ASTM D7783-21

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: D7783 − 21
Standard Practice for
Within-laboratory Quantitation Estimation (WQE)
This standard is issued under the fixed designation D7783; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope variation (that is, those sources that can reasonably be avoided
in routine sample measurements). Avoidable sources include,
1.1 This practice establishes a uniform standard for com-
but are not limited to, modifications to the sample, modifica-
puting the within-laboratory quantitation estimate associated
tions to the measurement procedure, modifications to the
with Z % relative standard deviation (referred to herein as
measurement equipment of the validated method, and gross
WQE ), and provides guidance concerning the appropriate
Z%
and easily discernible transcription errors (provided there is a
use and application.
way to detect and either correct or eliminate these errors in
1.2 WQE is computed to be the lowest concentration for
Z%
routine processing of samples).
which a single measurement from the laboratory will have an
estimated Z% relative standard deviation (Z% RSD, based on
1.4 The WQE applies to measurement methods for which
within-laboratory standard deviation), where Z is typically an
instrument calibration error is minor relative to other sources,
integer multiple of 10, such as 10, 20, or 30. Z can be less than
because this practice does not model or account for instrument
10 but not more than 30. The WQE is consistent with the
10 %
calibration error, as is true of most quantitation estimates in
quantitation approaches of Currie (1) and Oppenheimer, et al.
general. Therefore, the WQE procedure is appropriate when
(2).
the dominant source of variation is not instrument calibration,
but is perhaps one or more of the following:
1.3 The fundamental assumption of the WQE is that the
media tested, the concentrations tested, and the protocol
1.4.1 Sample Preparation, and especially when calibration
followed in developing the study data provide a representative
standards do not go through sample preparation.
and fair evaluation of the scope and applicability of the test
1.4.2 Differences in Analysts, and especially when analysts
method, as written. Properly applied, the WQE procedure
have little opportunity to affect instrument calibration results
ensures that the WQE value has the following properties:
(as is the case with automated calibration).
1.3.1 Routinely Achievable WQE Value—The laboratory
1.4.3 Differences in Instruments (measurement equipment),
shouldbeabletoattaintheWQEinroutineanalyses,usingthe
such as differences in manufacturer, model, hardware,
laboratory’s standard measurement system(s), at reasonable
electronics, sampling rate, chemical-processing rate, integra-
cost. This property is needed for a quantitation limit to be
tion time, software algorithms, internal signal processing and
feasible in practical situations. Representative data must be
thresholds, effective sample volume, and contamination level.
used in the calculation of the WQE.
1.3.2 Accounting for Routine Sources of Error—The WQE
1.5 Data Quality Objectives—For a given method, one
should realistically include sources of bias and variation that
typically would compute the WQE for the lowest RSD for
are common to the measurement process and the measured
which the data set produces a reliable estimate. Thus, if
materials.Thesesourcesinclude,butarenotlimitedtointrinsic
possible, WQE would be computed. If the data indicated
10%
instrument noise, some typical amount of carryover error,
that the method was too noisy, so that WQE could not be
10%
bottling, preservation, sample handling and storage, analysts,
computed reliably, one might have to compute instead
sample preparation, instruments, and matrix.
WQE , or possibly WQE . In any case, a WQE with a
20% 30%
1.3.3 Avoidable Sources of Error Excluded—The WQE
higherRSDlevel(suchasWQE )wouldnotbeconsidered,
50%
should realistically exclude avoidable sources of bias and
though a WQE with RSD < 10% (such as WQE ) could be
5%
acceptable. The appropriate level of RSD is based on the data
quality objective(s) for a particular use or uses. This practice
This practice is under the jurisdiction ofASTM Committee D19 on Water and
is the direct responsibility of Subcommittee D19.02 on Quality Systems,
allows for calculation of WQEs with user selected RSDs less
Specification, and Statistics.
than 30%.
Current edition approved Nov. 15, 2021. Published March 2022. Originally
approved in 2012. Last previous edition approved in 2013 as D7783–13. DOI:
1.6 This international standard was developed in accor-
10.1520/D7783-21.
2 dance with internationally recognized principles on standard-
Theboldfacenumbersinparenthesesrefertothelistofreferencesattheendof
this standard. ization established in the Decision on Principles for the
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D7783 − 21
Development of International Standards, Guides and Recom- (after optional outlier removal) from at least six independent
mendations issued by the World Trade Organization Technical measurements at a minimum of five concentrations.
Barriers to Trade (TBT) Committee.
