Spectrochimica Acta Part B 66 (2011) 105–113

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Spectrochimica Acta Part B

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Statistical behavior of ten million experimental detection limits

Edward Voigtman ⁎, Kevin T. AbrahamDepartment of Chemistry, University of Massachusetts, Amherst, 710 N. Pleasant St., Amherst, MA 01003-9336, United States

⁎ Corresponding author. Tel.: +1 413 545 0293; fax:E-mail address: voigtman@chem.umass.edu (E. Voig

0584-8547/$ – see front matter © 2011 Elsevier B.V. Aldoi:10.1016/j.sab.2011.01.008

a b s t r a c t

a r t i c l e i n f o

Article history:Received 13 November 2010Accepted 20 January 2011Available online 31 January 2011

Keywords:Limit of detectionDetection limitCurrieNon-central t distributionBootstrapping

Using a lab-constructed laser-excited fluorimeter, together with bootstrapping methodology, the authorshave generated many millions of experimental linear calibration curves for the detection of rhodamine 6Gtetrafluoroborate in ethanol solutions. The detection limits computed from them are in excellent agreementwith both previously published theory and with comprehensive Monte Carlo computer simulations. Curriedecision levels and Currie detection limits, each in the theoretical, chemical content domain, were found to besimply scaled reciprocals of the non-centrality parameter of the non-central t distribution that characterizesunivariate linear calibration curves that have homoscedastic, additive Gaussian white noise. Accurate andprecise estimates of the theoretical, content domain Currie detection limit for the experimental system, with5% (each) probabilities of false positives and false negatives, are presented.

+1 413 545 4490.tman).

l rights reserved.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

In 2008, Voigtman published a series of papers [1–4] presentingthe detailed theory of limits of detection of the simplest chemicalmeasurement systems. The theoretical developments were based onthe assumption of a univariate linear system in which the measure-ment noise was additive, Gaussian white noise (AGWN). Most of thework presented dealt with homoscedastic AGWN, but four commonheteroscedastic noise precision models were also investigated. Thedetection limits, and associated decision levels, were as per Currie'sprogram [5], thereby accommodating a priori specification and controlof both false positives and false negatives. So-called “traditional”detection limits, wherein false negatives are entirely ignored, wereshown to be just a special case of the Currie scheme in that the Curriedecision level served double duty as both the decision level itself andthe detection limit. It was also shown that all detection limits of thegeneral form ksblank/b, where k is a “coverage factor”, sblank is a samplestandard deviation of the blank, and b is a regression-determinedslope of an experimental linear calibration curve, are distributed as amodified non-central t distribution and that the k value is actually aproduct that depends on both critical t or z values and on preciselyhow blank-subtraction is performed.

The theory was tested extensively with Monte Carlo computersimulations. Literally tens of billions of calibration curves were gen-erated and used to challenge and test the derived theory and, inevery case, the computer simulations were in agreement with theory.

However, despite the outstanding agreement between the derivedtheory and the computer simulation results, there were no experi-mental results. This was due to what, at the time, appeared to be aninsurmountable obstacle, i.e., the need for an experimental systemthat met the fundamental assumptions upon which theory was basedand that could be used to generate millions of independent ex-perimental detection limits in a reasonable time period.

The solution to the dilemma was a powerful resampling statisticstechnique known as bootstrapping [6]. Ordinarily, calibration curvesare prepared by making measurements on a series of errorlessstandards. Even if every standard were to be measured only once, asis the case in all the work reported below, they would be measuredsequentially. Then ordinary least squares (i.e., linear regression)would be applied to the acquired data and the desired statisticalestimateswould be computed. The drawback of this customarymethodis its general slowness: generating a million or more calibration curvesis usually infeasible.

The bootstrapping solution is to measure each standard manytimes, using a suitable analog-to-digital converter (ADC) and com-puter, saving the resulting experimental data files on the computer.Then a given calibration curve, and its associated detection limit andother desired statistics, is computed by randomly selecting (withreplacement) a single measurement result from each of the desiredstandards files. The process is repeated many times: in the presentwork, ten million calibration curves, and their associated statistics,were computed in ten sets of one million each. One major advantageof the bootstrapping procedure is that all temporal correlations in thefiles for the standards are destroyed, thereby causing the measure-ment noise to be white. This is important because preamplifier noisesare never truly white due to low frequency noise and, possibly,interferences.

