LCMRL: Improved Estimation of Quantitation Limits

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1 LCMRL: Improved Estimation of Quantitation Limits John H Carson Jr., PhD Robert O Brien Steve Winslow Steve Wendelken David Pittcon 2015 CB&I Federal Services LLC CB&I Federal Services LLC USEPA, OGWDW USEPA, OGWDW (retired) A World of Solutions

2 What is the LCMRL? 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 A World of Solutions 1

3 Summary of some prior developments Currie (1968) Applied stat decision theory to detection limits (L C, L D ) Determination Limit (L Q ) -- 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 L C and L Q Rocke and Lorenzato (1995) Analytical error is combination of additive and multiplicative errors Error variance does not 0 as concentration 0 A World of Solutions 2

4 Horwitz Trumpet A World of Solutions 3

5 USEPA OGWDW Objectives 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 A World of Solutions 4

6 Start with an experiment and a picture Measured Concentration ug/l Trichloroethene--LCMRL Plot Data LCMRL = 57 ug/l Y=X Regression % Recovery Lower/Upper Prediction Limits Spike Concentration ug/l A World of Solutions 5

7 How to estimate probabilities? 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 A World of Solutions 6

8 Conditional Distribution of Response 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 A World of Solutions 7

9 Estimating variance function 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 2 c a b Fit by Nelder-Meade with invariant scale loss, such as m i 1 n sˆ 1 2 i i i sˆi A World of Solutions 8

10 Estimating mean response function 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 A World of Solutions 11

11 Why nonlinearity? 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 A World of Solutions 12

12 Mean squared error function 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) A World of Solutions 13

13 Prediction variance Need prediction error variance to compute Q μ Use pointwise maximum of variance and MSE functions Uncertainly about parameter estimates in response model f 2 2 x 1 x a b x v x a b x 1 x x m n n x x j 1 c e e x f x max x, x pred 2 2 MSE function v e c v j j A World of Solutions 14

14 Predictive distribution 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 almethods_ogwdw.cfm#four A World of Solutions 15

15 1,4-Dioxane LCMRL Plot Measured Concentration ug/l ,4-dioxane--LCMRL Plot Data LCMRL = ug/l Y=X Regression % Recovery Lower/Upper Prediction Limits Spike Concentration ug/l A World of Solutions 16

16 QC Interval Coverage Probability Probability of QC Interval Coverage ,4-dioxane--QC Interval Coverage Plot 1 LCMRL = ug/l Qual Lim: % Coverage Prob: Spike Concentration ug/l QC Interval is 50%- 150% recovery LCMRL is found on x-axis A World of Solutions 17

17 QRL Generalization of LCMRL 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 A World of Solutions 18

18 PoliMat Compositional Analyzer Application 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 A World of Solutions 19

19 QRL(1,95) A World of Solutions 20

20 QRL Diagnostic Plots Carbon C Response (mg) C Raw Residuals (mg) C True Mass (mg) C True Mass (mg) +/- 3 Sigma Limits in Red C Absolute Value of Residuals (mg) C Standardized Residuals C True Mass (mg) C True Mass (mg) +/- 3 Sigma Limits in Red A World of Solutions 21

21 What Next? 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 A World of Solutions 22

22 Acknowledgements This work has been funded by USEPA under contract (EP-C ) to Shaw Environmental, Inc (now CBI Federal Services LLC) and under contract (EP-C ) 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 A World of Solutions 23

23 Questions Questions? John H. Carson, Jr PhD Senior Statistician CB&I Federal Services LLC A World of Solutions 24

24 Contact For further information, contact John H. Carson, Jr PhD Senior Statistician CB&I Federal Services LLC A World of Solutions 25

25 Analytical Chemistry References Currie, L. A. (1968), Limits for Qualitative Detection and Quantitative Determination. Analytical Chemistry, Vol. 40, pp 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): 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 Hubaux, Andre and Gilbert Vos (1970), Decision and Detection Limits for Linear Calibration Curves. Analytical Chemistry. Vol. 42, No. 8, pp Rocke, D.M. and S. Lorenzato (1995), A Two-Component Model for Measurement Error in Analytical Chemistry. Technometrics. Vol. 37, No. 2, pp A World of Solutions 26

26 Statistical References Horn, Paul S. (1988) A Biweight Prediction Interval for Random Samples. Journal of the American Statistical Association. Vol. 83, No (Mar., 1988), pp 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 Nelder, J.A. and R. Mead (1965), A Simplex Method for Function Minimization, Computer Journal. Vol. 7, pp 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 A World of Solutions 27