# ASTM D7366-08 (Reapproved 2013)

Designation: D7366 − 08 (Reapproved 2013)Standard Practice forEstimation of Measurement Uncertainty for Data fromRegression-based Methods1This standard is issued under the fixed designation D7366; 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 standard for computing themeasurement uncertainty for applicable test methods in Com-mittee D19 on Water. The practice does not provide a single-point estimate for the entire working range, but rather relatesthe uncertainty to concentration. The statistical technique ofregression is employed during data analysis.1.2 Applicable test methods are those whose results comefrom regression-based methods and whose data are intra-laboratory (not inter-laboratory data, such as result fromround-robin studies). For each analysis conducted using such amethod, it is assumed that a fixed, reproducible amount ofsample is introduced.1.3 Calculation of the measurement uncertainty involves theanalysis of data collected to help characterize the analyticalmethod over an appropriate concentration range. Examplesources of data include: 1) calibration studies (which may ormay not be conducted in pure solvent), 2) recovery studies(which typically are conducted in matrix and include allsample-preparation steps), and 3) collections of data obtainedas part of the method’s ongoing Quality Control program. Useof multiple instruments, multiple operators, or both, andfield-sampling protocols may or may not be reflected in thedata.1.4 In any designed study whose data are to be used tocalculate method uncertainty, the user should think carefullyabout what the study is trying to accomplish and muchvariation should be incorporated into the study. General guid-ance on designing studies (for example, calibration, recovery)is given in Appendix A. Detailed guidelines on sources ofvariation are outside the scope of this practice, but generalpoints to consider are included in Appendix B, which is notintended to be exhaustive. With any study, the user must thinkcarefully about the factors involved with conducting theanalysis, and must realize that the computed measurementuncertainty will reflect the quality of the input data.1.5 Associated with the measurement uncertainty is a user-chosen level of statistical confidence.1.6 At any concentration in the working range, the measure-ment uncertainty is plus-or-minus the half-width of the predic-tion interval associated with the regression line.1.7 It is assumed that the user has access to a statisticalsoftware package for performing regression. A statisticianshould be consulted if assistance is needed in selecting such aprogram.1.8 A statistician also should be consulted if data transfor-mations are being considered.1.9 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2D1129 Terminology Relating to Water3. Terminology3.1 Definitions of Terms Specific to This Standard:3.1.1 confidence level—the probability that the predictioninterval from a regression estimate will encompass the truevalue of the amount or concentration of the analyte in asubsequent measurement. Typical choices for the confidencelevel are 99 % and 95 %.3.1.2 fitting technique—a method for estimating the param-eters of a mathematical model. For example, ordinary leastsquares is a fitting technique that may be used to estimate theparameters a0,a1,a2,…of the polynomial modely=a0+a1x+a2x2+ …, based on observed {x,y} pairs. Weighted leastsquares is also a fitting technique.1This practice is under the jurisdiction of ASTM Committee D19 on Water andis the direct responsibility of Subcommittee D19.02 on Quality Systems,Specification, and Statistics.Current edition approved Jan. 1, 2013. Published January 2013. Originallyapproved in 2008. Last previous approval in 2008 as D7366 – 08. DOI: 10.1520/D7366-08R13.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at service@astm.org. For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.1.3 lack-of-fit (LOF) test—a statistical technique whenreplicate data are available; computes the significance ofresidual means to replicate y variability, to indicate whetherdeviations from model predictions are reasonably accountedfor by random variability, thus indicating that the model isadequate; at each concentration, compares the amount ofresidual variation from model prediction with the amount ofresidual variation from the observed mean.3.1.4 least squares—fitting technique that minimizes thesum of squared residuals between observed y values and thosepredicted by the model.3.1.5 model—mathematical expression (for example,straight line, quadratic) relating y (directly measured value) tox (concentration or amount of analyte).3.1.6 ordinary least squares (OLS)—least squares, where