Trace Environmental Quantitative Analysis: Principles, Techniques, and Applications - Chapter 2 potx

TABLE 2.1Advantages and Limitations of the Various Modes of Instrument Calibration Used in TEQA Calibration Area% No standards needed; provides for a preliminary evaluation of sample

Analytical Data, Detection Limits, and Quality

Assurance/Quality Control

If you can measure that of which you speak, and can express it by a number, you know something of your subject, but if you cannot measure it, your knowledge is meager and unsatisfactory.

—Lord Kelvin

CHAPTER AT A GLANCE

Good laboratory practice 38

Error in laboratory measurement 41

Instrument calibration and quantification 45

Linear least squares regression 58

Uncertainty in interpolated linear least squares regression 64

Instrument detection limits 68

Limit of quantitation 81

Quality control 85

Linear vs nonlinear least squares regression 91

Electronic interfaces between instruments and PCs 104

Sampling considerations 112

References 117

Chromatographic and spectroscopic analytical instrumentation are the key determi-native tools to quantitate the presence of chemical contaminants in biological fluids and in the environment These instruments generate electrical signals that are related

to the amount or concentration of an analyte of environmental or environmental health significance This analyte is likely to be found in a sample matrix taken from the environment, or from body fluids Typical sample matrices drawn from the environment include groundwater, surface water, air, soil, wastewater, sediment, sludge, and so forth Computer technology has merely aided the conversion of an

analog signal from the transducer to the digital domain It is the relationship betweenthe analog or digital output from the instrument and the amount or concentration of

a chemical species that is discussed in this chapter The process by which an electricalsignal is transformed to an amount or concentration is called instrument calibration.Chemical analysis based on measuring the mass or volume obtained from chemicalreactions is stoichiometric Gravimetric (where the analyte of interest is weighed)and volumetric (where the analyte of interest is titrated) techniques are methods thatare stoichiometric Such methods do not require calibration Most instrumentaldeterminative methods are nonstoichiometric and thus require instrument calibration.This chapter introduces the most important aspect of TEQA for the reader Afterthe basics of what constitutes good laboratory practice are discussed, the concept

of instrumental calibration is introduced and the mathematics used to establish suchcalibrations are developed The uncertainty present in the interpolation of the cali-bration is then introduced A comparison is made between the more conventionalapproach to determining instrument detection limits and the more contemporaryapproaches that have recently been discussed in the literature.1–6 These more con-temporary approaches use least squares regression and incorporate relevant elementsfrom statistics.7 Quality assurance/quality control principles are then introduced Acontemporary statistical approach toward evaluating the degree of detector linearity

is then considered The principles that enable a detector’s analog signal to bedigitized via analog-to-digital converters are introduced Principles of environmentalsampling are then introduced Readers can compare QA/QC practices from twoenvironmental testing laboratories Every employer wants to hire an analyst whoknows of and practices good laboratory behavior

Good laboratory practice (GLP) requires that a quality control (QC) protocol fortrace environmental analysis be put in place A good laboratory QC protocol for anylaboratory attempting to achieve precise and accurate TEQA requires the followingconsiderations:

addition mode of instrument calibration is most appropriate for theintended quantitative analysis application

• Establishing a calibration curve that relates instrument response to analyteamount or concentration by preparing reference standards and measuringtheir respective instrument responses

• Performing a least squares regression analysis on the experimental bration data to evaluate instrument linearity over a range of concentrations

cali-of interest and to establish the best relationship between response andconcentration

• Computing the statistical parameters that assist in specifying the tainty of the least squares fit to the experimental data points

uncer-• Running one or more reference standards in at least triplicate as initialcalibration verification (ICV) standards throughout the calibration range

ICVs should be prepared so that their concentrations fall to within themid-calibration range.

• Computing the statistical parameters for the ICV that assist in specifyingthe precision and accuracy of the least squares fit to the experimental datapoints

• Determining the instrument detection limits (IDLs)

estab-lishing the percent recovery for a given analyte in both a clean matrix andthe sample matrix With some techniques, such as static headspace gaschromatography (GC), the MDL cannot be determined independentlyfrom the instrument’s IDL

every 5 or 10 samples This QC standard serves to monitor instrumentprecision and accuracy during a batch run This assumes that both cali-bration and ICV criteria have been met A mean value for the QC referencestandard should be obtained over all QC standards run in the batch The

standard deviation, s, and the relative standard deviation (RSD) should be

calculated

• Preparing the running QC surrogates, matrix spikes, and, in some cases,matrix spike duplicates per batch of samples A batch is defined in EPAmethods to be approximately 20 samples These reference standard spikesserve to assess extraction efficiency where applicable Matrix spikes andduplicates are often required in EPA methods

• Preparing and running laboratory blanks, laboratory control samples, andfield and trip blanks These blanks serve to assess whether samples mayhave become contaminated during sampling and sample transport

It has been stated many times by experienced analysts that in order to achieveGLP, close to one QC sample must be prepared and analyzed for nearly each andevery real-world environmental sample

AND STATISTICAL TREATMENT BE SUMMARIZED

BEFORE WE PLUNGE INTO CALIBRATION?

International Union of Pure and Applied Chemistry (IUPAC) recommendations, asdiscussed by Currie,1 is this author’s attempt to do just that The true amount that

is present in the unknown sample can be expressed as an amount such as a #nganalyte, or as a concentration [#µg analyte/kg of sample (weight/weight) or #µganalyte/L of sample (weight/volume)] The amount or concentration of true unknownsented by τ is shown in Figure 2.1 being transformed to an electrical signal y accomplished The signal y, once obtained, is then converted to the reported estimate

Yes, indeed Figure 2.1, adapted and modified, while drawing on recently published

present in either an environmental sample or human/animal specimen and

repre-Chapters 3 and 4 describe how the six steps from sampling to transducer are

x0, as shown in Figure 2.1 This chapter describes how the eight steps from calibration

to statistical evaluation are accomplished The ultimate goal of TEQA is then

real-ized, i.e., a reported estimate x0 with a calculated uncertainty using statistics in themeasurement expressed as ±u We can assume that the transduced signal varies

linearly with x, where x is the known analyte amount or concentration of a standard

reference This analyte in the standard reference must be chemically identical to theanalyte in the unknown sample represented by its true value τ x is assumed to beknown with certainty since it can be traced to accurately known certified referencestandards, such as that obtained from the National Institute of Standards and Tech-nology (NIST) We can realize that

where

y0 = the y intercept, the magnitude of the signal in the absence of analyte.

m = slope of the best-fit regression line (what we mean by regression will

be taken up shortly) through the experimental data points The slope also defines the sensitivity of the specific determinative technique

e y = the error associated with the variation in the transduced signal for a

given value of x We assume that x itself (the amount or concentration

of the analyte of interest) is free of error This assumption is used throughout the mathematical treatment in this chapter and serves to simplify the mathematics introduced

FIGURE 2.1 The process of trace environmental quantitative analysis (Adapted from

L Currie, Pure and Applied Chemistry, 67, 1699–1723, 1995.)

xo ± u

Sampling Sample preservation and storage Extraction Cleanup Injection Transducer

Calibration Quantification Verification Measure IDLs Calculate MDLs Conduct QA/QC Interpretation Statistical evaluation y

Signal (y) from transducer

that corresponds to τ; signal may or may not include background interferences;

requires quality analytical instrumentation, efficient sample preparation and competent analytical scientists and technicians

Reported estimate of

amount or concentration

of unknown targeted analyte (x0) with calculated uncertainty (±u) in the

measurement; the ultimate goal and limitation of TEQA

need to know, the

need for TEQA!

