Conditional Distribution of Response
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Start with resistant scale estimates at each spiking level as starting point
Construct robust estimates of scale at each spiking level,
Use M-estimator of scale or other robust estimator
Fit Replicate Variance (RV) model based on Horwitz’s power law with additive component also
Fit by Nelder-Meade with invariant scale loss, such as
Estimating variance function
2 ca b
2
1ˆ 1
m
i i iin s
is
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Non-constant variance + possible outliers use Iteratively Reweighted Least Squares (IRLS)
Weights are product of
– robust weights to minimize impact of ‘outliers’ and
– reciprocal of variance function
Possible nonlinearity at upper end use low order polynomial
Estimating mean response function
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Methods often designed to operate in linear range, BUT
Typical calibration experiments for analytical instruments – Usually do not include replication
– Usually have 5 or fewer design points
– Often will not detect nonlinearity in response function
Typical LCMRL experiment has – 4 replicates
– 7 or 8 design points in low working range
LCMRL experiment allows estimation of bias, including nonlinearity, not otherwise detectable
Why nonlinearity?
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Nonlinearity at low concentration caused by non-ideal processes in measurement such as: – Presence of analyte interferent
– Analyte absorption or degradation by instrument
– Loss of analyte in extraction step
– Matrix enhancement
Polynomial only handles loss of linearity at high end
Mean-squared error (MSE) function – Incorporates error due to nonlinearity at low end of curve
– Modeled using constant + power function
– Estimation similar to variance function
– Captures part of lack of fit (Type III error)
Mean squared error function
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Need prediction error variance to compute 𝑄 𝜇
Use pointwise maximum of variance and MSE functions
Uncertainly about parameter estimates in response model
Prediction variance
2
2
1
2
2
2 2 2
11
MSE function
max ,
v
e
m
j jj
c
v v
c
e e
pred
x xf x
n n x x
x a b x
x a b x
x f x x x
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Distributional family + mean response curve + prediction variance curve completely define an estimated distribution of replicate measurements at each true sample concentration.
Can directly estimate as function of sample concentration probability that sample recovery is between 50% and 150%.
At this point finding LCMRL is a numerical optimization problem.
EPA LCMRL Calculator software makes LCMRL usable for labs.
Calculator download and technical manual are at http://water.epa.gov/scitech/drinkingwater/labcert/analyticalmethods_ogwdw.cfm#four
Predictive distribution
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0 0.05 0.1 0.15 0.2 0.250
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
Mea
sure
d C
once
ntra
tion
ug/L
Data LCMRL = 0.042 ug/L Y=X Regression 50-150% Recovery Lower/Upper Prediction Limits
1,4-Dioxane LCMRL Plot
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QC Interval is 50%-150% recovery
LCMRL is found on x-axis
QC Interval Coverage Probability
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
Pro
ba
bilit
y o
f Q
C I
nte
rva
l C
ove
rag
e
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Quality Reporting Limit (QRL) defined as lowest true sample value such that measured result expected to be within 100 ± Q% of true value C% of the time—QRL(Q,C)
LCMRL is QRL(50,99)
In some cases, response is mass rather than concentration
Not always possible to have replicates at exactly the same values
– Compositional analysis
– Need a lot more data in this case, but it is doable
QRL – Generalization of LCMRL
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PoliMat application was for compositional analysis (CHNS) using Element Analyzer
Response was mass (mg) converted to composition
Composition of standard materials known exactly
But mass of sample increment not reproducible with needed degree of accuracy
Used many measurements to compensate
– from four designed studies
– Could have used routine QC data
Computed QC(Q,0.95) for Q=0.5, 0.25, 0.2, 0.05, 0.01
PoliMat Compositional Analyzer Application
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0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.5
1.5
2.5
3.5
C True Mass (mg)
C R
esponse (
mg)
0.5 1.0 1.5 2.0 2.5 3.0 3.5
-0.0
40.0
00.0
2
+/- 3 Sigma Limits in Red
C True Mass (mg)
C R
aw
Resid
uals
(m
g)
0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0
00.0
20.0
4
C True Mass (mg)
C A
bsolu
te V
alu
e o
f R
esid
uals
(m
g)
0.5 1.0 1.5 2.0 2.5 3.0 3.5
-20
-10
-50
5
+/- 3 Sigma Limits in Red
C True Mass (mg)
C S
tandard
ized R
esid
uals
Carbon
QRL Diagnostic Plots
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Affiliated and supporting procedures:
– MRL, essentially a programmatic LCMRL upper limit, is already done
– LCMRL/MRL quick validation procedure
– Revamping lab QC program to monitor LCMRL at similar cost
– Extending to other matrices through standard additions
Fully Bayesian implementation via MCMC
Multianalyte LCMRL method, requires Bayesian MCMC implementation
Development as ASTM standard practice
What Next?
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This work has been funded by USEPA under contract (EP-C-06-031) to Shaw Environmental, Inc (now CBI Federal Services LLC) and under contract (EP-C-07-022) to The Cadmus Group, Inc.
EPA OGWDW Program Managers—Steve Wendelken, David Munch (retired)
CBI statisticians—John Carson, Robert O’Brien
CBI principal analyst—Steve Winslow, extensive testing, feedback and supplying test data sets
CBI project manager—Mike Zimmerman
Acknowledgements
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Questions?
John H. Carson, Jr PhD
Senior Statistician
CB&I Federal Services LLC
+1 419-429-5519
Questions
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Contact
For further information, contact
John H. Carson, Jr PhD
Senior Statistician
CB&I Federal Services LLC
+1 419-429-5519
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Currie, L. A. (1968), “Limits for Qualitative Detection and Quantitative Determination.” Analytical Chemistry, Vol. 40, pp. 586-593.
Horwitz W, Kamps LR, Boyer KW. (1980) Quality assurance in the analysis of foods and trace constituents. Journal of the Association of Official Analytical Chemists. 63(6):1344-54.
Glaser, J. A., Foerst, D. L., McKee, G. D., Quane, S. A. and W. L. Budde (1981), “Trace Analyses for Wastewaters.” Environmental Science and Technology. Vol. 15, pp. 1426-1435.
Hubaux, Andre and Gilbert Vos (1970), “Decision and Detection Limits for Linear Calibration Curves.” Analytical Chemistry. Vol. 42, No. 8, pp. 849-855.
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.
Analytical Chemistry References
A World of Solutions
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Horn, Paul S. (1988) “A Biweight Prediction Interval for Random Samples.” Journal of the American Statistical Association. Vol. 83, No. 401. (Mar., 1988), pp. 249-256.
Kagan, A. M.; Linnik, Yu. V. and C. R. Rao (1973) Characterization Problems in Mathematical Statistics. John Wiley. New York. 499 pp.
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.
Nelder, J.A. and R. Mead (1965), “A Simplex Method for Function Minimization”, Computer Journal. Vol. 7, pp. 308-313.
Rousseeuw, P.J. and C. Croux (1993) “Alternatives to the Median Absolute Deviation” Journal of the American Statistical Association. Vol. 88, No. 424 (Dec., 1993), pp. 1273-1283.
Statistical References