106 E. Voigtman, K.T. Abraham / Spectrochimica Acta Part B 66 (2011) 105–113

2. Theoretical

2.1. Background

The fundamental assumptions, notations and theory of Currieschemedecision levels anddetection limits are as previously describedin considerable detail [1, pp 115–118]. Briefly, the univariate linearchemical measurement system is as described by Eq. (1):

y = α + βX + noise ð1Þ

where X is the analyte content (concentration, quantity or amount), αis the true intercept, β is the true slope, y is the measured response,and the noise is AGWN having σ0 population standard deviation.Thus, any specific measurement yi is distributed as yi~N :α+βX,σ0.The average of M such independent measurements, denoted by y,is distributed as y eN : α + βX;σ0 =M1=2. For the important specialcase ofM replicate measurements of an analytical blank, X=0, so thatyblank eN : α;σ0 =M1=2.

2.2. Blank-subtraction possibilities

If α were known, then the net response for the mean of Mfuture replicate measurements of a true blank would be computedas yfuture blank−α and would be distributed as yfuture blank−α eN :0;σ0 =M1=2. The canonical and most reasonable choice for M issimply unity, i.e., one future measurement of a true blank, in whichcase yfuture blank−α~N :0,σ0. Throughout the present work, it is as-sumed thatM=1. However, for generality, the relevant equationswillretain the general M.

If α is unknown, as is almost certainly the case, then blank-subtraction is performed by subtracting an unbiased experimentalestimate, denoted by α̂, from either yblank or yblank, depending on thevalue of M. Since α̂ is an estimate, the net response is noisier than thecorresponding uncorrected response. Furthermore, there are severalcommon ways to obtain an experimental value for α̂, so it becomesconvenient to define a factor that groups together the effects of M andthe error factor associated with α̂. This composite “blank-subtraction”factor is denoted by η1/2 and it has the general form

η1=2≡ M−1 + error factor due to α̂h i1=2 ð2Þ

One commonmethod for obtaining a value for α̂ is by preparing anexperimental calibration curve from N standards, performing linear

Fig 1. Block diagram of the laser-excited

regression, and then taking α̂ as the intercept of the calibration curve.In this case, η1/2 is given by

η1=2 = M−1 + N−1 + X2=SXX

� �1=2 ð3Þ

whereX is themeanof theN standards (forNN2) and SXX is the sumof the

squared differences: SXX≡ ∑N

i=1Xi−X� �2

, with Xi being the i-th standard.Note that

SXX = Nσ2X = N−1ð Þs2X ð4Þ

where σX2 is the population variance of the errorless standards and sX

2 isthe corresponding sample variance of the errorless standards. This blank-subtraction method is used in the present work.

Another common method for obtaining a value for α̂ is as theaverage ofM0 independent replicate measurements of a true blank. Inthis case, η1/2 is given by

η1=2 = M−1 + M−10

h i1=2 ð5Þ

For a more complicated blank-subtraction possibility, involvingpooling, see [4, p. 163].

2.3. Currie decision levels and detection limits

In Currie detection limit theory, M future independent replicatemeasurements are to be made on a true blank. The resulting averageis blank-subtracted with α or α̂, to yield the net response, whichis then compared to the Currie decision level in the net responsedomain. A false positive occurs if the net response exceeds thedecision level; otherwise the net response is correctly classified as atrue blank. Alternatively, and with equal validity, the comparisonmaybe performed in the chemical content domain by simply dividing boththe net response, and the decision level, by the slope of the calibrationcurve.

The present work focuses on the chemical content domain ratherthan on the net response domain, since the former is generally ofgreater interest to practicing analysts. In this case, the Currie decisionlevels (XDC and xDC ) and Currie detection limits (XD and xD) are:

XDC = zpη1=2σ0 = β & xDC = tpη

1=2s0 = b ð6Þ

molecular fluorescence instrument.

Table 2Fluorescence standards (rhodamine 6G tetrafluoroborate in ethanol).