τ

y = y0 + m x+ ey

y=y0+mx+e y

the amount or concentration at a trace level, represented by x0, with an uncertainty

u such that x0 could range from a low of x0 – u to a high of x0 + u Let us focus abit more on the concept of error in measurement

Let us digress a bit and discuss measurement error Each and every measurementincludes error The length and width of a page from this book cannot be measuredwithout error There is a true length of this page, yet at best we can only estimateits length We can measure length only to within the accuracy and precision of ourmeasuring device, in this case, a ruler or straightedge We could increase our pre-cision and accuracy for measuring the length of this page if we used a digital caliper

Currie has defined x0 as the statistical estimate derived from a set of observations

The error in x0 represented by e is shown to consist of two parts, systematic or

bias error represented by ∆ and random error represented by δ such that:8

∆ is defined as the absolute difference between a population mean represented

by µ (assuming a Gaussian or normal distribution) and the true value τ δ is defined

as the absolute difference between the estimated analytical result for the unknown sample x0 and the population mean µ δ can also be viewed in terms of a multiple

z of the population standard deviation σ, σ being calculated from a Gaussian or

normal distribution of x values from a population.

A RE U SED ?

Yes, indeed Bias, ∆, reflects systematic error in a measurement Systematic error

may be instrumental, operational, or personal.

Instrumental errors arise from a variety of sources such as:9

• Faulty calibration of scales

• Deterioration of electrical, electronic, or mechanical parts due to age orlocation in a harsh environment

Errors in this category are often the easiest to detect They may present achallenge in attempting to locate them Use of a certified reference standard might

help to reveal just how large the degree of inaccuracy as expressed by a percent relative error really is The percent relative error (%error), i.e., the absolute differ- ence between the mean or average of a small set of replicate analyses, xave, and thetrue or accepted value, τ divided by τ and multiplied by 100, is mathematicallystated (and used throughout this book) as follows:

It is common to see the expression “the manufacturer states that its instrument’saccuracy is better than 2% relative error.” The analyst should work in the laboratorywith a good idea as to what the percent relative error might be in each and everymeasurement that he or she must make It is often difficult if not impossible to knowthe true value This is where certified reference standards such as those provided bythe NIST are valuable High precision may or may not mean acceptable accuracy.Operational errors are due to departures from correct procedures or methods.These errors often are time dependent One example is that of drift in readings from

an instrument before the instrument has had time to stabilize A dependence ofinstrument response on temperature can be eliminated by waiting until thermalequilibrium has been reached Another example is the failure to set scales to zero

or some other reference point prior to making measurements Interferences can causeeither positive or negative deviations One example is the deviation from Beer’s law

at higher concentrations of the analyte being measured However, in trace analysis,

we are generally confronted with analyte concentration levels that tend toward theopposite direction

Personal errors result from bad habits and erroneous reading and recording ofdata Parallax error in reading the height of a liquid in a buret from titrimetic analysis

is a classic case in point One way to uncover personal bias is to have someone elserepeat the operation Occasional random errors by both persons are to be expected,but a discrepancy between observations by two persons indicates bias on the part

zero within a set of standard masses What if an analyst, who desires to prepare asolution of a reference standard to the highest degree of accuracy possible, dissolveswhat he thinks is 100 mg of standard reference (the solute), but really is only 89 mg,

in a suitable solvent using a graduated cylinder and then adjusts the height of thesolution to the 10-mL mark? Laboratory practice would suggest that this analyst use

a 10-mL volumetric flask Use of a volumetric flask would yield a more accuratemeasurement of solution volume Perhaps 10 mL turns out to be really 9.6 mL when

a graduated cylinder is used We now have inaccuracy, i.e., bias, in both mass and

in volume Bias has direction, i.e., the true mass is always lower or higher Bias is

usually never lower for one measurement and then higher for the next measurement.The mass of solute dissolved in a given volume of solvent yields a solution whoseconcentration is found from dividing the mass by the total volume of solution Thepercent relative error in the measurement of mass and the percent relative error inthe measurement of volume propagate to yield a combined error in the reportedconcentration that can be much more significant than each alone Here is where thecliché “the whole is greater than the sum of its parts” has some meaning

Random error, δ, occurs among replicate measurement without direction If wewere to weigh 100 mg of some chemical substance, such as a reference standard,

on the most precise analytical balance available and repeat the weighing of the samemass additional times while remembering to rezero the balance after each weighing,

we might get data such as that shown below:

Notice that the third replicate weighing yields a value that is less than the second.Had the values kept increasing through all five measurements, systematic error orbias might be evident

Another example for the systematic vs random error “defective,” this time usinganalytical instrumentation, is to make repetitive 1-µL injections of a referencestandard solution into a gas chromatograph (GC) A GC with an atomic emissiondetector (GC-AED) was used by this author to evaluate whether systematic errorwas evident for triplicate injection of a 20 ppm reference standard containing tetra-

chloro-m-xylene (TCMX) and decachlorobiphenyl (DCBP) dissolved in the solvent

iso-octane Both analytes are used as surrogates in EPA organochlorine cide/polychlorinated biphenyl (PCB)-related methods such as EPA Methods 608 and

pesti-8080 The atomic emission from microwave-induced plasma excitation of chlorineatoms, monitored at a wavelength of 837.6 nm, formed the basis for the transduced

Replicate No Weight (mg)

electrical signal Both analytes are separated chromatographically (refer to Chapter

4 for an introduction to the principles underlying chromatographic separations) and

appear in a chromatogram as distinct peaks, each with an instrument response The emitted intensity is displayed graphically in terms of a peak whose area beneath the curve is given in units of counts-seconds These data are shown below:

The drop between the first and second injections in the peak area along with therise between the second and third injections suggests that systematic error has beenlargely eliminated A few days before these data were generated a similar set oftriplicate injections was made using a somewhat more diluted solution containingTCMX and DCBP into the same GC-AED The following data were obtained:

The rise between the first and second injections in peak area followed by thedrop between the second and third injections suggests again that systematic errorhas been largely eliminated One of the classic examples of systematic error, andone that is most relevant to TEQA, is to compare the bias and percent relativestandard deviations in the peak area for five identical injections using a liquid-handling autosampler against a manual injection into a graphite furnace atomicabsorption spectrophotometer using a common 10-µL glass liquid-handling syringe

It is almost impossible for even the most skilled analyst around to achieve the degree

of reproducibility afforded by most automated sample delivery devices

Good laboratory practice suggests that it should behoove the analyst to eliminateany bias, ∆, so that the population mean equals the true value Mathematically stated:

∆ = 0 = µ − τ

∴ µ = τ

errors Mathematically stated:

TCMX (counts-seconds)

DCBP (counts-seconds)

1st injection 48.52 53.65 2nd injection 47.48 52.27 3rd injection 48.84 54.46

TCMX (counts-seconds)

DCBP (counts-seconds)

1st injection 37.83 41.62 2nd injection 38.46 42.09 3rd injection 37.67 40.70

δ= x0−µ

Random error alone becomes responsible for the absolute difference between

the reported estimate x0 and the statistically obtained population mean Randomproceed in this chapter to take a more detailed look at those factors that transform

AND VERIFICATION?