4.2 Retaineddataareanalyzedtoidentifyandfitoneoffour
proposedstandard-deviationmodels: constant, linear(straight-
2. Referenced Documents
line), hybrid (proposed by Rocke and Lorenzato (3)), and
2.1 ASTM Standards:
exponential. These models describe the relationship between
D1129Terminology Relating to Water
the within-laboratory standard deviation of measurements and
D2777Practice for Determination of Precision and Bias of
the true concentration, T. The selection process involves
Applicable Test Methods of Committee D19 on Water
evaluating the models, starting with the linear model and
E2586Practice for Calculating and Using Basic Statistics
performing statistical tests to choose the simplest model that
adequately fits the data. Evaluation includes statistical signifi-
2.2 BIPM Documents:
cance testing and residual analysis, and requires the judgment
GUM: JCGM 100:2008Evaluation of measurement data—
of a qualified chemist.
Guide to the expression of uncertainty in measurement
4.3 Once the standard-deviation model has been selected, it
3. Terminology
determines the fitting technique for the model of measured
3.1 Definitions—For definitions of terms used in this
concentration versus true concentration, referred to in this
practice, refer to Terminology D1129, Practice E2586, and the practice as the mean-recovery model. If the standard deviation
GUM.
is constant, then ordinary least squares may be used. If the
standard deviation is not constant, the predicted standard
3.2 Definitions of Terms Specific to This Standard:
deviations are used to generate weights for use in weighted
3.2.1 censored measurement, n—a measurement that is not
least squares. Regardless of the fitting technique, the mean-
reported numerically, but is stated as a “nondetection” or a
recovery model fits a straight line to the data.
less-than (for example, “less than 0.1 ppb”).
4.4 The linear regression (measured versus true) is evalu-
3.2.2 quantitation limit (QL) or limit of quantitation (LQ),
ated for statistical significance, for lack of fit, and for residual
n—a numerical value, expressed in physical units or
patterns.
proportion, intended to represent the lowest level of
quantitation, based on a set of criteria for quantitation.
4.5 Thesetwomodels(standarddeviationandrecovery)are
3.2.2.1 Discussion—The WQE is an example of a QL.
then used to calculate the WQE values. Either a direct
3.2.3 Z % within-laboratory quantitation estimate calculation or iterative algorithm (depending on the model) is
(WQE ), n—(in accordance with Currie (1))—The lowest used to compute WQE , the lowest true concentration with
10%
Z%
estimated RSD = 10% (Z = 10); WQE (%RSD = 20 = Z);
concentration for which a single measurement from the exam-
20%
ined laboratory will have an estimated Z% relative standard and WQE (%RSD = 30 = Z). If needed for particular
30%
data-qualityobjectives(DQOs),WQE maybecomputedfor
deviation (Z% RSD, based on the within-laboratory standard
Z %
deviation). some Z<10.Theparticular Z%selectedforuseshoulddepend
upon the data-quality needs and the realized performance.
4. Summary of Practices
Typically, either 10% or 20% is used in environmental water
testing. The 30% RSD approaches the criterion for detection.
4.1 The WQE procedure provides an estimate of the true
Z values greater than 30 should not be used. An RSD of 5%
concentration at which a desired relative precision is achieved.
approximates a level at which at least one sure significant digit
Whether from analysis of routine quality samples or from
has been achieved.
studies undertaken from time to time (or both), the first step is
toacquiredatarepresentativeofthelaboratoryperformancefor
5. Significance and Use
use in the WQE calculations. Such data must include concen-
trations suitable for modeling the precision and bias over a
5.1 Appropriate application of this practice should result in
range of concentrations. Each datum for a method/matrix/
a WQE achievable by the laboratory in applying the tested
analyte should represent an independent sample where routine
method/matrix/analytecombinationtoroutinesampleanalysis.
sources of measurement variability occur at typical levels of
That is, a laboratory should be capable of measuring concen-
influence. Outlying individual measurements should be
trations greater than WQE , with the associated RSD equal
Z%
eliminated, using an accepted, scientifically-based procedure
to Z% or less.
for outlier identification and a documented, scientific basis for
5.2 The WQE values may be used to compare the quanti-
removal of data from the data set, such as found in Practice
tation capability of different methods for analysis of the same
D2777. WQE computations must be based on retained data
analyte in the same matrix within the same laboratory.
5.3 The WQE procedure should be used to establish the
within-laboratory quantitation capability for any application of
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
a method in the laboratory where quantitation is important to
Standards volume information, refer to the standard’s Document Summary page on
data use. The intent of the WQE is not to impose reporting
the ASTM website.
limits. The intent is to provide a reliable procedure for
Available from https://www.bipm.org/en/publications/guides/, accessed May
2021. establishing the quantitative characteristics of the method (as
D7783 − 21
implemented in the laboratory for the matrix and analyte) and of replicates at each concentration. The range chosen, exclud-
thus to provide the laboratory with reliable information char- ing the zero value for purposes of the discussion of range,
acterizing the uncertainty in any data produced. Then the should extend from below the estimated detection level to
laboratory can make informed decisions about censoring data abovetheWQEofinterest(forexample,10%,20%,or30%),
and has the information necessary for providing reliable to avoid the need for extrapolation.
