True detection limits in an experimental linearly heteroscedastic system.: Part 2

Despite much different processing of the experimental fluorescence detection data presented in Part 1, essentially the same estimates were obtained for the true theoretical Currie decision levels (Y_C and X_C) and true Currie detection limits (Y_D and X_D). The obtained experimental values, for 5% probability of false positives and 5% probability of false negatives, were Y_C = 56.0 mV, Y_D = 125. mV, X_C = 0.132 μg/mL and X_D = 0.293 μg/mL. For 5% probability of false positives and 1% probability of false negatives, the obtained detection limits were Y_D = 158. mV and X_D = 0.371 μg/mL. Furthermore, by using bootstrapping methodology on the experimental data for the standards and the analytical blank, it was possible to validate previously published experimental domain expressions for the decision levels (y_C and x_C) and detection limits (y_D and x_D). This was demonstrated by testing the generated decision levels and detection limits for their performance in regard to false positives and false negatives. In every case, the obtained numbers of false negatives and false positives were as specified a priori.     

Statistical behavior of ten million experimental detection limits

Using a lab-constructed laser-excited fluorimeter, together with bootstrapping methodology, the authors have generated many millions of experimental linear calibration curves for the detection of rhodamine 6G tetrafluoroborate in ethanol solutions. The detection limits computed from them are in excellent agreement with both previously published theory and with comprehensive Monte Carlo computer simulations. Currie decision levels and Currie detection limits, each in the theoretical, chemical content domain, were found to be simply scaled reciprocals of the non-centrality parameter of the non-central t distribution that characterizes univariate linear calibration curves that have homoscedastic, additive Gaussian white noise. Accurate and precise estimates of the theoretical, content domain Currie detection limit for the experimental system, with 5% (each) probabilities of false positives and false negatives, are presented.     

Legislative limits below detection capability1

 A study of contaminant emission data collected for paper and pulp mills in the United Kingdom over a 1-year period is presented. For most sites and analytes, the data could not demonstrate positive compliance with absolute emission limits to water in the 1 g range in large (million tonne) effluent streams, even though some sites were apparently removing contaminants from their water supply. The implications are discussed, with particular reference to the implications of results quoted at 'detection limits' or 'reporting limits'. Received: 6 February 2000 / Accepted: 27 March 2000     

Limits of detection and decision. Part 1

Extensive Monte Carlo studies of instrumental limits of detection were performed on a simple univariate chemical measurement system having homoscedastic, Gaussian measurement noise and using ordinary least squares (OLS) processing of tens of millions of independent calibration curve data sets. It was found that prediction interval-based experimental detection limits were significantly negatively biased, in both the net response domain and the chemical content domain, resulting in substantially higher rates of false negatives than specified via customary critical t values. The diagnostic fix for the bias problem provided clear proof that hypothesis-based detection limits need not be unique, even as distributions of random variates, if the alternate hypothesis is non-unique. It was also demonstrated that hypothesis-based decision and detection limits have finite support that does not include the region near zero analyte content, so that both have finite moments and finite confidence intervals.     

Kinetic performance limits of constant pressure versus constant flow rate gradient elution separations. Part II: experimental

We report on a first series of experiments comparing the selectivity and the kinetic performance of constant flow rate and constant pressure mode gradient elution separations. Both water-methanol and water-acetonitrile mobile phase mixtures have been considered, as well as different samples and gradient programs. Instrument pressures up to 1200 bar have been used. Neglecting some small possible deviations caused by viscous heating effects, the experiments could confirm the theoretical expectation that both operation modes should lead to identical separation selectivities provided the same mobile phase gradient program is run in reduced volumetric coordinates. Also in agreement with the theoretical expectations, the cP-mode led to a gain in analysis time amounting up to some 17% for linear gradients running from 5 to 95% of organic modifier at ultra-high pressures. Gains of over 25% were obtained for segmented gradients, at least when the flat portions of the gradient program were situated in regions where the gradient composition was the least viscous. Detailed plate height measurements showed that the single difference between the constant flow rate and the constant pressure mode is a (small) difference in efficiency caused by the difference in average flow rate, in turn leading to a different intrinsic band broadening. Separating a phenone sample with a 20-95% water-acetonitrile gradient, the cP-mode leads to gradient plate heights that are some 20-40% smaller than in the cF-mode in the B-term dominated regime, while they are some 5-10% larger in the C-term dominated regime. Considering a separation with sub 2-μm particles on a 350 mm long coupled column, switching to the constant pressure mode allowed to finish the run in 29 instead of in 35 min, while also a larger peak capacity is obtained (going from 334 in the cF-mode to 339 in the cP-mode) and the mutual selectivity between the different peaks is fully retained.     