Standard Concentrationmg/100 mL

MeanV

Standard deviationV

1 0.03392 0.1091 0.29012 0.04240 0.1416 0.29133 0.0848 0.2779 0.28854 0.1272 0.4272 0.28975 0.2120 0.7374 0.28996 0.2968 1.041 0.29187 0.3604 1.261 0.28818 0.4240 1.481 0.28968a 0.6360 2.330 0.29089 0.8480 3.202 0.28769a 1.060 4.002 0.285910 1.272 4.847 0.2879

x=0.2893s=0.0017

107E. Voigtman, K.T. Abraham / Spectrochimica Acta Part B 66 (2011) 105–113

XD = zp + zq� �

η1=2σ0 = β & xD = tp + tq� �

η1=2s0 = b ð7Þ

where the equations on the left are the theoretical domain expres-sions (upper case X) and those on the right are the experimentaldomain expressions (lower case x). In the above equations, p = falsepositives probability, q = false negatives probability, zp and zq arethe critical z values for p and q respectively, tp and tq are the criticalt values for p and q respectively, b is the linear regression slope ofthe calibration curve, s0 is the sample standard deviation of the noise,and σ0, β and η1/2 are as defined previously. Note that these fourexpressions are exact, for all degrees of freedom.

Throughout the present work, and without loss of generality, itis assumed that p=q=0.05 exactly. It is also assumed that blank-subtraction involves the use of α only for the theoretical domainexpressions on the left side of Eqs. (6) and (7). However, for theexperimental domain expressions on the right side of Eqs. (6) and (7),it is assumed that blank-subtraction involves use of the experimentallinear regression intercept α̂. Hence the two theoretical expressionssimplify, since η1/2=1, but the experimental expressions have η1/2N1,as given by Eq. (3) with M=1. Thus

XDC = zpσ0 = β & xDC = tpη1=2s0 = b ð8Þ

XD = zp + zq� �

σ0 = β & xD = tp + tq� �

η1=2s0 = b ð9Þ

Note that if σ0, β and α were known, which is virtually never thecase, then the two theoretical expressions on the left would be thecomplete solution to the detection limit problem. Indeed, finding away to accurately estimate XDC and XD from the experimental dataalone is the outstanding detection limit problem.

3. Experimental

3.1. Experimental apparatus

For a fair experimental test of the previously published theoreticalresults, it was necessary to construct a chemical measurement systemthat was linear over a reasonably wide range of analyte concentra-

Table 1Components, reagents and instruments used.

Item Description

Laser diode Blu-ray, 405 nm, b100 mW, eBayLaser driver FlexMod2, Nautilus IntegrationCuvette Supracil, 1.00 cm, Janus Laboratories, Inc.Fluorophor Rhodamine 6G tetrafluoroborate, Exciton Chemical Co., Inc.Solvent Ethanol, 200 proof, Pharmco-Aaper, Pharmco Products Inc.Emission filter 20 CGA-550, 550 nm long pass, Oriel Corporation of AmericaPhotodiode UDT-10DP/SB, 0.5 A/W, United Detector TechnologyTransimpedancepreamplifier

Lab constructed, 1.10 MΩ transimpedance, 1.0 s output timeconstant

Noise generator Lab constructed, ~289 mV rmsSummingamplifier

Lab constructed, signal channel gain=+1.00,noise channel gain=+7.10

ADC hardware LabPro, 12 bit, ±10 V full scale, 1000 samples/s,Vernier Software & Technology

ADC software Logger Pro, v. 2.2, Vernier Software & TechnologyLaptop for dataacquisition

Macintosh Powerbook G3 w/ Mac OS 8.6, Apple Inc.

Desktopcomputer

Macintosh iMac 24 (white) w/ Mac OS 10.5.8, Apple Inc.

Spreadsheet Excel 2004 for Mac, v. 11.6, Microsoft Corp.Generalcomputation

LightStone, E. Voigtman's freeware add-on for Extend

Softwareplatform

Extend, v. 6.0.8, Imagine That, Inc.