estimates of uncertainty with reported data. 6.2.2.2 A single model (one of the four models in this
practice)shoulddescribethebehaviorofthestandarddeviation
6. Procedure in this range. The anticipated form of the relationship between
measurement standard deviation and true concentration, if
6.1 This procedure is described in stages as follows: Devel-
known, can help in choosing design spacing. Chemistry,
opment of Data, Data Screening, Modeling Standard
physics, empirical evidence, or informed judgment may make
Deviation, Fitting the Recovery Relationship, and Computing
one model more likely than others. Evaluation of interlabora-
the Quantitation Estimates.
tory method-validation studies may also provide information
6.2 Development of Data for Input to the Calculations—A
about these relationships. If a model of standard deviation is
single WQE calculation is performed per analyte, matrix/
likely to be one with curvature at lower concentrations (hybrid
medium and method. A minimum of five concentrations must
orexponential),thenasemi-geometricdesignisfavored.Ifthe
be used to allow for high-quality estimation of measured-
likely relationship is constant or linear, then equidistant spac-
versus-true concentration, and for modeling the relationship of
ing might be favored.
standard deviation to true concentration. At least six values at
6.2.2.3 Inclusion of additional concentrations, beyond the
each concentration are required to provide a high-quality
minimum of five concentrations, is strongly recommended
estimation of the standard-deviation and the recovery relation-
where knowledge of these relationships is unknown. Where
ships. Additional concentrations (especially additional
more than one order of magnitude is covered in the range
representative, independent samples at each concentration) are
selected (per range definition in 6.2.2.1), it is recommended
highly encouraged. Such inclusion will reduce the uncertainty
that four additional unique concentrations be added per addi-
in the estimate and better assure that after outlier removal, the
tional order of magnitude greater than one.
minimum requirements for concentrations and values will be
(1)Where ongoing quality-control (QC) information is
met. Data for each WQE calculation should come from only
available and it indicates that precision is good at the concen-
one laboratory and one method, and be for only one analyte in
tration of this quality control measure (for example, 5% RSD
one matrix/medium. Concentrations may be designed in
or less at higher concentrations), then establishing the maxi-
advance, or data already developed may be used.
mumconcentrationforthestudyatorbelowthatconcentration
6.2.1 Designing Concentrations—Where concentrations are
should be considered where the RSD criterion for the WQE is
being selected in advance of the collection of data, the
higher (for example, a WQE ).
10%
development of an optimized design should consider many
(2)Where ongoing QC demonstrates a high RSD (for
factors, including:
example,above30%),severalconcentrationsatandabovethe
6.2.1.1 Concentrations of available data, such as routine
concentration of the QC sample should be included.
quality-control samples.
NOTE 1—Where more than five concentrations are available, determi-
6.2.1.2 Potential use of the same data to calculate detection
nation of the WQE with and without the highest (and potentially the
limits and or other control limits.
lowest) concentration(s) included can provide insight into the effects of
6.2.1.3 The anticipated or previously determined WQE
thehighestconcentration(s)ontherecoveryrelationshipandthemodeling
(study range should exceed this value by at least a factor of 2).
of standard deviation. Calculation of the WQE values based on the most
appropriate and applicable concentrations, so long as minimums are met,
6.2.1.4 The potential need to eliminate the lowest concen-
is allowed.
tration(s) selected (see zero-concentration discussion above).
6.2.1.5 Where possible, select a WQE study design that has 6.2.2.4 The minimum of six independent values at each
concentration is required by this practice to provide a mini-
enoughdistinctconcentrationstoassessstatisticallackoffitof
the models (see Draper and Smith (4)). Recommended designs mally acceptable data set for calculation of standard deviation
are: (a) The semi-geometric design with five or more true at each concentration. Increasing the number of levels is
concentrations, T , T , and so forth, such as: 0, WQE /D , desirablewhereprojectconstraintsallow.Itisnotrequiredthat
1 2 0
WQE /D,WQE , D×WQE , D × WQE , where D is a the same number of replicates be used for each concentration;
0 0 0 0
number greater than or equal to 2 and WQE is an initial however, extreme differences (for example orders of magni-
estimate of the WQE, (b) equi-spaced design: 0, WQE /2, tude) should be avoided.
