# ASTM D6512-07 (Reapproved 2014)

Designation D6512 07 Reapproved 2014Standard Practice forInterlaboratory Quantitation Estimate1This standard is issued under the fixed designation D6512; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice establishes a uni standard for com-puting the interlaboratory quantitation estimate associated withZ relative standard deviation referred to herein as IQEZ,and provides guidance concerning the appropriate use andapplication. The calculations involved in this practice can bepered with DQCALC, Microsoft Excel-based softwareavailable from ASTM.21.2 IQEZis computed to be the lowest concentration forwhich a single measurement from a laboratory selected fromthe population of qualified laboratories represented in aninterlaboratory study will have an estimated Z relativestandard deviation Z RSD, based on interlaboratory stan-dard deviation, where Z is typically an integer multiple of 10,such as 10, 20, or 30, but Z can be less than 10. The IQE10 is consistent with the quantitation approaches of Currie 13and Oppenheimer, et al. 2.1.3 The fundamental assumption of the collaborative studyis that the media tested, the concentrations tested, and theprotocol followed in the study provide a representative and fairuation of the scope and applicability of the test aswritten. Properly applied, the IQE procedure ensures that theIQE has the following properties1.3.1 Routinely Achievable IQE ValueMost laboratoriesare able to attain the IQE quantitation perance in routineanalyses, using a standard measurement system, at reasonablecost. This property is needed for a quantitation limit to befeasible in practical situations. Representative laboratoriesmust be included in the data to calculate the IQE.1.3.2 Accounting for Routine Sources of ErrorThe IQEshould realistically include sources of bias and variation thatare common to the measurement process. These sourcesinclude, but are not limited to intrinsic instrument noise, some“typical” amount of carryover error; plus differences inlaboratories, analysts, sample preparation, and instruments.1.3.3 Avoidable Sources of Error ExcludedThe IQEshould realistically exclude avoidable sources of bias andvariation; that is, those sources that can reasonably be avoidedin routine field measurements. Avoidable sources wouldinclude, but are not limited to modifications to the sample;modifications to the measurement procedure; modifications tothe measurement equipment of the validated , and grossand easily discernible transcription errors, provided there wasa way to detect and either correct or eliminate them.1.4 The IQE applies to measurement s for whichcalibration error is minor relative to other sources, such aswhen the dominant source of variation is one of the following1.4.1 Sample Preparation, and calibration standards do nothave to go through sample preparation.1.4.2 Differences in Analysts, and analysts have little oppor-tunity to affect calibration results as is the case with automatedcalibration.1.4.3 Differences in Laboratories for whatever reasons,perhaps difficult to identify and eliminate.1.4.4 Differences in Instruments measurement equipment,such as differences in manufacturer, model, hardware,electronics, sampling rate, chemical processing rate, integra-tion time, software algorithms, internal signal processing andthresholds, effective sample volume, and contamination level.1.5 Data Quality ObjectivesTypically, one would com-pute the lowest RSD possible for any given dataset for aparticular . Thus, if possible, IQE10 would be com-puted. If the data indicated that the was too noisy, onemight have to compute instead IQE20 , or possibly IQE30 .In any case, an IQE with a higher RSD level such asIQE50 would not be considered, though an IQE with RSD0 though this constraintis irrelevant for the Hybrid Model. A value ofg 0, there is sufficient statistical evidence of curvature in therelationship between skand Tkto warrant the use of the Hybrid Model,Model C Q 0 ensures that the increase in skwith respect to Tkis fasterthan linear. If these conditions do not hold, then the Straight-line ModelModel B is the appropriate model to use. Proceed to 6.3.410 The Hybrid Model for the ILSD Model C can beused if there is evidence of curvature.TABLE 1 Bias-Correction Adjustment Factors for SampleStandard Deviations Based on n Measurements at a particularconcentrationAn2 3 4 5 6 7 8 910an1.253 1.128 1.085 1.064 1.051 1.042 1.036 1.031 1.028AFor each true concentration, Tk, the adjusted value skanskshould be modeledin place of sample standard deviation, sk. For n 10, use the ula, an14n11. See Johnson and Kotz 7.D6512 07 2014511 To uate the reasonableness of the Hybrid Model,Model C, the model must