Graphing IGOR Pro, v. 6.20, Wavemetrics, Inc.Oscilloscope HM 205-2, 20 MHz digital storage, Hameg Instruments GmbH

tions and that met the requirement of being limited by homoscedasticAGWN. Laser-excited molecular fluorescence of commonly used laserdyes is well known to meet the criterion of linearity easily, andmoderate power laser diodes are readily available to serve as theexcitation source, so this was the system chosen for testing. A blockdiagram of the experimental system is shown in Fig. 1, with com-ponent specifications given in Table 1.

Almost every component in the experimental system was lab con-structed, inexpensive or readily available in many laboratories, withnothing exotic or state-of-the-art. This was deliberate: for presentpurposes, the more ordinary and obvious the experimental apparatus,the better. Indeed, even without the large amount of noise deliberatelyadded for purposes of testing the detection limit theory, the fluorimeterin Fig. 1 was much inferior in detection power to any commercialfluorimeter.

3.2. Experiment protocol

A total of 12 laser dye solutions were carefully prepared, usingdigital pipettes and class A volumetric flasks. Starting with the mostdilute standard, 3 mL of solution was placed in a Supracil cuvette,which was then placed in the sample compartment. The output ofthe summing amplifier was digitized at the rate of 1000 samples/sfor a period of 12 s, thereby resulting in data files of 12 001 data perfile. The resulting 12 data files were then used in all subsequentbootstrapping experiments. The standards, their concentrations and

Fig. 2. Calibration curve for the calibration design using all 12 standards.

Fig. 3. The analog noise generator, transimpedance preamplifier and non-inverting summing amplifier circuitry.

Fig. 4. The power spectral density and probability density function (inset) of the noiseon the digitized fluorescence response files.

108 E. Voigtman, K.T. Abraham / Spectrochimica Acta Part B 66 (2011) 105–113

statistics (i.e., sample means and sample standard deviations foreach 12 001 point data file) are given in Table 2 and Fig. 2 shows thecalibration curve that results from using all 12 standards in the cali-bration design. Note that the error bars are±one standard deviation, asgiven in Table 2. Clearly, the standard deviations well approximate therequirement of homoscedasticity (“same scattering”), i.e., the standarddeviations are independent of the standard's concentrations.

3.3. Testing the noise: is it AGWN?

With regard to the remaining noise properties, a simple analognoise generator, shown in Fig. 3 (upper half), was designed to provideGaussian noise with approximately white power spectral density(PSD). The lower half of Fig. 3 shows the transimpedance preamplifierand non-inverting summing amplifier. Noise performance was eval-uated by first subtracting the mean values (i.e., fluorescence signals)from each of the acquired experimental data files for the standards.The resulting 12 files were then concatenated, starting with thatfrom the lowest standard (1) and working up to the highest standard(10), to yield a file of 144012 points. The merged noise file wasprocessed by computing and averaging 70 sequential 2048 pointunilateral, mean-subtracted PSDs, without zero-filling or apodization.The result, shown in Fig. 4, is not perfectly white, and power supplyinterferences are evident at 60, 180 and 300 Hz, but the resamplingprocess involved in bootstrapping automatically whitens the noise,as noted above. This was confirmed by direct test: a total of 700unilateral, mean-subtracted PSDs, of 16384 points each, were aver-aged, resulting in a white PSD having mean value 1.6734 V2/Hz andrms value of 0.2893 V.

The merged noise file was also binned into a histogram to obtainthe empirical normalized probability density function (PDF), shownin the inset in Fig. 4, together with an overplotted Gaussian. TheGaussian curve fit standard deviation was 0.2897 V, in excellentagreement with the directly determined 144012 point standarddeviation of 0.2893 V. Themagnitude of the added noisewas designedto be more than sufficient to dominate the relatively small intrinsicnoise (approximately 2.6 mV) of the transimpedance preamplifier.Hence, the requirement of AGWN is satisfied, since additivity isobviously satisfied by construction.

The time constant of the fluorimeter was one second, so thatthe fluorimeter's output was approximately the true fluorescence

response, while the sampling frequency was 1 kHz. Without thelarge added amount of approximately white noise, an additional gainstage would have been required before the ADC and the digitizedsamples would have been strongly correlated. However, propertesting of the theory required that the noise be both Gaussian anduncorrelated, and hence independent. Therefore, the small intrinsicnoise was deliberately submersed in a large excess of added noisewith approximately white PSD, in order to have maximum controlover, and knowledge of, the operative noise. As noted above, thebootstrapping process entirely eliminated correlations in the resultingsum. As a consequence, the digitized samples were both Gaussiandistributed and uncorrelated, as required to fairly test the theory.