WQE , (3/2) × WQE,2×WQE , (5/2) × WQE . Other 6.2.2.5 Known, routine sources of measurement variability,
0 0 0 0
designs with at least five concentrations—provided the design
consistentwiththoseofroutineanalysisofsamples,musthave
includes blanks, one concentration that approximates 2 × been in action at the time of the generation of the data to be
WQE , and at least one nonzero concentration belowWQE —
used, if the WQE is to be used for characterizing routine
0 0
should be adequate. performance. That is, in order for the WQE to represent
6.2.2 Considerations for All Concentration Selections:
routinely achieved quantitation, the data used for WQE calcu-
6.2.2.1 The range of the data, the number of unique lation must be generated under routine analytical conditions.
concentrations, and the spacing of the concentrations are the Representative within-laboratory variation can only be seen if
primary decisions for study design, in addition to the number the number of qualified analysts and qualified measurement
D7783 − 21
systems in the laboratory are represented. The data used and However,formanymethods,itmaynotbepossibletoconduct
the more combinations included, the less effect any specific an unbiased sampling of the zero (blank) concentration
bias in these pairings should have on the WQE estimate. samples, since instruments and software systems routinely
Similarly,samplemanagement(forexample,holdingtime)and smoothelectronicinformation(rawdata)fromthedetectorand
allowed variations in routine sample-processing procedures
through software settings that censor reported data. Through
mustbeincluded.Thetimeperiodspannedmustallowroutine, these automated processes, many testing instruments return to
time-dependent sources of variation to affect the testing. This
the operator a result value of “zero,” when, if these processes
consideration should include factors such as the frequency of had been turned off, a non-zero numeric result (positive or
calibration of instruments, introduction of newly prepared or
negative) would have been produced. These “false-zero” val-
purchased standards, reagents and supplies, and sample-
ues adversely affect the use of the zero-concentration data in
holding times. Historically, the failure to utilize representative
statistics and should not be used for WQE studies. Most
data in determination of quantitation limits has been a primary
chromatography systems (and many other types of computer-
component in over-statements of quality through quantitation-
assisted instruments) have instrument set-points (such as (digi-
limit values and should be strictly avoided (that is, garbage in,
tal) bunch rate, slope sensitivity, and minimum area counts)
garbage out). Ideally, each measurement would be a double-
that are operator-controllable. For purposes of this study,
blind measurement made by a different analyst, using a
generating as much uncensored low-level data as practical is
different (qualified) measurement system on a different day.
important and the presence of these processes as well as the
Optimally,datatobeusedshouldbeeithercompletelyblind,or
setting of any operator-controllable setting should be evalu-
fromknownbutcompletelyroutine,integratedtesting(suchas
ated.
routine quality-control data). In any case, the goal is to
NOTE 2—Qualitative criteria used by the method to identify and
minimize special treatment of the WQE test samples.
discriminate among analytes are separate criteria, and must be satisfied
6.2.2.6 Where the WQE is meant to represent the best
according to the method.
possible performance, and not routine performance, then opti-
6.2.3.1 Once true-concentration-zero measurements have
mized conditions for data generation would be appropriate.
beengenerated,andpriortouse,itisimportanttoexamineand
Similarly, if the performance of only a single process, instru-
evaluate these data. A graph of measured concentration by
ment system, analyst, etc., is of interest, only the applicable
frequency of occurrence may be helpful. However, unless a
variables should be included. It is the responsibility of the user
fairly large sample size is represented (for example, n > 20),
of this practice to assure that the appropriate data are utilized
the distribution may be distorted by the random nature of
for the end use(s) ofWQE.Where the end use is unknown, the
samplingalone.Asageneralrule,iftherewerenobias,thenon
data generator who is using the WQE needs to disclose the
average and over a large sampling, a truly uncensored set of
specificattributesofthedatausedinthecalculation(aswellas
zero-concentration (blank) data would have a mean of zero
the RSD), and thus of the WQE.
with approximately half of the results being negative values
6.2.2.7 Where preexisting, routine-source data (for
andhalfpositive,andbeNormallydistributed.Ifsomepositive
example, quality-control data) are used, care must be taken to
or negative bias were present, the percentages would shift.
assure that: (1) each data point represents a true and indepen-
However, in general the frequency should be higher near the
dent sampling of the population (as well as of the sample
mean of the values and should decline as the concentrations
mediumbeingexamined,whereapplicable)and(2)allsample-
move away from the mean, with approximately half of the
processing steps and equipment (for example, bottles,
non-mean data above and half below the mean.
preservatives, holding, preparation, cleanup) are represented.
(1)Blank data are considered suspect if: (1) there is no
Also,“true”concentrationlevelsmusteitherbeknown(thatis,
variation in these data, (2) there are an inordinate number of
true“spiked”concentrationlevels),orknowable,afterthefact.
zerovalues(andnonegativevalues)relativetothefrequencies
Aconcentrationisconsidered knownifreferencestandardscan
ofpositivevalues(6.2.3above),(3)ifthereisahighfrequency
be purchased or constructed, and knowable if an accurate
of the lowest value in the data set (for example, where
determination can be made.
minimum-peak-area rejection has been used) relative to the
6.2.2.8 Transformation of other types of data (such as
frequency of higher concentration values, and few or no lower
laboratory replicates, which under-represent the variability as
values, or (4) a frequency graphic does not begin to approxi-
compared to independent samples and usually do not have
mate a bell curve (when there are 20 or more samples).
known true concentrations), using scientifically and statisti-
(2)If the distribution of the data is suspect, the literature,
cally sound approaches is not prohibited by this practice.