It is very important and the most important task for the analyst who is responsible

for operation and maintenance of analytical instrumentation Calibration is followed

by a verification process in which specifications can be established and the analystcan evaluate whether the calibration is verified or refuted A calibration that hasbeen verified can be used in acquiring data from samples for quantitative analysis

A calibration that has been refuted must be repeated until verification is achieved,e.g., if, after establishing a multipoint calibration for benzene via a gas chromato-graphic determinative method, an analyst then measures the concentration of benzene

in a certified reference standard The analyst expects no greater than a 5% relativeerror and discovers to his surprise a 200% relative error In this case, the analystmust reconstruct the calibration and measure the certified reference standard again.Close attention must be paid to those sources of systematic error in the laboratorythat would cause the relative error to greatly exceed the minimally acceptable relativeerror criteria previously developed for this method

An analyst who expects to implement TEQA and begins to use any one of thevarious chromatography data acquisition and processing software packages available

in the marketplace today is immediately confronted with several calibration modes

available Most software packages will contain most of the modes of instrumental

tages as well as the overall limitations are given Area percent and normalizationpercent (norm%) are not suitable for quantitative analysis at the trace concentrationlevel This is due to the fact that a concentration of 10,000 ppm is only 1% (partsper hundred), so that a 10 ppb concentration level of, for example, benzene, indrinking water is only 0.000001% benzene in water Weight% and mole% are subsets

of norm% and require response factors for each analyte in units or peak area or peakwith its corresponding quantification equation Quantification follows calibrationand thus achieves the ultimate goal of TEQA, i.e., to perform a quantitative analysis

of a sample of environmental or environmental health interest in order to determinethe concentration of each targeted chemical analyte of interest at a trace concentra-

tion level Table 2.1 and Table 2.2 are useful as reference guides.

We now proceed to focus on the most suitable calibration modes for TEQA

Referring again to Table 2.1, these calibration modes include external standard (ES), internal standard (IS), to include its more specialized isotope dilution mass spectrom- etry (IDMS) calibration mode, and standard addition (SA) Each mode will be dis-

cussed in sufficient detail to enable the reader to acquire a fundamental understanding

calibration that appear in Table 2.1 For each calibration mode, the general

advan-height per gram or per mole, respectively Table 2.2 relates each calibration modeerror can never be completely eliminated Referring again to Figure 2.1, let us

y to x We focus on those factors that transform to y in Chapters 3 and 4

TABLE 2.1

Advantages and Limitations of the Various Modes of Instrument Calibration Used in TEQA

Calibration

Area% No standards needed; provides for a

preliminary evaluation of sample

composition; injection volume precision

not critical

Need a nearly equal instrument response for all analytes so peak heights/areas all uniform; all peaks must be included in

calculation; not suitable for TEQA

Norm% Injection volume precision not critical;

accounts for all instrument responses for

all peaks

All peaks must be included; calibration standards required; all peaks must be

calibrated; not suitable for TEQA

ES Addresses wide variation in GC detector

response; more accurate than area%,

norm%; not all peaks in a chromatogram

of a given sample need to be quantitated;

compensates for recovery losses if

standards are taken through sample prep

in addition to samples; does not have to

add any standard to the sample extract for

calibration purposes; ideally suited to

TEQA

Injection volume precision is critical; instrument reproducibility over time is critical; no means to compensate for

a change in detector sensitivity during a batch run; needs a uniform matrix whereby standards and samples should have similar matrices

IS Injection volume precision not critical;

instrument reproducibility over time not

critical; compensates any variation in

detector sensitivity during a batch run;

ideally suited to TEQA

Need to identify a suitable analyte to serve

as an IS; bias is introduced if the IS is not added to the sample very carefully; does not compensate for percent recovery losses during sample preparation since IS

is usually added after both extraction and cleanup are performed

IDMS Same as for IS; injection volume precision

not critical; instrument reproducibility

over time not critical; compensates for

analyte percent recovery losses during

sample preparation since isotopes are

added prior to extraction and cleanup;

eliminates variations in analyte vs

internal standard recoveries; ideally

suited to TEQA

Need to obtain a suitable isotopically labeled analog of each target analyte; isotopically labeled analogs are very expensive; bias is introduced if the labeled isotope is not added to the sample very carefully; needs a mass spectrometer

to implement; mass spectrometers are expensive in comparison to element- selective GC detectors or non-MS LC detectors

SA Useful when matrix interference cannot be

eliminated; applicable where analyte-free

matrix cannot be obtained; commonly

used to measure trace metals in “dirty”

environmental samples

Need two aliquots of same sample to make one measurement; too tedious and time consuming for multiorganics quantitative analysis

Source: Modified and adapted from Agilent Technologies GC-AED Theory and Practice, Training Course

from Diablo Analytical, Inc., 2001.

— concentration of analyte i in the unknown sample (the ultimate goal of TEQA)

— area of ith peak in unknown sample;

N — total number of peaks in chromatogram

— concentration of analyte i after analyte i (standard) is added to the unknown sample

— response of unknown analyte and blank, both associated with unknown sample

— response of unknown analyte and blank, in spiked or standard added known sample

C unk i A unk i

A i

i N

i i

A unk i i

i N

i

IS i IS i

unk

spike i m

m unk i unk i

SA i unk i

bl spike i

of the similarities and differences among all three Correct execution of calibration

on the part of a given analyst on a given instrument is a major factor in achieving GLP

The ES mode uses an external reference source for the analyte whose concentration

in an unknown sample is sought A series of working calibration standards areprepared that encompass the entire range of concentrations anticipated for theunknown samples and may include one or more orders of magnitude For example,let us assume that a concentration of 75 ppb of a trihalomethane (THM) is anticipated

in chlorinated drinking water samples A series of working calibration standardsshould be prepared whose concentration levels start from a minimum of 5 ppb to amaximum of 500 ppb each THM The range for this calibration covers two orders

of magnitude Six standards that are prepared at 5, 25, 50, 100, 250, and 500 ppbfor each THM, respectively, would be appropriate in this case Since these standardswill not be added to any samples, they are considered external to the samples, hencedefining this mode as ES The calibration curve is established by plotting the analyteresponse against the concentration of analyte for each THM

The external standard is appropriate when there is little to no matrix effectbetween standards and samples To illustrate this elimination of a matrix effect,consider the situation whereby an aqueous sample is extracted using a nonpolarsolvent The reference standard used to construct the ES calibration is usuallydissolved in an organic solvent such as methanol, hexane, or iso-octane The analytes

of interest are now also in a similar organic solvent ES is also appropriate whenthe instrument is stable and the volume of injection of a liquid sample such as anextract can be reproduced with good precision A single or multipoint calibrationcurve is usually established when using this mode

