Limits of detection and decision. Part 2

Extensive Monte Carlo studies of instrumental limits of detection (LODs) 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 experimental decision and detection limits in the content domain were distributed as scaled reciprocals of noncentral t variates. In the response domain, the decision and detection limits were distributed as scaled χ variates. Rates of false negatives were found to be as expected statistically and no bias was found. However, use of detection limit expressions based on critical values of the noncentrality parameter of the noncentral t distribution were found to be significantly biased, resulting in substantial bias in rates of false negatives.     

Multiphonon energy transfer in atom-surface scattering

The probability density P(ω,q) of transferring the energy ω and the parallel momentum q to the surface is determined by the maximum entropy subject to constraints procedure. The two important constraints are identified as the recoil energy and the mean parallel energy transfer. Having determined P(ω,q) one can evaluate all observable quantities, e.g., the triple differential reflection probability d³σ/dE_tdΩ or the trapping probability. A three-parameter model for d³σ/dE_tdΩ is derived by assuming a parameterized form for the recoil energy. This model may be regarded as an extension of the hard-cube model, because it reduces to the latter if the third parameter, the speed of sound c, is set to infinity. The comparison of the predicted velocity and angular distributions with recent experiments of Hurst et al. is excellent considering the simplicity of the model.     

Tutorial on evaluation of type I and type II errors in chemical analyses: From the analytical detection to authentication of products and process control

Uncertainty is inherent in all experimental determinations. Nevertheless, these measurements are used to make decisions including the performance of the own measurement systems. The link between the decision and the true implicit system that generates the data (measurement system, production process, category of samples, etc.) is a representation of this uncertainty as a probability distribution. This representation leads to the probabilistic formalization of the possibility of making errors. In the context of regulations established by official agencies, it is important to use these statistical decision methods in some cases because the own norm makes them mandatory and, in general, because this is the way of reasonably evaluating whether a working hypothesis is rejected on the basis of the experimental data.The aim of the present tutorial is to introduce some ideas and basic methods for the critical analysis of experimental data. With this goal, the basic elements of the Neyman-Pearson theory of hypothesis testing are formally introduced in connection with the common problems in chemical analysis and, if this is the case, their relation to the norms of regulatory agencies. The notion of decision with ‘enough quality’ is modelled when explicitly considering: (1) the null, H₀, and alternative, H₁, hypotheses. (2) The significance level of the test, which is the probability, α, of rejecting H₀ when it is true, and the power of the test, 1 − β, β being the probability of accepting H₀ when it is false. (3) The difference between H₀ and H₁ that has to be detected with experimental data. (4) The needed sample size. These four concepts should be explicitly defined for each problem and, under the usual assumption of normal distribution of the data, the mathematical relations among these concepts are shown, which allow the analyst to design a decision rule with pre-set values of α and β.To illustrate the unifying character of this inferential methodology, several situations are exposed along the tutorial: the design of a hypothesis test to decide on the performance characteristics of analytical methods, the capability of detection of both quantitative and qualitative analytical methods (including its generalization to the case of multivariate and/or multiway signals), the analytical sensitivity with multivariate signals, the class-modelling and the process control.     

Measurement uncertainty as a tool for evaluating the ‘grey zone’ to reduce the false negatives in immunochemical screening of blood donors for infectious diseases

The risk of misclassifying infected individuals as healthy constitutes a crucial challenge when screening blood donors by means of immunoassays. This risk is especially challenging when the numerical results are close to the clinical decision level, i.e. in the ‘grey zone’. The concept of using measurement uncertainty for evaluating the ‘grey zone’ has previously not been systematically applied in this context. This article explains methods, models and empirical (top-down) approaches for the calculation of measurement uncertainty using results from a blood bank according to the internationally accepted GUM principles, focusing on uncertainty sources in the analytical phase. Of the different approaches available, the intralaboratory empirical approaches are emphasised since modelling (bottom-up) approaches are impracticable due to the lack of reliable model equations for immunoassays. Different methods are applied to estimate the measurement uncertainty for the Abbott Prism^® HCV immunoassay. The expanded uncertainty obtained at the clinical decision level from the intralaboratory empirical approach was 36 %. The estimated uncertainty was used to set acceptance and rejection zones following the procedure set in the Eurachem guideline, emphasising the need to minimise the occurrence of false negatives.     