plus instrument-software and equipment manuals, should be
However, care must be taken and the validity of these trans-
consulted. These documents can provide an understanding of:
formations tested. It is also critical that any standards used to
(1) the theory of operation of the detection system, (2) the
prepare study samples be completely independent of the
signal processing, calibration, etc., and (3) other aspects of the
standards used to calibrate the instrument.
conversion of response to reported values. Judgment will be
6.2.2.9 Blank correction should not be performed, unless
needed to determine whether to use some or all of the
the method requires this correction to calculate result values.
true-concentration-zero (blank) data, or to exclude the data
6.2.3 True-Concentration Zero (Blank) Data Discussion— from the calculations. In general, if less than 10% of the
Where possible, it is preferable to include data from samples zero-concentration data are: (1) censored, (2) suspect, or (3)
with true concentration of zero (for example, blanks). false-zeros, then these “problem” data should be removed.
D7783 − 21
Only the remaining blank data are used in the WQE calcula- is required. See Carroll and Ruppert (6) for further discussion
tions; there must be at least six replicates. Where the zero of standard-deviation modeling. The models are as follows:
concentration is excluded or is not possible to obtain, it is
Model C Constant WLSD Model :□s 5 g1error (1)
~ !
importanttoincludeatrueconcentrationascloseaspossibleto
where:
zero in the study design.
s = standard deviation of measurement results, and
(3)Where 75% or less of the data are censored or
g = model parameter.
smoothed, and there are at least six remaining values, it is
reasonable to use statistical procedures to simulate the distri-
Under Model C, standard deviation does not change with
bution that is missing or smoothed. Software procedures are
concentration, resulting in a relative standard deviation that
commercially available. Additionally, procedures such as log-
declines with increasing concentration, T.
normal transformation may be used to accommodate data that
Model L ~Linear WLSD Model!:□s 5 g1h 3 T1error (2)
are not normally distributed. The decision about inclusion or
exclusion of zero-concentration data in aWQE data set should where:
weigh:(1)thenumberofotherconcentrationsavailable,(2)the
s = standard deviation of measurement results,
range of the other concentrations, and (3) the risk of extrapo-
T = true concentration, and
lation of the WQE outside the data-set concentration range
g and h = model parameters.
against the quality of the zero-concentration data.
Under Model L, standard deviation increases linearly with
6.2.3.2 True Concentrations Near Zero—As with concen-
concentration, resulting in an asymptotically constant relative
tration zero, true concentrations very near to zero may also
standard deviation as T increases.
have been censored, smoothed, and contain false-zeros. Ex-
2 2
Model H Hybrid WLSD Model :□s 5 =g 1 h 3 T 1error(3)
~ ! ~ !
amination of these very low concentrations, as above for zero
concentration, is important. The likelihood of occurrence and
where:
the percentage of data affected decreases with increasing
s = standard deviation of measurement results,
concentration.
T = true concentration, and
g and h = model parameters.
6.3 Data Screening, Outlier Identification, and Outlier Re-
moval: Under Model H, within-laboratory standard deviation in-
creases with concentration in such a way that the relative
6.3.1 Data that are to be the input to the WQE calculation
standard deviation declines as T increases, approaching an
should be screened for compliance with this practice’s
asymptote of h.
conditions, appropriateness for the intended use of the WQE,
obvious errors, and individual outliers. Graphing of the data
Model E Exponential WLSD Model :□s 5 g 3exp h 3 T 1error
~ ! ~ !
(true versus measured) is recommended as an assistive visual
(4)
tool.
where:
6.3.2 Outlying individual measurements must be evaluated;
s = standard deviation of measurement results,
if determined to be erroneous, they should be eliminated using
T = true concentration, and
scientifically-based reasoning. Identification of potential outli-
g and h = model parameters.
ers for data evaluation and validation may be accomplished
using statistical procedures or through visual examination of a Under Model E, within-laboratory standard deviation in-
graphical representation of the data. WQE computations must creasesexponentiallywithconcentration,resultinginarelative
be based on retained data from at least six independent standarddeviationthatmayinitiallydeclineas Tincreases,but
measurements at each of at least five concentration levels.The eventually increases with T, so that there is at most a bounded
data removed and the percentage of data removed must be quantitation range within which the RSD is less than Z ⁄100.
recorded and retained to document the WQE calculations. 6.4.1.1 The procedures for estimating the parameters of
each model and their uncertainties are given inAppendix X2 –
6.4 Modeling Standard Deviation Versus True
Appendix X5. Model L should be tried first, since its fitting
Concentration—Thepurposeistopredictthewithin-laboratory
procedure (in Appendix X2) provides criteria for choosing
measurement standard deviation (WLSD) as a function of true
among the various models.
concentration, σˆ(T).The relationship is used for two purposes:
6.4.1.2 In all cases, it is assumed that g > 0.Avalue of g <
(1)toprovideweights(ifneeded)forfittingthemean-recovery
0 has no practical interpretation and may indicate that a
model and (2) to provide the within-laboratory standard
different WLSD model should be used. Furthermore, it is
deviation estimates crucial to determining the WQEs.
assumed that g is not underestimated by censored data among
NOTE 3—See Caulcutt and Boddy (5) for more discussion of standard
measurementsofblanksorotherlow-concentrationsamples.If
deviation modeling and weighted least squares (WLS) in analytical
h < 0, it must not be statistically significant, and Model C
chemistry.
should be evaluated.