For a single-point calibration, the concept of a response factor, R F, becomesimportant The use of response factors is valid provided that it can be demonstrated

that the calibration curve is, in fact, a straight line If so, the use of R F values serves

to greatly simplify the process of calibration R F is fixed and is independent of the

concentration for its analyte for a truly linear calibration A response factor for the

ith analyte would be designated as R F

i

For example, if 12 analytes are to be calibrated

and we are discussing the seventh analyte in this series, i would then equal 7 The magnitude of R F

i

does indeed depend on the chemical nature of the analyte and on

the sensitivity of the particular instrument The definition of R F for ES is given asfollows (using notation from differential calculus):

(2.1)

A response factor for each analyte (i.e., the ith analyte) is obtained during the

calibration and is found by finding the limit of the ratio of the incremental change

in peak area for the ith analyte, ∆A S

S

i

S i

A

0

the ith analyte in the reference standard, ∆C S

i, as ∆CS

i

approaches zero Quantitativeanalysis is then carried out by relating the instrument response to the analyte con-centration in an unknown sample according to

(2.2)

Equation (2.2) is then solved for the concentration of the ith analyte, Cunknowni , in theunknown environmental sample Refer to the quantification equation for ES inFigure 2.2 graphically illustrates the ES approach to multipoint instrumentcalibration Six reference standards, each containing Aroclor 1242 (AR 1242), wereinjected into a gas chromatograph that incorporates a capillary column appropriate

to the separation and an electron-capture detector This instrumental configuration

is conveniently abbreviated C-GC-ECD Aroclor 1242 is a commercially producedmixture of 30 or more polychlorinated biphenyl (PCB) congeners The peak areasunder the chromatographically resolved peaks were integrated and reported as anarea count in units of microvolts-seconds (µV-sec) The more than 30 peak areas

are then summed over all of the peaks to yield a total peak area, A T1242, according to

FIGURE 2.2 Calibration for Aroclor 1242 using an external standard.

The total peak area is then plotted against the concentration of Aroclor 1242expressed in units of parts per million (ppm) The experimental data points closelyapproximate a straight line This closeness of fit demonstrates that the summation

of AR 1242 peak areas varies linearly with AR 1242 concentration expressed interms of a total Aroclor These data were obtained in the author’s laboratory andnicely illustrate the ES mode Chromatography processing software is essential toaccomplish such a seemingly complex calibration in a reasonable time frame Thisauthor would not want to undertake such a task armed with only a slide rule

3.2 H OW D OES THE IS M ODE OF I NSTRUMENT C ALIBRATION W ORK

The IS mode is most useful when it has been determined that the injection volumecannot be reproduced with good precision This mode is also preferred when theinstrument response for a given analyte at the same concentration will vary overtime Both the analyte response and the IS analyte response will vary to the sameextent over time; hence, the ratio of analyte response to IS response will remainconstant The use of an IS thus leads to good precision and accuracy in construction

of the calibration curve The calibration curve is established by plotting the ratio ofthe analyte to IS response against the ratio of the concentration of analyte to eitherthe concentration of IS or the concentration of analyte In our THM example, 1,2-dibromopropane (1,2-DBP) is often used as a suitable IS The molecular formulafor 1,2-DBP is similar to each of the THMs, and this results in an instrument responsefactor that is near to that of the THMs The concentrations of IS in all standards andsamples must be identical so that the calibration curve can be correctly interpolatedfor the quantitative analysis of unknown samples Refer to the THM example aboveand consider the concentrations cited above for the six-point working calibrationstandards 1,2-DBP is added to each standard so as to be present at, for example,

200 ppb This mode is defined as such since 1,2-DBP must be present in the sample

or is considered internal to the sample A single-point or multipoint calibration curve

is usually established when using this mode

The IS mode to instrument calibration has become increasingly important overthe past decade as the mass spectrometer (MS) replaced the element-selective detec-tor as the principal detector coupled to gas chromatographs in TEQA The massspectrometer is somewhat unstable over time, and the IS mode of GC-MS calibrationquite adequately compensates for this instability

We return now to the determination of clofibric acid (CF) in wastewater This

QA A plot of the ratio of the CF methyl ester peak area to that of the internalstandard 2,2′,4,6,6′-pentachlorobiphenyl (22′466′PCBP) against the concentrationregression line was established and drawn as shown (we will take up least squaresregression shortly) The line shows a goodness of fit to the experimental data points

i

1242=∑

case study was introduced in Chapter 1 as one example of trace enviro-chemical

This plot demonstrates adequate linearity over the range of CF methyl ester centrations shown Any instability of the GC-MS instrument during the injection ofthese calibration standards is not reflected in the calibration Therein lies the valueand importance of IS.

con-For a single-point calibration approach, a relative response factor is used:

(2.3)

Quantitative analysis is then carried out by relating the ratio of analyte instrumentresponse for an unknown sample to that of IS instrument response to the ratio ofunknown analyte concentration to IS concentration according to

(2.4)

Equation (2.4) is then solved for the concentration of analyte i in the unknown

sample,

using high-energy detectors such as mass spectrometers The ratio

FIGURE 2.3 Calibration for CFME using 2,2′,4,6,6′PCBP as IS.

#ppm Clofibric acid methyl ester (CFME)

S i

IS i

S i

IS i

S i S i

A A C C

A

C C

i

i i

i

unknown IS

unknown IS

remains fixed over time This fact establishes a constant and hence preserves

the linearity of the internal standard mode of instrument calibration Equation (2.4)

con-centration of the ith analyte in the unknown,

Figure 2.4 graphically illustrates the internal standard approach to multipoint

instrument calibration for trichloroethylene (TCE) using perchloroethylene (PCE)

(or tetrachloroethylene) as the IS An automated headspace gas chromatograph

incorporating a capillary column and ECD (HS-GC-ECD) was used to generate the

in the author’s laboratory A straight line is then drawn through the experimental

data points whose slope is m Rewriting Equation (2.4) gives the mathematical

equivalent for this calibration plot:

FIGURE 2.4 Calibration for TCE using PCE as the internal standard.

TCE

=

Quantitative analysis is accomplished by interpolating the calibration curve Thisyields a value for the concentration for TCE expressed in units of ppm The con-centration of TCE in a groundwater sample obtained from a groundwater aquiferthat is not known to be contaminated with this priority pollutant volatile organiccompound (VOC) can then be found

have a significant impact on the analytical result Three strategies, shown in Scheme2.1, have emerged when considering the use of the IS mode of calibration.10 In thefirst strategy, internal standards are added to the final extract after sample prep steps

Internal standards are extracted from samples;

calibration is established

in appropriate solvent;

e.g EPA method 525.2

Internal standard mode of instrument calibration for TEQA

Isotope dilution

ICP-MS (metals); e.g.