Identification of the geometrical isomers of α-linolenic acid using gas chromatography/mass spectrometry with a binary decision tree

Gas chromatography, using a highly polar column, low energy (30 eV) electron ionization mass spectrometry and multivariate curve resolution, are combined to obtain the mass spectra of all eight geometrical isomers of α-linolenic acid. A step by step Student's t-test is performed on the m/z 50-294 to identify the m/z by which the geometries of the double bonds could be discriminated. The most intense peak discriminates between cis (m/z 79) and trans (m/z 95) at the central (carbon 12) position. The configuration at carbon 15 is then distinguished by m/z 68 and 236, and finally the geometry at carbon 9 is determined by m/z 93, 173, 191 and 236. A three-question binary tree is developed based on the normalized intensities of these ions by which the identity of any given isomer of α-linolenic is accurately determined. Application of Bayes theorem to data from independent samples shows that the complete configuration is determined correctly with a minimum probability of 87%.     

Determination of uncertainty in analytical measurements from collaborative study results on the analysis of a phenoxymethylpenicillin sample

The correct interpretation of a measurement result requires knowledge about its uncertainty. Depending on the conditions under which the analyst is operating, different operational definitions of uncertainty have been proposed. They include: within-laboratory uncertainty, reproducibility uncertainty, bias-included uncertainty and absolute uncertainty. Here we consider the evaluation of the reproducibility uncertainty derived from the results obtained in an inter-laboratory experiment. Nine laboratories participated in an inter-laboratory study for the analysis of phenoxymethylpenicillin. The analyses consisted of a Karl-Fischer water determination, an acid-base titration to assay phenoxymethylpenicillin and a liquid chromatography (LC) method to determine 4-hydroxyphenoxymethylpenicillin and other impurities. The experimental set-up allowed to obtain for each determination s_r² and s_L² as estimates of the repeatability variance (σ_r²) and the between-laboratory variance (σ_L²), respectively. The reproducibility uncertainties for the different assays were then derived from these estimates.     

Status of uncertainty assessment in k 0-NAA measurement: anything still missing?

Several approaches to quantifying measurement uncertainty in k ₀-based neutron activation analysis (k ₀-NAA) are reviewed, comprising the original approach, the spreadsheet approach, the dedicated computer program involving analytical calculations and the two k ₀-NAA programs available on the market. Two imperfectness in the dedicated programs are identified, their impact assessed and possible improvements presented for a concrete experimental situation. The status of uncertainty assessment in k ₀-NAA is discussed and steps for improvement are recommended. It is concluded that the present magnitude of measurement uncertainty should further be improved by making additional efforts in reducing uncertainties of the relevant nuclear constants used.     

A model for transitional plane Couette flow

A simplified model of plane Couette flow is derived by means of a cross-stream (y) Galerkin expansion in terms of trigonometric functions appropriate for idealized stress-free boundary conditions at the plates. A set of partial differential equations is obtained, governing the in-plane (x–z) space-dependence of a velocity field taken in the form: u=U₀(x,z,t)+[1+U₁(x,z,t)]sin(πy/2), v=V₁(x,z,t)cos(πy/2), w=W₀(x,z,t)+W₁(x,z,t)sin(πy/2). Beyond Lorenz-like Waleffe's modeling (Waleffe 1997), this Swift–Hohenberg type of approach is expected to give an access to the microscopic mechanism of spatiotemporal intermittency typical of the transition to turbulence in plane Couette flow (Pomeau 1986, Bergé et al. 1998).     

Experimental and theoretical examinations of n→σ* negative hyperconjugation in pyridinium dichlorophosphinomethylide

An X-ray crystal structure investigation as well as theoretical DFT (hybrid B3LYP/6-31G** level) calculations of 2-ethylpyridinium dichlorophosphino-ethoxycarbonylmethylide confirm the existence of significant n→σ* negative (anionic) hyperconjugation between the ylidic charge and the σ*_P-Cl bond in the most stable conformer, which has the phosphorus lone pair oriented orthogonally to the ylidic p_z orbital, resulting in elongation of the P-Cl bond parallel to the p_z orbital but shortening of the C-P bond.     

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.     

An improved model of the fringing fields of a quadrupole mass filter

The fringing field region of a quadrupole mass filter is modelled using an iterative finite difference technique to solve Laplace's equation for the electrostatic potential. The results are used to formulate an expression f(z) such that the electrostatic potential of the fringing field, F(x, y, z, t) may be written in the form F(x, y, z, t) = f(z) φ(x, y, t), where φ(x, y, t) is the hyperbolic field of the quadrupole. The function f(z) is expressed in the form f(z) = 1 - exp {- az - bz²}), where a and b are constants and z is in units of r_O, where 2r_O is the spacing of the quadrupole rods. The effect of the distance from the quandrupole rods to the end plate of the quadrupole mass filter, d on f(z) is investigated and the results presented show that for d < 0.125 r_O the function f(z) does not alter significantly.     

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.     