6.4.1 This practice uses four models as potential fits for the 6.4.2 If a model other than the best fit is chosen, the reason
Within-Laboratory Standard Deviation (WLSD) model. The for the choice should be scientific and should be recorded to
selection process considers a linear model first, performing document the WQE.
statistical tests to decide whether a simpler constant model can 6.4.2.1 It is recommended that the relationship of measure-
beusedorwhetheroneofthemorecomplicatedcurvedmodels ment standard deviation to true concentration be graphed and
D7783 − 21
used for visual verification of the appropriateness of each 7. Review, Documentation and Reporting
model and of the model selected for use.
7.1 The WQE analysis report should include: (1) the iden-
tification of laboratory and (2) identification of analytical
6.5 Fitting the Mean-Recovery Relationship (Measured ver-
method,analyte(s),matrix(ormatrices),sampleproperties(for
sus True Concentration)—Based on the standard-deviation
example,volumeormass)andspecificmethodoptions(ifany)
model selected (constant versus other models), the mean-
utilized. Where the laboratory uses standard operating proce-
recoveryconcentrationisfittedversustrueconcentration,using
dures(SOPs)toimplementmethodsormethodprotocols,these
ordinaryleastsquaresorweightedleastsquares,asappropriate.
SOPs should be referenced, including the identification of any
The mean recovery is evaluated for statistical significance and
revision/version.Documentationofeachdatumusedshouldbe
lack of fit.Agraph of mean recovery (along with the “calibra-
equivalent to that of reported data (for example, instrument,
tion” line) and a graph of the residuals should also be visually
analyst, date, etc.). There should be a description of all
examined. Many off-the-shelf statistical software packages
data-screening procedures employed, all results obtained, all
may be used for this purpose.
individualvaluesomittedfromfurtheranalysis(thatis,outliers
NOTE 4—Regression coefficients should not be used to assess goodness
that have been removed), all missing values, and the percent-
of fit.
ageofdatautilizedinthecalculationsrelativetotheinitialdata
set.Anyanomaliesencounteredshouldbelisted,includingand
6.5.1 The mean-recovery regression (measured versus true
anomalous calibration or quality control sample results (for
concentration) model is a straight line,
example, data validation qualifiers or flags). The data (statis-
Model R:□Y 5 a1bT1error (5)
tical) analysis should be included or referenced and the WQE
values determined recorded. The selected standard-deviation
The fitting procedure depends on the chosen standard-
model, plus the coefficient estimates for this model and for
deviationmodel.Iftheconstantmodel,ModelC,wasselected,
mean-recovery model, should also be recorded. Where a
then ordinary least squares (OLS) can be used to fit Model R
statistical model other than the mathematical best fit has been
for mean recovery. In all other cases, weighted least squares
chosen, the reasoning should be described.
(WLS) should be used. WLS approximately provides the
minimum-variance unbiased linear estimate of the parameters,
8. Report
a and b. The WLS procedure is described in Appendix X6.
8.1 The analysis report should at a minimum contain:
6.6 Compute the WQE for each Z (%RSD)—Using the
8.1.1 Identification of laboratory,
mean-recovery regression line determined above, the most
8.1.2 Analytical method,
appropriate model of the relationship of relative standard
8.1.3 Analyte(s),
deviation to true concentration (also determined above), and
8.1.4 Matrix (or matrices),
the Z value desired, the user obtains the WQE, which is an
8.1.5 Sample properties (for example, volume),
estimate of the lowest true concentration (corresponding to the
8.1.6 Study design,
measuredconcentration)atwhichthedesiredRSDisachieved.
Procedures for calculating theWQE and estimating its relative 8.1.7 Analyst, method, and date of testing for each study
sample,
standarduncertaintyaregiveninAppendixX2–AppendixX5.