EPA method 6800; Sb, B, Ba, Cd, Ca, Cr,

Cu, Fe, Pb, Mg, Hg, Mo, Ni,

K, Se, Ag, Sr, TI, V, Zn

Other analytical methods; e.g liquid scintillation, radio- immunoassay, mass spectrometry without prior separation

GC-MS (organics):

isotopically labeled

priority pollutants are

used; e.g EPA method

is established from extracted standard solutions; e.g EPA method 524.2

Internal standards are

not extracted from

analytical result for that is lower than the true concentration for the ith

analyte in the original sample since percent recovery losses are not accounted for.This strategy is widely used in analytical method development The second strategyfirst calibrates the instrument by adding standards and ISs to appropriate solvents,and then proceeds with the calibration ISs are then added in known amounts tosamples prior to extraction and cleanup According to Budde:10

The measured concentrations will be the true concentrations in the sample if the extraction efficiencies of the analytes and ISs are the same or very similar This will

be true even if the actual extraction efficiencies are low, for example, 50%.

Again, according to Budde:10

The system is calibrated using analytes and ISs in a sample matrix or simulated sample matrix, for example, distilled water, and the calibration standards are processed through the entire analytical method … [this strategy] is sometimes referred to as calibration with procedural standards.

3.2.1 What Is Isotope Dilution?

Scheme 2.1 places isotope dilution under the second option for using the IS mode

of instrument calibration The principal EPA methods that require isotope dilution mass spectrometry (IDMS) as the means to calibrate a GC-MS, LC-MS, MS (without

a separation technique interfaced), or ICP-MS are shown in Scheme 2.1 Otheranalytical methods that rely on isotope dilution as the chief means to calibrate and

to quantitate are liquid scintillation counting and various radioimmunoassay niques that are not considered in this book

tech-TEQA can be implemented using isotope dilution The unknown concentration

of an element or compound in a sample can be found by knowing only the natural

isotope abundance (atom fraction of each isotope of a given element) and, after an

enriched isotope of this element has been added, equilibrated, and measured, bymeasuring this altered isotopic ratio in the spiked or diluted mixture This is thesimple yet elegant conceptual framework for isotope dilution as a quantitative tool

3.2.2 Can a Fundamental Quantification Equation Be

Derived from Simple Principles?

Yes, indeed, and we proceed to do so now The derivation begins by first definingthis altered and measured ratio of isotopic abundance after the enriched isotope(spike or addition of labeled analog) has been added and equilibrated Only twoisotopes of a given element are needed to provide quantification Fassett andPaulsen11 showed how isotope dilution is used to determine the concentration attrace levels for vanadium in crude oil, and we use their illustration to develop theprinciples that appear below

Cunknowni

are complete The quantification equation for IS shown in Table 2.2 would yield an

The third strategy depicted in Scheme 2.1 corrects for percent recovery losses

Let us start by defining R m as the measured ratio of each of the two isotopes of

a given element in the spiked unknown The contribution made by 50V appears in

the numerator, and that made by 51V appears in the denominator Fassett and Paulsen

obtained this measured ratio from mass spectrometry Mathematically stated,

(2.5)

concentration of vanadium in the sample as the 50V and the weight of sample This

is expressed as follows:

(2.6)

The natural isotopic abundances for the element vanadium are 0.250% as 50V

and 99.750% as 51V, so that f51 = 0.997512 for the equations that follow

Equation (2.6) can be abbreviated and is shown rewritten as follows:

(2.7)

In a similar manner, we can define the amount of the higher isotope of vanadium

in the unknown as follows:

Equation (2.5) can now be rewritten using the symbolism defined by Equation

(2.7) to Equation (2.10) and generalized for the first isotope of the ith analyte (i, 1) and for the second isotope of the ith analyte (i, 2) according to

((unknownsample)][weight unknownsample( )]

amt V f native C unk W

V unk

amt V f native C unk W

V unk

i unk i unk spike

i

spike i spike

unk i unk i u

R m = isotope ratio (dimensionless number) obtained after an aliquot of the unknown sample has been spiked and equilibrated by the enriched

isotope mix This is measurable in the laboratory using a determinative

technique such as mass spectrometry The ratio could be found by taking the ratio of peak areas at different quantitation ions (quant ions

or Q ions) if GC-MS was the determinative technique used

= natural abundance (atom fraction) of the ith element of the first isotope

in the unknown sample This is known from tables of isotopic dance

abun-= natural abundance (atom fraction) of the ith element of the second

isotope in the unknown sample This is known from tables of isotopic abundance

= concentration [µmol/g, µg/g] of the ith element or compound in the

unknown sample This is unknown; the goal of isotope dilution is to find this value.

= concentration [µmol/g, µg/g] of the ith element or compound in the

spike This is known

Wunk = weight of unknown sample in g This is measurable in the laboratory

Wspike = weight of spike in g This is measurable in the laboratory

yield the quantification equation:

(2.12)

achieve TEQA when a GC-MS is the determinative technique employed Methodsthat determine polychloro-dibenzo-dioxins (PCDDs), polychloro-dibenzo-difurans(PCDFs), and coplanar polychlorinated biphenyls (cp-PCBs) require IDMS IDMScoupled with the use of high-resolution GC-MS represents the most rigorous andhighly precise trace organics analytical techniques designed to conduct TEQA knowntoday

3.2.3 What Is Organics IDMS?

Organics IDMS makes use of 2H-, 13C-, or 37Cl-labeled organic compounds Theselabeled analogs are added to environmental samples or human specimens Labeledanalogs are structurally identical except for the substitution of 2H for 1H, 13C for

12C, or 37Cl for 35Cl A plethora of labeled analogs are now available for most prioritypollutants or persistent organic pollutants (POPs) that are targeted analytes To

unk

spike i m

m unk i unk i

illustrate, the priority pollutant or POP phenanthracene and its deuterated form, i.e.,

2H, or D, isotopic analog, are shown below:

Polycyclic aromatic hydrocarbons (PAHs), of which phenanthracene is a ber, have abundant molecular ions in electron-impact MS The molecular weight forphenanthracene is 178, while that for the deuterated isotopic analog is 188 (phen-d10)

mem-If phenanthracene is diluted with itself, and if an aliquot of this mixture is injectedinto a GC-MS, the native and deuterated forms can be distinguished at the same

retention time by monitoring the mass to charge ratio, abbreviated m/z at 178 and

3.3.1 Can We Derive a Quantification Equation for SA?

goal of TEQA, i.e., the concentration of the ith analyte, such as a metal in the

D D

D D

C unk i

We have seen the use of phen-d10 (Table 1.8) as an internal standard to quantitate

unknown environmental sample or human specimen Also assume that

repre-sents the concentration of the ith analyte in a spike solution After an aliquot of the

spike solution has been added to the unknown sample, an instrument response of

the ith analyte for the standard added sample, whose concentration must be

let us prove this The proportionality constant k must be the same between the concentration of the ith analyte and the instrument response, such as a peak area in

atomic absorption spectroscopy The following four relationships must be true:

(2.13)

(2.14)

(2.15)

(2.16)

Solving Equation (2.15) for and substituting this into Equation (2.14) leads

to the following ratio:

(2.17)

(2.18)

For real samples that may have nonzero blanks, the concentration of the ith

(2.19)

C spike i

C SA R

i SA i

R unk i

,

R SA C

i unk i

,

C unk i kR

unk i

=

C spike kR

i spike i

=

R SA R R

i unk i spike i

C SA i k R R

unk i spike i

R spike i

C C

R

unk i

spike i

unk i

SA i unk i

=

−

C unk i

i spike i

,

R SA

i

R unk i

R spike i

SA i unk i

bl unk i

−

the spike solution and accounts for any contribution that the spike makes to the

If a multipoint calibration is established using SA, the line must be extrapolated

across the ordinate (y axis) and terminate on the abscissa (x axis) The value on the

abscissa that corresponds to the amount or concentration of unknown analyte yieldsthe desired result Students are asked to create a multipoint SA calibration tographite furnace atomic absorption spectroscopy (GFAA) routinely incorporates SA

as well as ES modes of calibration Autosamplers for GFAA easily can be grammed to add a precise aliquot of a standard solution containing a metal to anaqueous portion of an unknown sample that contains the same metal

pro-Most comprehensive treatments of various analytical approaches utilizing SA

as the principal mode of calibration can be found in an earlier published paper byBader.13

REALLY MEAN?

Ideally, a calibration curve that is within the linearity range of the instrument’sdetector exhibits a straight line whose slope is constant throughout the range ofconcentration taken By minimizing the sum of the squares of the residuals, a straight

line with a slope m and a y intercept b is obtained This mathematical approach is

called a least squares (LS) fit of a regression line to the experimental data Thedegree of fit expressed as a goodness of fit is obtained by the calculation of acorrelation coefficient The degree to which the least squares fit reliably relatesdetector response and analyte concentration can also be determined using statistics.Upon interpolation of the least squares regression line, the concentration or amount

of analyte is obtained The extent of uncertainty in the interpolated concentration

or amount of analyte in the unknown sample is also found In the next section,equations for the least squares regression will be derived and treated statistically toobtain equations that state what degree of confidence can be achieved in an inter-polated value These concepts are at the heart of what constitutes GLP

The concept starts with a definition of a residual for the ith calibration point The residual Q i is defined to be the square of the difference between the experimentaldata point

illustrates a residual from the author’s laboratory where a least squares regression

line is fitted from the experimental calibration points for

N,N-dimethyl-2-amino-ethanol using gas chromatography Expressed mathematically,

y i

e

i c

Q i y i e y

i c

blank Equation (2.19) is listed in Table 2.2 as the quantification equation for SA

and the calculated data point from the best-fit line y Figure 2.5

quantitate both Pb and anionic surfactants in Chapter 5 Contemporary software for

where y c is found according to

with m being the slope for the best-fit straight line through the data points and b being the y intercept for the best-fit straight line x i is the amount of analyte i or the

FIGURE 2.5 Experimental vs calculated ith data point for a typical ES calibration showing

concentration of analyte i x i is obtained from a knowledge of the analytical referencestandard used to prepare the calibration standards and is assumed to be free of error.

There are alternative relationships for least squares regression that assume x i is notfree of error To obtain the least squares regression slope and intercept, the sum of

the residuals over all N calibration points, defined as Q, is first considered:

The total residual is now minimized with respect to both the slope m and the intercept b:

Next, substitute for b from Equation (2.22) into Equation (2.23):

Upon simplifying, we obtain

(2.24)

i e i c

i N

i e i i

Recall the definition of a mean:

Rearranging in terms of summations gives

(2.25)

(2.26)

Upon substituting Equations (2.25) and (2.26) into Equation (2.24), we arrive

at an expression for the least squares slope m in terms of only measurable data points:

(2.27)

Defining the sum of the squares of the deviations in x and y calculated from all

N pairs of calibration points gives

The sum of the products of the deviation s is given by

The slope for the least squares regression can then be expressed as

(2.28)

and the y intercept can be obtained by knowing only the slope m and the mean value

of all of the x data and the mean value of all of the y data according to

(2.29)

y N

i N

xy

xx

=

b= −y mx

Equations (2.28) and (2.29) enable the best-fit calibration line to be drawn

through the experimental x, y points Once the slope m and the y intercept b for the

least squares regression line are obtained, the calibration line can be drawn by anyone of several graphical techniques A commonly used technique is to use EXCEL®,

the calibration curve These equations can also be incorporated into computer grams that allow rapid computation

pro-A quantitative measure of the degree to which the dependent variable (i.e., theanalytical signal) depends on the independent variable, the concentration of analyte

i, is denoted by the correlation coefficient r according to

Using the previously defined terms, the correlation coefficient can be describedas

close to 1 The square of r is called the coefficient of determination Several

com-mercially available chromatography software packages, such as Total Chrom Elmer) or ChemStation (Agilent), are programmed to calculate only the coefficient

(Perkin-of determination following the establishment (Perkin-of the least squares regression

calibra-tion curve Equacalibra-tion (2.28), which expresses the least squares slope m in terms of

a ratio of sums of the square, can be compared to Equation (2.30) If this is done,

it becomes obvious that the true nature of r is merely a scaled version of the slope (i.e., the slope estimate multiplied by a factor to keep r always between –1 and +1),7

How reliable are these least squares parameters? To answer this question, we firstneed to find the standard deviation about the least squares best-fit line by summation

xx

yy

=

over N calibration points of the square of the difference between experimental and

calculated detector responses according to

(2.31)

where s y/x represents the standard deviation of the vertical residuals and N – 2 represents the number of degrees of freedom N is the number of x, y data points

used to construct the best-fit calibration line less a degree of freedom used in

determining the slope and a second degree of freedom used to determine the y

intercept

The uncertainty in both the slope and intercept of the least squares regressionline can be found using the following equations to calculate the standard deviation

in the slope, s m , and the standard deviation in the y intercept, s b:

For a given instrumental response such as for the unknown, y0, the corresponding

value x0 from interpolation of the best-fit calibration is obtained and the standard

deviation in x0 can be found according to

where s x0 represents the standard deviation in the interpolated value x0 and L sents the number of replicate measurements of y0 for a calibration having N, x, y

repre-data points

Upon replacing the summate with S xx, the standard deviation in the interpolated

value x0 yields a most useful expression:

(2.32)

s

y y N

y x

i e i e

=

−

∑

/ 2

x0 Equation (2.32) shows that

the uncertainty, s x0 , in the interpolated value, x0, is largely determined by minimizing

the ratio of s y/x to m A small value for s y/x infers good to excellent precision in

establishment of the least squares regression line A large value for m infers good

to excellent detector sensitivity The standard deviation in the interpolated value x0

is also reduced by making L replicate measurements of the sample The standard deviation can also be reduced by increasing the number, N, of calibration standards

used to construct the calibration curve

The determination of x0 for a given detector response, y0, is, of course, the most important outcome of trace quantitative analysis x0, together with an estimate of its

degree of uncertainty, represents the ultimate goal of trace quantitative analysis; that is, it answers the questions, how much of analyte i is present, and how reliable

is this number in a given environmental sample? For TEQA, it is usually unlikelythat the population mean for this interpolated value µ(x0) can ever be known andthat the standard deviation in this population mean, σ(x0), can ever be known TEQArequires the following:

• A determinative technique whereby the concentration of a priority ant VOC, such as vinyl chloride, can be measured by an instrument Itcan be assumed that for a given analyte such as vinyl chloride, repeatedlyinjected into a gas chromatograph, µ(x0) and σ(x0) are known

pollut-• However, the concentration of vinyl chloride in the environmental samplemay involve some kind of sample preparation to get the analyte into theappropriate chemical matrix for the application of the appropriate deter-minative technique, and we can assume that both µ(x0) and σ(x0) areunknown

These two constraints require that the confidence limits be found when the

standard deviation in the population mean is known This is where the Student’s t

statistics have a role to play Who was “Student”? Anderson has introduced a littlehistory (pp 70–72):7

Unfortunately, we do not know the true standard deviation of our set; we have only an

estimate s based on L observation Thus, the distribution curve for x is an estimate of– the true distribution curve A research chemist, W.S Gosset, worked out the relationship between these estimates of the distribution curve and the true one so that we can use

s in place of σ Gosset was employed by a well-known Irish brewery which did not want its competition to know it was using statistical methods However, he was per- mitted to publish his work under the pen name “Student,” and the test for differences

in averages based on his work is known as Student’s t test Student’s t test assumes

that the average is based on data from a normal population.

Because we know s x0 (the standard deviation in the interpolated value x0), we

can use the Student’s t statistics to estimate to what extent x0 estimates the populationmean µx0 This can be determined according to

C, to be written that allow for rapid calculation of s

Conf Int for

level of α for N – 2 degrees of freedom (df ), where N is the number of x, y data

1–α2,dfs x0] is called the

confidence interval and represents the range of x values within which one can be

(1 – α/2) 100% confident that the mean value for x0 will approximate the populationmean µx0

author The program not only finds the least squares regression line through a set

of N calibration points, but also finds decision and detection limits, and for a given instrument response, y0, the program finds x0 and the confidence interval for µx0.Figure 2.6 is this author’s attempt to graphically represent the uncertainty present

in an interpolated instrument response y0 A segment of what might be a typical ES

or IS calibration plot shrouded with its corresponding confidence limits both aboveand below the regression line is shown The confidence interval is shown to be

equidistant from the regression line This assumption represents the homoscedastic

case, i.e., a regression line having a constant variance, over the range of analyteconcentration used to construct the linear regression line Those sections of thecalibration that are highlighted in bold reveal that horizontal movement to either

confidence limit at y0 results in equal confidence intervals The confidence interval

FIGURE 2.6 Interpolation of a least squares regression line showing confidence intervals.

points used to construct the calibration curve The term [±t

Appendix C shows a computer program written in GWBASIC by this

can be related to the product of Student’s t and the standard deviation in the interpolated value, s x0 Mathematically,

earlier by MacTaggert and Farwell.14

The prediction interval about x L (or xlower, as shown in Figure 2.6) can bemathematically defined as follows:

where

b = y intercept of the linear regressed line

m = slope of the linear regressed line

t = two-tailed value for Student’s t for N – 2 degrees of freedom, where N

is the number of x, y data points used to construct the calibration Refer again to Figure 2.6 We see that if the value for y L is taken and the height

of this interval added to it, y0 is obtained This enables two relationships to emerge

that can both be set equal to y0 These two relationships are given below for y0:

Both equations being equal to y0 can be set equal to each other such that

Eliminating b, setting this equation equal to zero, squaring both sides, and then solving for x L yields the following quadratic equation:

Here g is used to collect terms and greatly simplify the expression

Solving the above quadratic equation via the quadratic formula and incorporating

the result for x U yields a discrimination interval about the interpolated value for x0:

If L replicates are made of the y0 value, the following equation may be used to

define discrimination intervals about the interpolated value for x0 according to

The expression for g above can be further simplified by substituting for S xx usingEquation (2.28) so that

This relationship shows that g is a measure of the statistical significance of the slope value Further use of g can be made to derive a simpler equation for the discrimination interval Let us assume that g << 1 such that

This equation shows a symmetrical interval about x0 and is comparable to a predictioninterval, as introduced earlier We have thus mathematically shown that the standard

deviation in the vertical residual, s y/x, can approximate the discrimination interval.The following quote reinforces this notion:15

Analytical chemists must always emphasize to the public that the single most

charac-teristic of any result obtained from one or more analytical measurements is an adequate statement of its uncertainty interval Lawyers usually attempt to dispense with uncer-

tainty and try to obtain unequivocal statements; therefore, an uncertainty interval must

be clearly defined in cases involving litigation and/or enforcement proceedings Otherwise, a value of 1.001 without a specified uncertainty, for example, may be viewed

as legally exceeding a permissible level of 1.

g t s m

The concept of a confidence interval around the least squares calibration linewill again become important in the calculation of an instrument’s detection limit for

a given analyte In other words, how reliable is the detection limit for a given analyte?

INSTRUMENT DETECTION LIMITS?

The IDL for a specific chemical analyte is defined to be the lowest possible tration that can be reliably measured Experimentally, if lower and lower concentrations

concen-of a given analyte are measured, the smallest concentration that is barely detectableconstitutes a rough estimate of the IDL for that analyte using that particular instru-ment For years, EPA has required laboratory analysts to first measure a blankreplicate a number of times The most recent guidelines appeared in 40 Code of

16 The steps that are recommendedare listed as follows:

1 Prepare a calibration curve for the test with standards

2 Analyze seven laboratory water blanks

3 Record the response of the test for the blanks

4 Prepare the mean and standard deviation of the results from the blanks

as above

5 The IDL is three times the standard deviation on the calibration curve.Fortunately, a bit more thought has been given to the calculation of the IDL thanthe meager guidelines given above The average signal that results from these

replicate measurements yields a mean signal, Sblank The analyst then calculates the

standard deviation of these replicate blanks, sblank, and finds the sum of the mean

signal and a multiple k (often k = 3) of the standard deviation to obtain a minimum detectable signal SIDL according to17

SIDL= Sblank+ ksblank

If a least squares calibration curve has been previously established, the equation

for this line takes on the common form, with S being the instrument response, m being the least squares slope, and x being the concentration according to

sensitivity It becomes obvious that minimizing xIDL requires that the signal at the

detection limit, SIDL, be maximized while the noise level remains minimized, andthat the steepness of the calibration curve be maximized

The use of Equation (2.34) to quantitatively estimate the IDL for chromatographsand for spectrometers has been roundly criticized over the years There have beenreported numerous attempts to find alternative ways to calculate IDLs This authorwill comment on this most controversial topic in the following manner The approachencompassed in Equation (2.34) clearly lacks a statistical basis for evaluation, andhence is mathematically found to be inadequate As if this indictment is not enough,IDLs calculated based on Equation (2.34) also ignore the uncertainty inherent in theleast squares regression analysis of the experimental calibration, as presented earlier