first be fitted using nonlinear leastsquares NLLS, either by Newtons- iteration pre-sented in the appendix, or another NLLS .12 The fit from the Hybrid Model should be uated.Aplot of the residuals, in log , should be constructed plot rkversus Tk, whererk5 lnsk2 lnsk, 8and kis the predicted value of skusing the model. The plotshould show no systematic behavior for example, curva-ture. If the fit satisfies both types of uation, go to 6.3.4.Otherwise, a different and possibly more complex modelmay be used, such as the exponential model s g exphT1 error. If there are enough true concentrations, amodel with more coefficients could be considered; possibili-ties include quadratic strictly increasing with increasingconcentration, or even cubic.6.3.4 Fit the Mean-Recovery ModelThe mean-recoverymodel is a simple straight line,Model RY 5 a1bT1error. 9The fitting procedure depends on the model selection from6.3.3. If the constant model, Model A, was selected for ILSD,then OLS can be used to fit Model R for mean recovery see theleft column of Table 2, or Caulcutt and Boddy 5. If anonconstant ILSD model was selected, such as the Straight-line Model Model B, or the Hybrid Model Model C, thenweighted least squares WLS should be used to fit meanrecovery. The WLS approximately provides the minimum-variance unbiased linear estimate of the coefficients, a and b.The WLS procedure is described in 6.3.4.1.6.3.4.1 Weighted Least Squares Procedure, Using the Inter-laboratory Standard Deviation ILSD Model1 Using the ILSD model and coefficient estimates from6.3.3, compute the predicted interlaboratory standarddeviation, k, for each true concentration, TkModel Bsk5 g1hTk10Model Csk5 g21hTk21/2112 Compute weights for WLSwk5 sk22. 12Note that if WLS is carried out using computer software, thedefault setting for weights may be different. For example,instead of supplying the values, k2, as weights, the soft-ware may require the user to supply values kork2asweights that are internally transed by the software.3 Carry out WLS computations analogous to OLS com-putations. See Table 2 or Caulcutt and Boddy 5. The resultwill be coefficient estimates, a and b, for the mean-recoverymodel, Model R. Appendix II describes three approximateapproaches to WLS commonly practiced, but not acceptablefor this application.4 After fitting, the mean-recovery model should be u-ated for reasonableness and lack of fit. This uation shouldbe done by ensuring the following 1 The fit is statisticallysignificant overall p-value 5 ; 3A plot of the residuals shows no obvious systematic curvaturefor example, quadratic-like behavior. If the mean-recoverymodel fails the uation, then the study supervisor will haveto determine if only a subset of the data should be analyzedperhaps the model fails for the higher concentrations, or ifmore data are needed.6.4 Compute the IQEThe IQE is computed using theILSD model to estimate the interlaboratory standard deviation,and using the mean-recovery model to scale the standarddeviation. For any computed IQE to be valid, it must lie withinthe range of concentrations used in the study. The general of the computation is to find the solution, LQ within the rangeof concentrations used in the study, to the following equationT 5 100/ZGT 13where function GT is the estimated interlaboratory stan-dard deviation in concentration units of true value, T, and Zis taken to be 10, 20, or 30, in increasing order. That is, the firstattempt is to compute IQE10 .IfIQE10 does not exist or isoutside the range of concentrations used in the study, thenIQE20 is computed, if possible. If IQE20 does not exist or isoutside the range of concentrations used in the study, thenIQE30 is computed, if possible. If appropriate for a particularuse, IQEZcan be computed for any value of Z 30is not recommended. Thus, the IQE computation depends onthe of the ILSD model, which is part of function G. Theratio, Z100h/b, represents the limit of the RSD achievable.Therefore the strictest IQE achievable by the analytical studied is IQEZ . For example, if Z 1000.17/1.0 17, thenthe strictest IQE achievable would be the IQE20 according tothe nearest higher multiple of 10.6.4.1 ILSD Constant Model Model AIn this case, g;hence GT g/b and LQ 100/Zg/b. Thus,IQEZ 5 100/Zg/b 146.4.2 ILSD “Straight-line” Model Model BIn this case, ghT; hence GT g h T/b. To find the IQE, onemust solve for T T 100/ZghT/b. The solution isTABLE 2 Ordinary Least Squares OLS and Weighted LeastSquares WLS Computations to Estimate Straight-line ModelCoefficientsComputations shown for convenience and contrastOLS WLST51noi51nTi, Tw5oi5lnwiTi/oi5lnwiy 51noi51nyiyw5oi5lnwiyi/oi5lnwiSTT5oi5lnsTi2 Td2SwTT5oi51nwisTi2 Td2STY5oi5lnsTi2 Tdsyi2 yd SwTY5oi5lnwisTi2 