8.1.8 Any anomalies in the study, including QA/QC sample
6.6.1 Given the standard-deviation model, its estimated
results,
parameters, and the mean-recovery regression line, there is a
8.1.9 Data-screening results, individual values and labora-
lower limit, Z , below which WQE cannot be calculated
lim Z%
tories omitted from further analysis, and missing values,
because there is no true concentration at which RSD equals
8.1.10 WLSD model selected, and
Z%.For Z> Z ,theWQEcanbecalculatedbutitstruevalue
lim
may nevertheless be extremely uncertain, especially when Z is 8.1.11 CoefficientestimatesfortheWLSDmodelandmean-
recovery model.
near Z . If the relative standard uncertainty, u (WQE),
lim rel
exceeds 25%, the calculated WQE should be considered an
8.2 The report should be given a second-party review to
unreliable estimate. The actual reliability of the WQE also
verify that:
depends on the adequacy of the standard-deviation model,
8.2.1 The data transcription and reporting have been per-
which is more difficult to quantify.
formed correctly,
8.2.2 The analysis of the data and the application of this
NOTE 5—Under Model C, Z is zero. For the other models described
lim
here, Z is always positive. standard have been performed correctly, and
lim
8.2.3 The results of the analysis have been used
6.6.2 The measured concentration (Y ) at which the desired
Q
appropriately, including assessment of assumptions necessary
RSDwasachievedmayalsobeofinterestforsomeuses.This
to compute a WQE.
value is the level at which the required RSD was obtained in
measured concentration units (that is, the value, paired with a
NOTE 6—Reviewer(s) should be qualified in one or more of the
following areas: (1) applied statistics, (2) metrology, and (3) analytical
WQE, that has not been corrected for bias through the
chemistry.
mean-recovery regression). Where the Y and the WQE are
Q
equal (following application of significant figures and
8.3 A statement of the review and the results of the review
rounding), there is no apparent bias at theWQE concentration. should accompany the report.
D7783 − 21
FIG. 1 Sample Standard Deviations Versus True Concentration, with Linear Fit, Hybrid Model Fit, and Residuals from Linear Fit (Lower
Plot), All in ppb
9. Rationale relative error of any measurement of a true concentration
greater than theWQE will not exceed 63 × Z%. For example,
9.1 The basic rationale for the WQE is contained in Currie
a measurement above the WQE (and assumed to have true
10%
(1). The WQE is a performance characteristic of an analytical
concentration above the WQE) could be reported as 6 ppb
measurement process, to paraphrase Currie. Like an estimated
(630%) = (6 6 2) ppb, with a high degree of certainty.
detection limit, theWQE is helpful for the planning and use of
chemical analyses. The WQE is another benchmark indicating
9.3 There are several real-world complications to this ide-
whether the method can adequately meet measurement needs.
alized situation. See Maddalone et al. (7), Gibbons (8), and
9.2 The idealized definition of WQE is that it is the Coleman et al. (9). Some of these complications are listed as
Z %
lowest concentration, L , that satisfies: L = (100 / Z) σ follows:
Q Q L
Q
(where σ istheactualstandarddeviationofwithin-laboratory
L 9.3.1 Analyte recovery is not perfect; the relationship be-
Q
measurements at concentration L ); this definition is equiva-
Q
tween measured values of concentrations and true concentra-
lent to satisfying RSD = σ ⁄ L = Z %. In other words,
L
Q
Q tions cannot be assumed to be trivial. There is bias between
WQE is the lowest concentration with Z% RSD (assuming
Z %
true and measured values. Recovery can and should be
suchaconcentrationexists).If,asiscommonlythecase, RSD
modeled. Usually, a straight line will suffice. In practice, when
declines with increasing true concentration, then the relative
both the standard deviation and recovery models are used,
uncertaintyofanymeasurementofatrueconcentrationgreater
WQEZ% is calculated to be the lowest concentration L that
Q
than the WQE will not exceed 6Z%. The range, 63σ ,isan
L
Q
satisfies L 5 100 ⁄ Z σˆ L ⁄ b.
~ ! ~ !
Q Q
approximate prediction or confidence interval very likely to
9.3.2 Variation is introduced by different laboratories,
contain the measurement, which is assumed to be normally
distributed. This assertion is based on critical values from the analysts, models, and pieces of equipment; environmental
factors; flexibility/ambiguity in a test method; contamination;
normal distribution (or from the student’s t-distribution if σ is
estimated rather than known). Then, with high confidence, the carryover; matrix influence; and other factors. It is intractable
D7783 − 21
to model these factors individually, but their collective contri- where the standard deviation used in the 10σ formula is
butions to measurement WLSD can be observed, if these estimated at a detection critical value, and then is taken to be
contributions are part of how a study is designed and con- a constant (over a trace-level range of concentrations) for the
ducted. 10σ computation. In contrast, WQE follows the “more
10%
statistically and conceptually rigorous” approach described by
9.3.3 The standard deviation of measurements is generally
unknown, and may change with true concentration, possibly Gibbons et al. (8), and contained in Currie (1). This greater
rigor comes at the risk of: (a) possibly being unattainable for
because of the physical principle of the test method.To ensure
that a particular RSD is attained at or above the WQE, there somemethods(forwhichonlyalessstrictlevelofRSDcanbe
ensured); (b) having uncertainty that is potentially complex,
must be a way to predict the WLSD at different true concen-
trations. Short of severely restricting the range of concentra- and depends both on the model used and on the data.