In other words, what if the analyte is reported to be absent when, in fact, it is present(a false negative)? In the subsections that follow, a more contemporary approach tothe determination of IDLs is presented and starts first with the concept of confidenceintervals about the regression line

The least squares regression line is seen to be shrouded within confidence intervals,developed by Hubaux and Vos18 over 30 years ago The upper and lower confidence

limits for y that define the confidence interval for the calibration are obtained for any x (concentration) and are given as follows:

(2.35)

represents the slope of the linear least squares regression t corresponds to a

prob-ability of 1 – α for the upper limit and 1 – β for the lower limit Vy represents the

variance in the instrument response y for a normal distribution of instrument responses at a given value of x This equation represents another way to view the

least squares regression line and its accompanying confidence interval The term inbrackets is the calculated response based on least squares regression The variance

in y, V y , is composed of two contributions The first is the variance in y calculated

from the least squares regression, and the second is the residual variance σ2.Expressed mathematically,

(2.36)

and the variance in the least squares calculated y can be viewed as being composed

of a variance in the mean value for y and a variance in the x residual according to

The variance in the least squares regression line can further be broken down

expressed in terms of V m according to

for the particular case where x = 0 is obtained from Equation (2.41), in which a

1 – α probability exists that the normal distribution of instrument responses falls to

within the mean y at x = 0 Mathematically, the instrument response y C, often termed

a critical response, is defined as

(2.42)

V y

V N

=

−

2 2

α

Equation (2.42) can be viewed as being composed of two terms This is expressed

as follows:

The y intercept of the least squares regression line is given by

The value for y0 cannot be reduced because it is derived from the least squares

regression; however, the second term, [P][s], may be reduced and hence lead to lower IDLs P is defined as

Figure 2.7 provides a graphical view of the terms used to define the decision

limit, x Cs , and detection limit, x D The decision limit, x C, is a specific concentrationlevel for a targeted analyte above which one may decide whether the result of

analytical measurement indicated detection The detection limit, x D, is a specificconcentration level for a targeted analyte above which one may rely upon to lead

to detection A third limit, known as a determination limit, or using more recent

FIGURE 2.7 Graphical estimate of the decision limit, x C , and the detection limit, x D, for a hypothetical LS regression (not drawn to scale).

Least squares regression fit of calibration

XC XD

YC

Concentration of analyte

jargon from the EPA, the practical quantitation limit, is a specific concentration atwhich a given procedure is sufficiently precise to yield satisfactory quantitativeanalysis.5 The determination limit will be introduced in more detail later.

C

the lower confidence limit in which a 1 – β probability exists that the normal

distribution of instrument responses falls to within the mean at this value of x The ordinate that corresponds to this value of x, x D, can be found according to

s Factors P and Q can be varied so as to minimized y D

Graphically, it is straightforward to interpolate from y C to the IDL x D

Numeri-cally, Equations (2.42) and (2.43) can be solved for the critical limit x C and for the

IDL, x D The equation for the calibration line can be solved for x as follows:

When the instrument response y is set equal to y C, the concentration that

corre-sponds to a critical concentration limit is defined This critical level x C, definedalmost 20 years ago,5,18 is a decision limit at which one may decide whether theresult of an analysis indicates detection The critical level has a specifically definedfalse positive (type I) error rate, α, and an undefined false negative (type II) errorrate, β; xC is found according to

−

−

∑1

−

−

∑1

m

P s m

C C

Referring to Figure 2.7, y can merely be drawn horizontally until it meets with

In a similar manner, the IDL that satisfies the statistical criteria discussed isobtained according to

(2.45)

The IDL denoted by x D and given in Equation (2.45) is the true concentration

at which a given analytical procedure may be relied upon to lead to detection.5,18 Afalse negative rate can now be defined A normal distribution of instrument responses

at a concentration of zero and at a concentration x D overlaps at x C and gives rise to

α and β Figure 2.8 graphically depicts this Gaussian distribution and stems fromapplication of Equation (2.41), whereas confidence intervals can be defined thatshroud the linear least squares regression line The intersection of the upper confi-

dence band with the y axis at y C , where x = 0, corresponds to the highest signal that

could be attributed to a blank 100 (1 – α)% of the time This is represented by a tdistribution with a one-sided tailed α for y at x = 0 The intersection of yC with the

lower confidence band at y L in Figure 2.8 corresponds to the lowest signal that could

be attributed to an analyte concentration x D 100 (1 – β)% of the time This is

represented by the t distribution with a one-tailed β for y at x = x D The mean for

the distribution of signals at x D is y D Equation (2.45) leads to a quadratic equationwhose root is given below:19

(2.46)

FIGURE 2.8 Gaussian distributions about the y intercept, y0, and about y D while indicating

the importance of y C in properly defining α and β.

s P Q m

D D

x S

5.2 W HAT I S W EIGHTED L EAST S QUARES AND H OW D OES T HIS

close to the IDL, have nonconstant variances about the linear regressed line

Chro-matographic processing software queries the user as to whether or not a weighted least squares regression (WLS) or OLS regression is to be used It behooves the

analyst to have at least a cursory understanding of WLS

Burdge et al.19 put in this way:

Application of OLS to data with non-constant variance (heteroscedasticity) yields confidence bands and thus detection limits that will not accurately reflect measurement capability WLS facilitates the determination of realistic detection limits for heterosce- dastic data by yielding confidence bands that directly reflect the changing variance.… Application of OLS to significantly heteroscedastic data results in the construction of unnecessarily broad confidence bands at the low end of the calibration curve The detection limit derived from OLS treatment of such data will not be a fair representation

of a measurement method’s detection capability.

Hence, weighting factors or simply weights are used to give more emphasis or

weight to calibration data points, usually at the lower end of the calibration Thiscan be done without changing the raw data However, implementing WLS requiresthat we modify the previously developed mathematical relationships used to deriveOLS

Let us return to Equation (2.31) for the standard deviation of the vertical residuals

for N – 2 degrees of freedom, as introduced earlier A weighted residual would look

like the following:

(2.47)

is obtained from the WLS regressed line according to:

The WLS slope can similarly be viewed and compared to Equation (2.28):

s

wi y y N

wy x

e wi c

wxx

=

The weighted parameters have replaced the unweighted parameters (s, xave, S xx),

the sum of the weights has replaced N, and t (t–α/2,N–p–2) is the (1 – α/2) 100 percentage

point of the Student’s t distribution on N – p – 2 degrees of freedom (where p is the number of parameters used to model the weights) Also, the inverse weight (1/w j)

at x j has replaced 1 in the unweighted equation [Equation (2.41)] The reader, at thispoint, should compare Equation (2.48) with Equation (2.41) while noting similaritiesand differences between OLS and WLS approaches to instrument calibration.Three general approaches are introduced to find appropriate weights:

• Define a weight as being inversely proportional to the standard deviation

for the ith calibration data point:

• Define a weight as being inversely proportional to the variance for the ith

calibration data point:

• Plot the standard deviation as a function of analyte concentration and fitthe data to either a quadric or exponential of a two-component model

An instrument detection limit based on WLS has been recently reported.19

i i

= 1

w s

i i