3.4. Bootstrapping protocol in the experiments

Computation of the bootstrapped experimental calibration curves,and associated statistics, takes place in the computer program shown

109E. Voigtman, K.T. Abraham / Spectrochimica Acta Part B 66 (2011) 105–113

in Fig. 5. Up to twelve standards files may be imported, allowinggeneration of experimental calibration curves with as few as 3standards and as many as 12 (an arbitrarily chosen upper limit).Additional inputs required are the number of standards to use (N),the concentration values of the standards (i.e., the Xi values), andthe values of η1/2, tp, and tq. A simple Excel spreadsheet template,an example of which is shown in Fig. 6, was developed in order tostreamline computation of necessary quantities such as η1/2 and SXX

1/2.In the present work, ten simulations of one million steps each

were performed for each desired calibration curve design. Thisgenerated ten million experimental calibration curves, each with itsown tslope test statistic and xD detection limit, computed via Eq. (9),since each standard was measured only once per calibration curveand therefore s0= sr (the sample standard error about regression).These were separately binned into histograms and the histogramsexported as ASCII text files for subsequent analysis and plotting.The required simulation times were less than 10 minutes on a firstgeneration iMac 24.

Fig. 5. The bootstrapping program used to compute ten million ex

3.5. Estimation of the experimental non-centrality parameter

For any given calibration curve design, the files of binned tslopevariates were then fitted with the non-central t distribution with N-2degrees of freedom. This provided an estimate, denoted by δ̂, of thenon-centrality parameter, δ, of the non-central t distribution. Thereason δ is important is that the theoretical PDF for xD is distributedas a modified non-central t variate [2, Eq. (26)]:

pxD xDð Þ = tp + tq� �

η1=2S1=2xx = x2D� �

t tp + tq� �

η1=2S1=2xx = xD jN−2; δ� �

ð10Þ

where δ≡β/σb and σb is the population standard error of the slope.Thus, the normalized histogram of experimental xD variates mustbe compared with Eq. (10) and an estimate of δ is essential. Thiscomparison was done using the computer program shown in Fig. 7,

perimental calibration curves, xD variates, and tslope variates.

Fig. 6. The MS Excel template for computation of decision and detection limits and necessary calibration design values.

110 E. Voigtman, K.T. Abraham / Spectrochimica Acta Part B 66 (2011) 105–113

with k≡(tp+ tq)η1/2 and “scale factor” = SXX1/2=N1/2σX=(N−1)1/2sX,

using Eq. (4).

3.6. Computer simulation protocol

The program in Fig. 7 also allowed for over-plotting of xD variatesgenerated by computer simulation. Computer simulated calibrationcurves, plus xDC and xD variates, were generated as before [1–4]. Aseach simulated calibration curve was generated, a separate true blankwas also generated along with an xDC variate and an xD variate. Thisallowed for immediate testing of the specified probabilities of falsepositives and false negatives. As was true for the experimental xDvariates, ten million computer simulated xD variates, in ten sets of onemillion each, were generated for each calibration curve design and the

Fig. 7. The program that overplots the theoretical detection limit PDFs with the normal

required simulation times were less than ten minutes. The computerprogram for generation of the simulated data is shown in Fig. 8.

4. Results and discussion

4.1. Results for four standards

Fig. 9 shows an overplot of histogram data and theory for thecalibration design using standards 1, 4, 8a, and 10. For this calibrationdesign, the number of standards and their specific concentrations(see Table 2) result in tp= tq=2.919986 (since N=4 and p=q=0.05), η1/2=1.235313, SXX1/2=0.984630, and k≡(tp+tq)η1/2=7.214191,so that, using Eq. (10), xD is distributed as

pxD xDð Þ = 7:10331 = x2D� �

t 7:10331= xD j2;12:838ð Þ ð11Þ

ized ten million event histograms of experimental and simulation detection limits.