alldata points are given equal weight.3.1.7 prediction interval—a pair of prediction limits (an“upper” and “lower”) used to bracket the “next” observation ata certain level of confidence.3.1.8 p-value—the statistical significance of a test; theprobability value associated with a statistical test, representingthe likelihood that a test statistic would assume or exceed acertain value purely by chance, assuming the null hypothesis istrue (a low p-value indicates statistical significance at a level ofconfidence equal to 1.0 minus the p-value).3.1.9 regression—an analysis technique for fitting a modelto data; often used as a synonym for OLS.3.1.10 residual—error in the fit between observed andmodeled concentration; response minus fit.3.1.11 root mean square error (RMSE)—an estimate of themeasurement standard deviation (that is, inherent variation inthe measurement system).3.1.12 significance level—the likelihood that a measured orobserved result came about due to simple random behavior.3.1.13 uncertainty (of a measurement)—the lack of exact-ness in measurement (for example, due to sampling error,measurement variation, and model inexactness); a statisticalinterval within which the measurement error is believed tooccur, at some level of confidence.3.1.14 weight—coefficient assigned to observations in orderto manipulate their relative influence in subsequent calcula-tions. For example, in weighted least squares, noisy observa-tions are weighted downwards, while precise data are weightedupwards.3.1.15 weighted least squares (WLS)—least squares, wheredata points are weighted inversely proportional to their vari-ance (“noisiness”).4. Summary of Practice4.1 Key points of the statistical protocol for measurementuncertainty are:4.1.1 Within the working range of the method’s data set, theestimate of the method uncertainty at any given concentrationis calculated to be plus-or-minus the half-width of the predic-tion interval.4.1.2 The total number of data points in any designed studyshould be kept high. Blanks may or may not be included,depending on the data-quality objectives of the test method.4.1.3 In applying regression to any applicable data set, theproper fitting technique (for example, ordinary least squares(OLS) or weighted least squares (WLS)) must be determined(for fitting the proposed model to the data).4.1.4 The residual pattern and the lack-of-fit test are used toevaluate the adequacy of the chosen model.4.1.5 The magnitude of the half-width of the predictioninterval must be evaluated, remembering that accepting orrejecting the amount of uncertainty is a judgment call, not astatistical decision.5. Significance and Use5.1 Appropriate application of this practice should result inan estimate of the test-method’s uncertainty (at any concentra-tion within the working range), which can be compared withdata-quality objectives to see if the uncertainty is acceptable.5.2 With data sets that compare recovered concentrationwith true concentration, the resulting regression plot allows thecorrection of the recovery data to true values. Reporting ofsuch corrections is at the discretion of the user.5.3 This practice should be used to estimate the measure-ment uncertainty for any application of a test method wheremeasurement uncertainty is important to data use.6. Procedure6.1 Introduction:6.1.1 For purposes of this practice, only regression-basedmethods are applicable. An example of a module that is notregression-based is a balance. If an object is placed on abalance, the readout is in the desired units; that is, in units ofmass. No user intervention is required to get to the neededresult. However, for an instrument such as a chromatograph ora spectrometer, the raw data (for example, peak area orabsorbance) must be transformed into meaningful units, typi-cally concentration. Regression is at the core of this transfor-mation process.6.1.2 One additional distinction will be made regarding theapplicability of this protocol. This practice will deal only withintralaboratory data. In other words, the variability introducedby collecting results from more than one lab is not beingconsidered. The examples that are shown here are for onemethod with one operator. If the user wishes, additionaloperators may be included in the design, to capture multiple-operator variability.6.1.3 A brief example will help illustrate the importance ofestimating measurement uncertainty. A sample is to be ana-lyzed to determine if it is under the upper specification limit of5 (the actual units of concentration do not matter).The final testresult is 4.5. The question then is whether the sample shouldpass or fail. Clearly, 4.5 is less than 5. If the numbers aretreated as being absolute, then the sample will