# ASTM D6512-07 (Reapproved 2014)

Designation: D6512 − 07 (Reapproved 2014)Standard 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 uniform 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 beperformed with DQCALC, Microsoft Excel-based softwareavailable from ASTM.21.2 IQEZ%is 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 (1)3and 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 fairevaluation of the scope and applicability of the test method aswritten. Properly applied, the IQE procedure ensures that theIQE has the following properties:1.3.1 Routinely Achievable IQE Value—Most laboratoriesare able to attain the IQE quantitation performance 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 Error—The 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 Excluded—The 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 method, 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 methods for whichcalibration error is minor relative to other sources, such aswhen the dominant source of variation is one of the following:1.4.1 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 Objectives—Typically, one would com-pute the lowest % RSD possible for any given dataset for aparticular method. Thus, if possible, IQE10 %would be com-puted. If the data indicated that the method 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 Model(Model B) is the appropriate model to use. Proceed to 6.3.4(10) 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 particularconcentration)An2 3 4 5 6 7 8 910a n1.253 1.128 1.085 1.064 1.051 1.042 1.036 1.031 1.028AFor each true concentration, Tk, the adjusted value sk=a ns kshould be modeledin place of sample standard deviation, s k. For n 10, use the formula, a n=1+[4(n−1)]−1. See Johnson and Kotz (7).D6512 − 07 (2014)5(11) To evaluate the reasonableness of the Hybrid Model,Model C, the model must first be fitted using nonlinear leastsquares (NLLS), either by Newton’s-Method iteration (pre-sented in the appendix), or another NLLS method.(12) The fit from the Hybrid Model should be evaluated.Aplot of the residuals, in log form, should be constructed: plot rkversus Tk, where:rk5 lnsk2 lnsˆk, (8)and ŝkis the predicted value of skusing the model. The plotshould show no systematic behavior (for example, curva-ture). If the fit satisfies both types of evaluation, go to 6.3.4.Otherwise, a different (and possibly more complex) modelmay be used, such as the exponential model: s = g exp{hT}·(1 + 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 Model—The mean-recoverymodel is a simple straight line,Model R:Y 5 a1bT1error. (9)The 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) Model:(1) Using the ILSD model and coefficient estimates from6.3.3, compute the predicted interlaboratory standarddeviation, ŝk, for each true concentration, Tk:Model B:sˆk5 g1hTk(10)Model C:sˆk5 ~g21@hTk#2!~1/2!(11)(2) Compute weights for WLS:wk5 ~sˆk!22. (12)Note that if WLS is carried out using computer software, thedefault setting for weights may be different. For example,instead of supplying the values, (ŝk)−2, as weights, the soft-ware may require the user to supply values (ŝk)or(ŝk)2asweights that are internally transformed 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 evalu-ated for reasonableness and lack of fit. This evaluation shouldbe done by ensuring the following: (1) The fit is statisticallysignificant (overall p-value 5 %); (3)A plot of the residuals shows no obvious systematic curvature(for example, quadratic-like behavior). If the mean-recoverymodel fails the evaluation, then the study supervisor will haveto determine if only a subset of the data should be analyzed(perhaps the model fails for the higher concentration(s)), or ifmore data are needed.6.4 Compute the IQE—The 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 formof the computation is to find the solution, LQ (within the rangeof concentrations used in the study), to the following equation:T 5 ~100/Z!·G~T! (13)where function G(T) 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, IQEZ%can be computed for any value of Z 30is not recommended. Thus, the IQE computation depends onthe form of the ILSD model, which is part of function G. Theratio, Z =100·h/b, represents the limit of the %RSD achievable.Therefore the strictest IQE achievable by the analytical methodstudied is IQEZ %. For example, if Z = 100·0.17/1.0 = 17, thenthe strictest IQE achievable would be the IQE20 %(according tothe nearest higher multiple of 10).6.4.1 ILSD Constant Model (Model A)—In this case, ŝ =g;hence G(T) = g/b and LQ = (100/Z)·g/b. Thus,IQEZ %5 ~100/Z!·g/b (14)6.4.2 ILSD “Straight-line” Model (Model B)—In this case,ŝ =g+hT; hence G(T) = (g + h T)/b. To find the IQE, onemust solve for T: T = (100/Z)·(g+hT)/b. The solution is:TABLE 2 Ordinary Least Squares (OLS) and Weighted LeastSquares (WLS) Computations to Estimate Straight-line ModelCoefficients(Computations shown for convenience and contrast)OLS WLST¯51noi51nTi, T¯w5oi5lnwiTi/oi5lnwiy¯ 51noi51nyiy¯w5oi5lnwiyi/oi5lnwiSTT5oi5lnsTi2 T¯d2SwTT5oi51nwisTi2 T¯d2STY5oi5lnsTi2 T¯dsyi2 y¯d SwTY5oi5lnwisTi2 T¯dsyi2 y¯dslope5 b 5 STY/STTslope5 b 5 SwTY/SwTTintercept5 a 5 y¯ 2 bT¯intercept5 a 5 y¯w2 bT¯wD6512 − 07 (2014)6IQEZ %5 g/~b·~Z/100! 2 h!. (15)6.4.3 ILSD Hybrid Model (Model C)—(additive and multi-plicative error, in accordance with Rocke and Lorenzato (3)).In this case, ŝ =(g2+[h·T]2)(1/2); hence G(T) =(g2+[h·T]2)(1/2)/b. To find the IQE, one must solveT 5 ~~100/Z!/b!~g21@h·T#2!~1/2!(16)This solution is derived by squaring each side of the equationand solving to get: IQEZ%= g/[(b·Z/100)2− h2]1/2, where thepositive square root is taken.6.5 Non-Trivial Amount of Censored Data—More 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 form 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 that:8.1.1.1 The data transcription and reporting have beenperformed correctly,8.1.1.2 The analysis of the data has been performedcorrectly, 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. Reviewer(s) should be qualified in oneor both of the following areas: (1) applied statistics, and (2)analytical chemistry.9. Rationale9.1 The basic rationale for the IQE is contained in Currie(1). The IQE is a performance characteristic of an analyticalmethod, 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 method canadequately meet measurement needs.9.2 The idealized definition of IQEZ%is that it is the lowestconcentration, LQ, that satisfies: T = (100/Z) σΤ(where σTisthe actual standard deviation of interlaboratory measurementsat concentration T), which is equivalent to satisfying, %RSD =σΤ/T = Z %. In other words, IQEZ%is 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, 63σLQ, 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 Student’s t distribution if σ 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 63·Z %. For example, a measurement above theIQE10 %(and assumed to have true concentration above theIQE) could be reported as 6 ppb (630%)=6(62) 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 asfollows:9.3.1 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 method; 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 method. 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