Detectors and detection limits in INAA

Expressions have been derived which link detection limits in INAA with the Ge-detector specifications, provided by the vendors: relative efficiency, peak-to-Compton ratio and resolution. Situations have been distinguished in which a peak has to be detected on a Compton background, or on a natural background. The expressions allow for a direct indication of the improvement in detection limits, or the shortening of tumaround-time to attain equal detection limits, when a given detector is replaced by a detector with better specifications.     

Detectors and detection limits in INAA

The kinetic behavior of deuteriation of 2-chloro-, 3-chloro- and 4-chlorobenzoic acids in the presence of homogeneous platinum salt catalyst in a medium containing solution of deuteriated acetic acid in heavy water has been studied at 130°C. The quasiunimolecular H/D exchange rate constants for particular position of aromatic ring hydrogens were determined by¹H NMR integration signal.     

Kinetic treatment of unsegmented flow systems : Part 3. Flow-injection system with gradient chamber evaluated with a linearly responding detector

A variable-time kinetic model is evaluated for a flow-injection sample-processing system that includes a gradient chamber. Quantification of triiodide, including reaction with thiosulfate, is used as a model system. A thin-layer electrochemical detector consisting of a platinum working electrode and glassy carbon counter electrode with a 200-mV difference imposed between them yields linear current response for triiodide concentrations between 0.1 and 1.2 mM. Equations based on the kinetic model are evaluated for situations in which neither the flow stream nor gradient chamber contain reactant (thiosulfate) initially, only the flow stream containes reactant, and both the flow stream and gradient chamber contain reactant. Both calibration data and response curves exhibit very good agreement between theory and experiment except for the situation in which only the flow stream contains reactant. In this case, the theoretical treatment accurately predicts the general nature of responses but predicts slightly longer time intervals for completion of the process than are observed experimentally.     

Shape of the proton potential in an intramolecular hydrogen-bonded system. Part II

The crystals of 5,5′-dibromo-3-diethylaminomethyl-2,2′-biphenol N-oxide were studied by X-ray and FT-IR spectroscopy. Within this molecule two short OHO intramolecular hydrogen bonds are formed. The NO?H⁺?O⁻ bond between the OH and the N-oxide groups is very strong, of 2.419(7) Å between the oxygen atoms. The proton potential of this hydrogen bond is flat, broad and has probably no barrier—consequently it could not be located from X-ray diffraction data. The other hydrogen bond formed between two hydroxyl groups appears asymmetrical from FT-IR spectra, and shows also relatively limited proton polarizability. The molecular conformation is non-planar, due to strong overcrowding effect between the oxygen atoms involved in the hydrogen bonds.     

Evaluation of detection limits in emission spectroscopy

Summary The didactical method for defining the practical noise level considered in Part I is supplemented by a treatment that avoids the disruption of the statistical combination of the error due to the graininess of the emulsion and the error originating from intensity fluctuations in the source. The relationships between the basic standard deviations occurring in the alternative treatments are discussed and the inherent detection limits are compared. The procedure finally adopted is demonstrated with examples.

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Zusammenfassung Anlehnend an die didaktische Methode zur Erfassung der praktischen Nachweisgrenze [I. Teil, diese Z. 220, 241 (1966)] bringt die vorliegende Arbeit eine ergänzende Theorie, welche das Unterbrechen des statistischen Zusammenwirkens von Körnigkeitsstreuung und Intensitätsschwankungen in der Lichtquelle umgeht. Die Beziehungen zwischen den Standardabweichungen, welche den beiden Betrachtungsweisen zugrunde liegen, werden diskutiert, und die zugehörigen Nachweisgrenzen werden mit einander verglichen. Das endlich gewählte Verfahren wird an Hand von Beispielen demonstriert.