pass. However,such a judgment call ignores the variability that always existswith a measurement. The width of any measurement’s uncer-tainty interval depends not only on the noisiness of the data,but also on the confidence level the user wishes to assume.ThisD7366 − 08 (2013)2latter consideration is not a statistical decision, but a reasoneddecision that must be based on the needs of the customer, theintended use of the data, or both. Once the confidence level hasbeen chosen, the interval can be calculated from the data. Inthis example, if the uncertainty is determined to be 61.0, thenthere is serious doubt as to whether the sample passes or not,since the true value could be anywhere between 3.5 and 5.5.On the other hand, if the uncertainty is only 60.1, then thesample could be passed with a high level of comfort. Only bymaking a sound evaluation of the uncertainty can the userdetermine how to apply the sample estimate he or she hasobtained. The following protocol is designed to answer ques-tions such as: 4.5 6 ?6.2 Regression Diagnostics for Recovery Data:6.2.1 Analysts who routinely use chromatographs and spec-trometers are familiar with the basics of the regression process.The final results are: 1) a plot that visually relates the responses(on the y-axis) to the true concentrations (on the x-axis) and 2)an equation that mathematically relates the two variables.6.2.2 Underlying these results are two basic choices: (1)amodel, such as a straight line or some sort of curved line, and(2) a fitting technique, which is a version of least squares. Themodeling choices are generally well known to most analysts,but the fitting-technique choices are typically less well under-stood. The two most common forms of least-squares fitting arediscussed next.6.2.2.1 Ordinary least squares (OLS) assumes that thevariance of the responses does not trend with concentration. Ifthe variance does trend with concentration, then weighted leastsquares (WLS) is needed. InWLS, data are weighted accordingto how noisy they are. Values that have relatively low uncer-tainty are considered to be more reliable and are subsequentlyafforded higher weights (and therefore more influence on theregression line) than are the more uncertain values.6.2.2.2 Several formulas have been used for calculating theweights. The simplest is 1/x (where x = true concentration),followed by 1/x2. At each true concentration, the reciprocalsquare of the actual standard deviation has also been used.However, the preferred formula comes from modeling thestandard deviation. In other words, the actual standard-deviation values are plotted versus true concentration; anappropriate model is then fitted to the data. The reciprocalsquare of the equation for the line is then used to calculate theweights. The simplest model is a straight line, but more precisemodeling should be done if the situation requires it. (Inpractice, it is best to normalize the weight formula by dividingby the sum of all the reciprocal squares. This process assuresthat the root mean square error is correct.)6.2.2.3 In sum, two choices, which are independent of eachother, must be made in performing regression. These twochoices are a model and a fitting technique. In practice, theoptions for the model are typically a straight line or a quadratic,while the customary choices for the fitting technique areordinary least squares and weighted least squares.6.2.2.4 However, a straight line is not automatically associ-ated with OLS, nor is a quadratic automatically paired withWLS. The fitting technique depends solely on the behavior ofthe response standard deviations (that is, do they trend withconcentrations). The model choice is not related to thesestandard deviations, but depends primarily on whether the datapoints exhibit some type of curvature.6.2.3 Once an appropriate model and fitting technique havebeen chosen, the regression line and plot can be determined.One other very important feature can also be calculated andgraphed. That feature is the prediction interval, which is an“envelope” around the line itself and which reports theuncertainty (at the chosen confidence level) in a future mea-surement predicted from the line. An example is given in Fig.1. The solid red line is the regression line; the dashed red linesform the prediction interval.6.2.4 While the concept of a model is familiar to mostanalysts, the statistically sound process for selecting an ad-equate model typically is not.Aseries of regression diagnosticswill guide the user. The basic steps are as follows, and can becarried out with most statistical software packages that arecommercially available:NOTE 1—The interval in the above plot is nearly parallel to the regression line. This geometry will typically