Fig. 8. The Monte Carlo computer simulation program, with automatic testing of probabilities of false positives and false negatives.

111E. Voigtman, K.T. Abraham / Spectrochimica Acta Part B 66 (2011) 105–113

where δ̂=12.838 was used as the estimate of δ in Eq. (10). A plot ofthis expression is shown in Fig. 9 as a solid (green) curve. As notedabove, the value for δ̂ was obtained by curve-fitting the experimentalhistogram of ten million binned tslope variates for this calibrationdesign.

The ten million binned experimental xD are overplotted in Fig. 9 insolid square (blue) markers and the ten million binned simulation xDare also overplotted in solid circular (red) markers. For the computersimulations, the necessary inputs are the specific values of thestandards, tp, tq, β, and σ0. An unbiased estimate of β, denoted by b,was obtained by performing linear regression on the 12001 pointmeans versus concentrations (given in Table 2) for standards 1, 4, 8a,and 10. The resulting slope, b, is the estimate of β. This gaveβ≈b=3.83236 V/(mg/100 mL). Then σ0 was estimated by noting that

δ≡ β = σb ≈ b= σb = bS1 = 2XX = σ r = bS1 = 2

XX = σ0 ð12Þ

where σr is the population standard error about regression and onlyone measurement is made per standard, so that σ0=σr. Hence,

Fig. 9. The overplotted theoretical (solid green line) detection limit PDF for xD(Eq. (10)), together with the normalized ten million event histograms of experimental(blue filled squares) and simulation (red filled circles) detection limits for calibrationwith 4 standards.

σ̂0≈bS1=2XX = δ̂. For this calibration design, σ̂0≈ 0.293928 V and theagreement among experiment, theory and computer simulation isessentially perfect.

Very importantly, σ̂0is higher than the average of the four stan-dard deviations for the standards 1, 4, 8a, and 10 (i.e., 0.28961 V) andalso higher than the average standard deviation for all twelve stan-dards (i.e., 0.28927 V). Neither of these latter values is an accurate oracceptable substitute for σ̂0 in the computer simulations.

4.2. Results for 3 through 12 standards

Fig. 10 shows the results for calibration designs having 3 through12 standards, with the standards used shown in Table 3. Remarkably,this figure shows 100 million experimental LODs plus 100 millioncomputer simulation LODs: for each of the 10 calibration designs, theslightly fuzzy “curves” are triple plots of theoretical PDFs, experi-mental histograms, and computer simulation histograms, withmethodology as described in Section 4.1.

Fig. 10. The overplotted detection limits for theory (solid black lines), experiment (bluefilled squares) and simulation (red open circles), for calibration designs with 3 through12 standards.

Table 3Calibration designs, required factors and estimates of δ.

Number ofstandards

Standardsused

R2 kk=(tp+ tq)η1/2

SXX1/2 δ̂

3 1, 6, 10 0.9997 16.312 0.922490 11.9714 1, 4, 8a, 10 0.9997 7.2142 0.984630 12.8385 1, 4, 7, 8a, 10 0.9994 5.6455 0.994578 12.9916 1, 3, 5,

7, 8a, 9a0.9994 4.9987 0.872667 11.363

7 2, 4, 6, 8,8a, 9, 10

0.9993 4.7402 1.06206 13.912

8 1, 2, 3, 4,6, 8, 9, 10

0.9994 4.3167 1.18632 15.509

9 1, 2, 3, 4, 6,7, 8, 9, 10

0.9992 4.1816 1.18667 15.515

10 1, 2, 3, 5, 6,7, 8a, 9, 9a, 10

0.9996 4.1268 1.33775 17.572

11 1, 2, 3, 4, 5, 6,7, 8a, 9, 9a, 10

0.9996 4.0131 1.38047 18.147

12 1, 2, 3, 4, 5, 6,7, 8, 8a, 9, 9a, 10

0.9994 3.9534 1.38074 18.103

112 E. Voigtman, K.T. Abraham / Spectrochimica Acta Part B 66 (2011) 105–113

Clearly, the results are in outstanding agreement in all cases. The 3point calibration design is very broad and monotonically decreasingas xD increases. All the other calibration designs have modes wellabove zero and it is clear that as the number of standards increases,the PDFs becomes narrower, taller and more symmetric. Presumably,if the number of standards were to increase without limit, the xD PDFwould become a delta function at XD, the theoretical content domainCurrie detection limit. This hypothesis has not yet been tested.