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Part I: see this journal220, 241 (1966).

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Dedicated to Prof. Dr. H.Kaiser on his 60th birthday.     

Evaluation of detection limits in emission spectroscopy

Summary The paper considers the evaluation of detection limits in photographic emission spectroscopy in case of the analytical line being superimposed on a blank line, caused by either a molecular band or a residual impurity in the electrodes, the admixture, or the atmosphere. The theory developed is a generalization of Kaiser's treatment of detection limits for a smooth background. The theory permits to compare the ideal detection limit, attainable in the absence of a blank with the practical detection limit, reached when a given blank interferes, all other conditions being equal. The relation between the evaluation of the practical detection limit and the determination of the blank concentration (addition method) is discussed.

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Zusammenfassung Die Auswertung von Nachweisgrenzen in der photographischen Emissionsspektroskopie wird für den Fall betrachtet, daß die Analysenlinie einer Blindlinie überlagert ist, die durch eine Molekülbande oder eine Verunreinigung in den Elektroden, im Zusatz oder in der Atmosphäre verursacht wird. Die aufgestellte Theorie ist eine Verallgemeinerung der Kaiserschen Berechnung von Nachweisgrenzen für glatten Untergrund. Sie ermöglicht es, die praktische Nachweisgrenze, erreicht mit einer gegebenen Blindwertstörung, und die ideale Nachweisgrenze, erzielbar ohne Blindwert, aber unter sonst gleichen Bedingungen, miteinander zu vergleichen. Die Beziehung zwischen der Auswertung der praktischen Nachweisgrenze und der Bestimmung der Blindkonzentration (Restgehaltsbestimmung nach dem Zugabeverfahren) wird diskutiert.

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Nomenclature G nominal concentration (or absolute amount) of an analysis element in a sample or standard - G detection limit - ¯G _B average concentration (or absolute amount) of the analysis element in the blank - I photographic intensity (= exposure) - I _L intensity of an analytical line corrected for the background and without a contribution from the blank - I _L antilogarithm of Y _L - I _U intensity of the background at the wavelength of an analytical line - ¯I _U antilogarithm of ¯Y _U - I _LU sum of intensities of analytical line and background - I _B deintensity emitted by a blank at the wavelength of an analytical line - ¯I _B antilogarithm of ¯Y _B - I _B antilogarithm of Y _B - I _BU sum of intensities contributed by blank and background at the wavelength of an analytical line - I _LB intensity at the wavelength of an analytical line resulting from the contributions from the sample and the blank, but corrected for the background - I _LBU sum of intensities I _LB and I _U - Y logarithmic intensity - Y _L logarithm of I _L - Y _L smallest significant value evaluated for log I _L in a sample - Y _U logarithm of I _U - ¯Y _U mean value of log I _U for a series of spectra taken under standard conditions characteristic for the analysis method - Y _LU logarithm of I _LU - Y _LU smallest significant reading of log I _LU in case there is a smooth background - Y _B logarithm of I _B - ¯Y _B mean value of log I _B calculated from measurements in at least twenty blank spectra taken under standards conditions - ¯Y _B minimum value of Y _B defined by equation (20) - Y _BU logarithm of I _BU - Y _LB logarithm of I _LB - Y _LBU logarithm of I _LBU - Y _LBU smallest significant reading of log I _LBU - Y _U Y _L-¯Y_U, noise level in case there is a smooth background, to be connected with a calibration curve normalized to the background [equation (11)] - (Y _U ) _B Y _L-¯Y_U, noise level in case there is a blank, to be connected with a calibration curve normalized to the background [equation (11)] - Y _B Y _L-¯Y_B, noise level in case there is a blank, to be connected with a calibration curve normalized to the blank [equation (36)] - Y _L,U Y _L-¯Y_U, logarithm of intensity of analytical line normalized to the background, i.e. logarithm of intensity ratio I _L/¯I_U - ¯Y _B-¯Y_U, logarithm of ratio ¯I _B/¯I_U - Y_L,B Y_L-¯Y_B logarithm of intensity of analytical line normalized to the blank, i.e. logarithm of intensity ratio I _L/¯I_B - _U standard deviation in logarithmic intensity due to the graininess - _B standard deviation of Y _B - slope of working curve log G vs Y _L [equations (9) and (31)] - K constant in the working curve log G vs Y _L [equations (9) and (31)] - n number of independent readings - P 32 _U - q 3 _B - r ¯I _b/¯I_U The authors acknowledge Prof. Dr. H. Kaiser and his co-workers for their readiness to discuss the subject and for their critical remarks.     