Tdsyi2 ydslope5 b 5 STY/STTslope5 b 5 SwTY/SwTTintercept5 a 5 y 2 bTintercept5 a 5 yw2 bTwD6512 07 20146IQEZ 5 g/bZ/100 2 h. 156.4.3 ILSD Hybrid Model Model Cadditive and multi-plicative error, in accordance with Rocke and Lorenzato 3.In this case, g2hT21/2; hence GT g2hT21/2/b. To find the IQE, one must solveT 5 100/Z/bg21hT21/216This solution is derived by squaring each side of the equationand solving to get IQEZ g/bZ/1002 h21/2, where thepositive square root is taken.6.5 Non-Trivial Amount of Censored DataMore than10 of the data for at least one true concentration may havebeen reported as nondetects or less-thans. Despite the attemptin 6.2.3.1 to reduce or eliminate reported nondetects orless-thans, they may still occur at a level that disrupts the dataanalysis presented in 6.3 and 6.4. If there is excessivecensoring, the study supervisor should contact laboratorieswith such measurements to see whether the data can beextracted in uncensored from data archives. If this effortis not a sufficient remedy, serious consideration should begiven to augmenting the IQE study with measurements ofsamples at new and different concentrations generally, higher.7. Data Analysis7.1 The data analysis for eliminating data is given in Section10 of Practice D2777.7.2 The data analysis involved in computing an IQE isshown by example in Section 10 of this practice.8. Report8.1 The analysis report should be structured as in AnnexA1.8.1.1 The report should be given a second-party review toverify that8.1.1.1 The data transcription and reporting have beenpered correctly,8.1.1.2 The analysis of the data has been peredcorrectly, and8.1.1.3 The results of the analysis have been usedappropriately, including assessment of assumptions necessaryto compute an IQE.8.1.2 A statement of the review and the results shouldaccompany the report. Reviewers should be qualified in oneor both of the following areas 1 applied statistics, and 2analytical chemistry.9. Rationale9.1 The basic rationale for the IQE is contained in Currie1. The IQE is a perance characteristic of an analytical, to paraphrase Currie. As with the InterlaboratoryDetection Estimate IDE described in Practice D6091, theIQE is vital for the planning and use of chemical analyses. TheIQE is another benchmark indicating whether the canadequately meet measurement needs.9.2 The idealized definition of IQEZis that it is the lowestconcentration, LQ, that satisfies T 100/Z where Tisthe actual standard deviation of interlaboratory measurementsat concentration T, which is equivalent to satisfying, RSD /T Z . In other words, IQEZis the lowest concentrationwith Z RSD assuming such a concentration exists. If, as iscommonly the case, RSD declines with increasing trueconcentration, then the relative uncertainty of any measure-ment of a true concentration greater than the IQE will notexceed 6Z . The range, 63LQ, is an approximate predictionor confidence interval very likely to contain the measurement,which is assumed to be Normally distributed. This assertion isbased on critical values from the Normal distribution or fromthe Students t distribution if is estimated rather than known.Then, with high confidence, the relative error of any measure-ment of a true concentration greater than the IQE will notexceed 63Z . For example, a measurement above theIQE10 and assumed to have true concentration above theIQE could be reported as 6 ppb 630662 ppb, witha high degree of certainty.9.3 There are several real-world complications to this ide-alized situation. See Maddalone et al. 8, Gibbons 9, andColeman et al. 10. Some of these complications are listed asfollows9.3.1 Analyte recovery is not perfect; the relationship be-tween measured values of concentrations and true concentra-tions cannot be assumed to be trivial. There is bias betweentrue and measured values. Recovery can and should bemodeled. Usually a straight line will suffice.9.3.2 Variation is introduced by different laboratories,analysts, models and pieces of equipment; environmentalfactors; flexibility/ambiguity in a test ; contamination;carryover; matrix influence; and other factors. It is intractableto model these factors individually, but their collective contri-butions to measurement ILSD can be observed, if thesecontributions are part of how a study is designed and con-ducted.9.3.3 The interlaboratory standard deviation of measure-ments is generally unknown, and may change with trueconcentration, possibly because of the physical principle of thetest . To ensure that a particular RSD is attained at orabove the IQE, there must be a way to predict the ILSD atdifferent true concentrations. Short of severely restricting therange of concentrations for a study, prediction is accomplishedb