4.3. Toward accurate estimates of XD

Accurate estimation of XD is the foremost goal of detection limitinvestigations. The reason is simple: XD is an errorless real number,greater than zero, that is entirely independent of calibration designand, therefore, ideally suited for its purpose of serving as an invariantfigure of merit for comparison purposes. It does depend upon theanalyst's specified probabilities of false positives and false negatives,but these could certainly be standardized by IUPAC or others of thatilk. In the present work, these probabilities were arbitrarily defined as5% each. Note in Table 4, however, that the second column containsestimates of XD, for the ten calibration designs shown in Fig. 10, andthese average 0.2516 mg/100 mL, with a standard deviation of only0.0010 mg/100 mL. Accordingly, the 99% confidence interval for XD is0.2516±0.0010mg R6G/100 mL ethanol. The value of 0.252 plotted asa vertical line in Fig. 10 and as small vertical line in Fig. 9. The widthsof these lines are actually wider than the 99% confidence interval.

4.4. How the XD estimates were obtained

These remarkably consistent estimates were obtained by firstnoting that the theoretical PDFs in Fig. 10 are in excellent agreement

Table 4Estimates of XD and simulation probabilities of false positives and negatives.

Number ofstandards

X̂D

mg/100 mLPercent areabelow X̂D

% false positives (left);% false negatives (right)

3 0.2535 15.9 5.002±0.015; 5.007±0.0164 0.2523 18.8 4.989±0.014; 5.012±0.0265 0.2519 20.5 4.989±0.022; 4.996±0.0226 0.2527 21.7 4.996±0.018; 4.991±0.0267 0.2511 21.5 5.014±0.022; 4.998±0.0208 0.2516 25.8 5.009±0.023; 5.002±0.0239 0.2516 26.5 5.005±0.028; 5.011±0.02610 0.2504 26.2 4.984±0.021; 5.009±0.01211 0.2503 27.7 5.012±0.017; 4.994±0.02212 0.2509 27.7 4.999±0.019; 5.001±0.021

with their corresponding experimental data histograms. Hence, δ̂ ,which was obtained from the bootstrapped experimental tslope data,must be a relatively accurate estimate of δ. Likewise, the agreementbetween computer simulated data and both theory and actualexperimental data was also a direct result of the accuracy of δ̂. But,by definition,

XD ≡ zp + zq� �

σ0 = β ð13Þ

assuming, as we have done above, thatM=1 and α is used for blank-subtraction. Then, if only one measurement is made per standard,σ0=σr, and it is always true that σr=σbSXX

1/2. Therefore

XD = zp + zq� �

σ0 = β = zp + zq� �

σ r = β

= zp + zq� �

σbS1 = 2XX = β = zp + zq

� �S1 = 2XX = δ ð14Þ

The estimate of XD, denoted by X̂D, is obtained merely by sub-stituting δ̂ for δ in Eq. (14), because everything else is constant andknown as soon as the calibration design is set by the analyst, beforeany measurements have even been made. Therefore

XD ≅ X̂D = zp + zq� �

S1=2XX = δ̂ = zp + zq� �

N1=2σX = δ̂

= zp + zq� �

N−1ð Þ1=2sX = δ̂ ð15Þ

using Eq. (4). The values of X̂D for the ten calibration designs shown inFig. 10 are given in Table 4. Note that XDC is similar:

XDC ≅ X̂DC = zpS1=2XX = δ̂ = zpN

1=2σX = δ̂ = zp N−1ð Þ1=2sX = δ̂ ð16Þ

so that if false negatives are of no concern, i.e., if one prefers the“traditional” detection limits, then these are, by definition, simply theCurrie decision levels and Eqs. (8) and (16) are used.