tions for a study, prediction is accomplished by an empirical
10. Keywords
WLSD model. In all of the respects discussed in 9.1 – 9.3,
WQE is similar to the AML developed by Gibbons et al. 10.1 critical limits; matrix effects; precision; quantitation;
10%
(10). However, the AML follows an approximate approach, quantitation limits
APPENDIXES
(Nonmandatory Information)
X1. GLOSSARY OF KEY SYMBOLS, ACRONYMS, AND LABELS
∆g—one iteration’s change in the estimate of parameter g, g—model parameter representing the intercept of the
the intercept parameter in the Hybrid model, using nonlinear prediction curve, equal to the predicted measurement standard
least squares deviation at zero concentration
∆h—one iteration’s change in the estimate of parameter h, g —initial estimate of g
the slope parameter in the Hybrid model, using nonlinear least g'—ln g, the natural logarithm of g
squares h—model parameter accounting for the dependence of the
∆m—bias-correction term for the natural logarithm of (the measurement standard deviation on concentration (Models L,
numerical value of) the sample standard deviation of m H, and E)
normally distributed results h —initial estimate of h
σ—true within-laboratory standard deviation j—index used to number iteration steps when using New-
σˆ—predicted value of the within-laboratory measurement ton’s Method, for example when estimating the parameters of
standard deviation the Hybrid Model (Model H)
σˆ(T)—predicted value of the within-laboratory measure- k—index used for different concentrations, T , and associ-
k
ment standard deviation at concentration T ated statistics
ζ —variance of the logarithm of (the numerical value of) LQ—another designation for theWQE, in accordance with
m
thesamplestandarddeviationof mnormallydistributedresults Currie’s notation
ζ—quotient of an estimated value and its standard uncer- Model C—Constant model for ILSD
tainty Model L—Linear model for ILSD; within-laboratory stan-
a—estimateoftheslopeinthemean-recoverycurve(linear dard deviation increases linearly with concentration
model) Model H—Hybrid model for ILSD; combines additive and
'
a —bias-correction factor for the sample standard devia- multiplicative error, with within-laboratory standard deviation
m
tion of a random sample of m normally distributed results that increases with increasing concentration, according to the
AML—Alternative Minimum Level, a quantitation limit model proposed by Rocke and Lorenzato
that is similar to the WQE (and compatible in approach) Model E—Exponential model for ILSD; within-laboratory
b—estimateoftheslopeinthemean-recoverycurve(linear standard deviation increases exponentially with concentration,
model) ensuring at most a finite quantitation range
c–intermediate variable used in estimating g and h for the Model R—the linear model for the mean-recovery curve
Hybrid model by nonlinear least squares; similar to d, p, q, u, n—number of true concentrations in the study (n ≥ 5)
and v NLLS—nonlinear least squares, where coefficients in a
d—similar to c nonlinearmodelarecomputedtominimizethe(weighted)sum
e—Euler’s number, 2.71828… of the squares of the residuals (that is, the differences between
e —the residual associated with T from a NLLS precision the observed and predicted values)
k k
model fit; defined as the difference in log sample standard OLS—ordinaryleastsquares,afittingtechniqueforalinear
deviation and log predicted standard deviation (that is, additive) model that minimizes the sum of the squares
f(T)—the natural log of the current estimate of the within- of the residuals (that is, the differences between observed and
laboratory standard deviation at concentration T predicted values)
D7783 − 21
p—similar to c u(x, y)—estimated measurement covariance of x and y
q—similar to c u (x)—relativestandarduncertaintyof x,equalto u(x)⁄|x|
rel
th 2
th
q —k residual for the formal regression of T versus T
w —statistical weighting factor used for the k data point
k
k
QL—quantitation limit (also called practical quantitation
when estimating model parameters by weighted linear or
limit, PQL); see L
Q nonlinear least squares
RSD—relative standard deviation, that is, the standard
WLS—weighted least squares, a modified form of ordinary
deviation divided by the concentration (both generally esti-
least squares that incorporates nonuniform variability in the
mated)
data
s —observed value of the within-laboratory standard de-
k
WLSD—within-laboratory standard deviation
viation at concentration T
k
WQE —within-laboratory quantitation estimate associ-
Z %
*
s —observed value of the standard deviation at concentra-
k
ated with approximately Z % RSD
tion T , corrected for bias
k
Y—random variable representing a measurement result
s —maximumsampleILSD:equaltomax(s ,s ,.,s )
max 1 2 n
Z—numerical value of RSD expressed as a percentage
s —pooled standard deviation estimate
p
Z —the estimated lowest limit of %RSD achievable,
lim
SS —residual sum of squares
e
based on study results, for a particular measurement system,
T—true concentration
th matrix, and analyte
T —k value of true concentra
...