Sample preparation for evaluation of detection limits in X-ray fluorescence spectrometry.

The influence of analyte mass concentration on determination of detection limits in X-ray fluorescence spectrometry has been investigated experimentally. Both the total reflection X-ray fluorescence (TXRF) and the conventional energy-dispersive X-ray fluorescence techniques have been used to derive the dependence of analyte mass concentration on the values of detection limits. Results obtained indicate that values of detection limits are optimum, or in other words, they are closer to the true detection limit of the technique, when analyte concentrations are in the range of 10 times of the detection limit.     

Confidence limits,error bars and method comparison in molecular modeling. Part 1: The calculation of confidence intervals

Computational chemistry is a largely empirical field that makes predictions with substantial uncertainty. And yet the use of standard statistical methods to quantify this uncertainty is often absent from published reports. This article covers the basics of confidence interval estimation for molecular modeling using classical statistics. Alternate approaches such as non-parametric statistics and bootstrapping are discussed.     

True detection limit of rare earth elements in atomic emission analysis with an arc two-jet plasmatron

The “true detection limit” C_L,true was measured by a method proposed by Boumans et al. for 10 prominent lines of La, Ce, Pr, Nd and Sm, the most abundant rare earth elements (REEs) in geological samples. It is demonstrated how spectral interferences can increase of C_L,true. The spectra were excited in an argon arc plasma jet with evaporation powders of five complex natural samples. The data show that there is a significant increase of C_L,true in real samples containing large concentrations of such elements as Ti, Zr, Cr, Nb, Ta and REEs emitting complex spectra.     

True detection limit of rare earth elements in atomic emission analysis with an arc two-jet plasmatron

The “true detection limit” C_L,true was measured by a method proposed by Boumans et al. for 10 prominent lines of La, Ce, Pr, Nd and Sm, the most abundant rare earth elements (REEs) in geological samples. It is demonstrated how spectral interferences can increase of C_L,true. The spectra were excited in an argon arc plasma jet with evaporation powders of five complex natural samples. The data show that there is a significant increase of C_L,true in real samples containing large concentrations of such elements as Ti, Zr, Cr, Nb, Ta and REEs emitting complex spectra. Received: 17 June 1997 / Revised: 20 October 1997 / Accepted: 26 October 1997     

Improvement of detection limits in flow injection analysis

Summary When using a conventional UV/VIS-Spectrophotometer with a large sample compartment as a detector for FIA, flow cells with longer pathlengths can be applied. Despite of a unavoidable band broadening a significant improvement of detection limit can be achieved compared to standard cells. Band broadening is compensated by slight modifications of the manifold, so that the dispersion is in the same range for long path cells and standard flow cells. With a 5 cm flow cell phosphate can be detected down to 0.005 mg/l P.     

Improving detection limits in chromatography by signal averaging

Ensemble or signal averaging has been used to increase detection limits in gas chromatography. The sample is injected several times into a gas chromatograph and the chromatograms generated are summed. The resulting chromatogram is treated as a separate chromatogram with the benefit of an improved signal-to-noise ratio. Use of the technique enables reliable quantitation, the time invested being used to convert an unacceptable signal-to-noise ratio into useful data. A discussion and some applications of the technique are described.     

Is odd-even effect reflected in detection limits?

It is observed for the first time from an extensive compilation that the detection limit ratios for odd/even atomic number neighbour elements are generally less than 1 for nearly all-stable elements of the periodic table (most are even less than 0.5), which is very similar to the odd-even effect in cosmic abundances. This systematic behaviour of detection limits can be explained by a simple relationship with physicochemical and environmental-geological factors. We propose that such a behaviour should be used in future as a fundamental criterion to evaluate the highly variable detection limit data for all chemical elements by a given analytical technique and the estimation of detection limits should be based on at least 30 or more measurements.