4.5. Ramifications

Precisely as expected, the detection limit obtained in the presentwork is very poor due to the large amount of deliberately added noiseand the underlying unexceptional fluorimeter itself. However, evenif the noise were reduced by several orders of magnitude, thefundamental curves in Fig. 10 would not change except for scaling ofthe axes. In other words, experimental detection limits, xD, arerandom samples from intrinsically broad distributions unless largenumbers of standards are used in construction of the calibrationcurves. Table 4 shows the approximate percent of the area, under therelevant PDFs, that lies below X̂D. For the present system, these areroughly the probabilities that any given experimental detection limitsmight fall below the true theoretical detection limit XD. Clearly, it isconsiderably more probable that a given experimental detection limitwill be above XD, even with twelve standards.

Note that Table 3 also shows the “coverage factor” k for each of thecalibration designs in the present work. These depend on η1/2, andtherefore upon how blank-subtraction is performed. Table 4 showsthat using the properly defined k values in Table 3, which are trivial tocompute, yields excellent agreement between the expected 5%probabilities of false positives and false negatives and the MonteCarlo computer simulation results discussed above.

5. Conclusions

Thanks to bootstrapping, the moderately sized experimental datafiles were used to produce high quality detection limit histograms, often million events each, comparable to those produced by computer

113E. Voigtman, K.T. Abraham / Spectrochimica Acta Part B 66 (2011) 105–113

simulation. All of the histograms were found to be in excellentagreement with previously published theory. From Eqs. (15) and(16), it is then clear that detection limits and decision levels aresimply scaled reciprocals of the non-centrality parameter of themodified non-central t distribution that characterizes univariatelinear calibration curves that have homoscedastic AGWN. Hence,any technique that accurately estimates δ automatically estimatesXC and XD accurately, and thereby solves the general detectionlimit estimation problem for the system described above. What isremarkable is that there is only one parameter that needs to beestimated, and, with hindsight, this should have been obvious:detection limits and decision levels depend only on the quotient ofσ0 and β, not on the individual values. Furthermore, even if blank-subtraction is effected by subtracting an estimate of α, rather than αitself, this just causes η1/2 to be other than unity: η1/2 is still easilycomputed from the calibration design parameters, Eqs. (6) and (7)are controlling and the rest of the analysis proceeds as before,mutatis mutandis.

There can be little doubt, then, that the previously publishedtheory [1–4] provides an accurate description of the fundamentalstatistical behavior of univariate linear chemical measurement sys-tems that are characterized by homoscedastic AGWN. These systemsare ubiquitous in modern analysis and it has been a vexing problemthat Currie's program [5] could not be brought to fruition until now.Many labored to find the key to accurately estimating XD, but therewere too many false leads, statistical errors, notational confusions andcompletely untested assumptions, the latter an irony since computershave been available since before 1968. With hindsight, it is obvious

that the non-central t distribution was lurking at the heart of thewhole matter and it should have been unmasked many years sooner.

The bootstrapping techniquewas crucial to the success of the abovework. When properly used, it is a powerful tool for investigating thedistribution of random variates, which, in the present case, were thexD variates. Bootstrapping experiments are currently in progress toprovide an experimental test of the heteroscedastic theory [3,4] andto clear up the limit of quantitation chaos so clearly reviewed byMermet [7].

Acknowledgements

This research is dedicated to the memory of Professor MitchellEvan “Mitch” Johnson.

References

[1] E. Voigtman, Limits of detection and decision, Part 1, Spectrochim. Acta Part B 63(2008) 115–128.

[2] E. Voigtman, Limits of detection and decision, Part 2, Spectrochim. Acta Part B 63(2008) 129–141.

[3] E. Voigtman, Limits of detection and decision, Part 3, Spectrochim. Acta Part B 63(2008) 142–153.

[4] E. Voigtman, Limits of detection and decision, Part 4, Spectrochim. Acta Part B 63(2008) 154–165.

[5] L.A. Currie, Limits for qualitative and quantitative determination — application toradiochemistry, Anal. Chem. 40 (1968) 586–593.

[6] B. Efron, R.J. Tibshirani, An Introduction to the Bootstrap, 1st Ed Chapman and Hall/CRC, Boca Raton, FL8 © 1994.

[7] J.-M. Mermet, Limit of quantitation in atomic spectrometry: an unambiguousconcept? Spectrochim. Acta Part B 63 (2008) 166–182.