LC method for the determination of R-timolol in S-timolol maleate: Validation of its ability to quantify and uncertainty assessment

This article presents the validation results of a chiral liquid chromatographic (LC) method previously developed for the quantitative determination of R-timolol in S-timolol maleate samples. A novel validation strategy based on the accuracy profiles was used to select the most appropriate regression model, to assess the method accuracy within well defined acceptance limits and to determine the limits of quantitation as well as the concentration range.The validation phase was completed by the investigation of the risk profiles of various acceptable regression models in order to ensure the risk of obtaining the future measurements outside the acceptance limits fixed a priori.On the other hand, the present paper also shows how data used in this validation approach can be used to estimate the measurement uncertainty. The uncertainty derived from β-expectation tolerance interval (), which is equal to the uncertainty of measurements as well as the expanded uncertainty (U_x) using a coverage factor k = 2 was estimated. The uncertainty estimates obtained from validation data were finally compared with those obtained from interlaboratory and robustness studies.     

Dynamic surface tension,critical micelle concentration,and activity coefficients of aqueous solutions of nonyl phenol ethoxylates

Dynamic surface tensions, σ(t) for aqueous solutions of nonyl phenol ethoxylates (NPEOs) at the temperature 298.15 K were measured using a Lauda drop volume tensiometer. The non-ionic surfactants analyzed in this work were Tergitol NP-9, NP-35 and NP-40. By using the classical Ward and Torday equation, the diffusion coefficient for each bulk surfactant concentration was calculated. The equilibrium surface tension values were determined by extrapolating the dynamic surface tension to t → ∞ on the σ(t) vs. t^−1/2 curves. These values were used to determine the critical micelle concentrations (CMC) of the surfactant aqueous solutions as well as to calculate the infinite dilution activity coefficient of the surfactant, following a model that combines the Volmer surface equation of state and the Gibbs adsorption equation.     

GC-MS Studies on Derivatization of Creatinine and Creatine by BSTFA and Their Measurement in Human Urine

In consideration of its relatively constant urinary excretion rate, creatinine (2-amino-1-methyl-5H-imidazol-4-one, MW 113.1) in urine is a useful endogenous biochemical parameter to correct the urinary excretion rate of numerous endogenous and exogenous substances. Reliable measurement of creatinine by gas chromatography (GC)-based methods requires derivatization of its amine and keto groups. Creatinine exists in equilibrium with its open form creatine (methylguanidoacetic acid, MW 131.1), which has a guanidine and a carboxylic group. Trimethylsilylation and trifluoroacetylation of creatinine and creatine are the oldest reported derivatization methods for their GC-mass spectrometry (MS) analysis in human serum using flame- or electron-ionization. We performed GC-MS studies on the derivatization of creatinine (d₀-creatinine), [methylo-²H₃]creatinine (d₃-creatinine, internal standard) and creatine (d₀-creatine) with N,O-bis(trimethylsilyl)trifluoroacetamide (BSTFA) using standard derivatization conditions (60 min, 60 °C), yet in the absence of any base. Reaction products were characterized both in the negative-ion chemical ionization (NICI) and in the positive-ion chemical ionization (PICI) mode. Creatinine and creatine reacted with BSTFA to form several derivatives. Their early eluting N,N,O-tris(trimethylsilyl) derivatives (8.9 min) were found to be useful for the precise and accurate measurement of the sum of creatinine and creatine in human urine (10 µL, up to 20 mM) by selected-ion monitoring (SIM) of m/z 271 (d₀-creatinine/d₀-creatine) and m/z 274 (d₃-creatinine) in the NICI mode. In the PICI mode, SIM of m/z 256, m/z 259, m/z 272 and m/z 275 was performed. BSTFA derivatization of d₀-creatine from a freshly prepared solution in distilled water resulted in formation of two lMate-eluting derivatives (14.08 min, 14.72 min), presumably creatinyl-creatinine, with the creatininyl residue existing in its enol form (14.08 min) and keto form (14.72 min). Our results suggest that BSTFA derivatization does not allow specific analysis of creatine and creatinine by GC-MS. Preliminary analyses suggest that pentafluoropropionic anhydride (PFPA) is also not useful for the measurement of creatinine in the presence of creatine. Both BSTFA and PFPA facilitate the conversion of creatine to creatinine. Specific measurement of creatinine in urine is possible by using pentafluorobenzyl bromide in aqueous acetone.     

Optimization of a technique for the estimation of the composition of edible oil blends

The fatty acid, sterol and tocopherol contents of edible oils were used to determine the formulation of blends of oils, which are themselves complex mixtures, by means of a weighted least-squares estimator with backwards elimination. A t-test was used to determine whether an oil should be eliminated from the blend. Initially, the minimum t-value was set at 2.7, which corresponds to a probability of ca. 0.01 if normal distribution is assumed. Some compounds are present in one oil but cannot be detected in others and are valuable in differentiating between various oils. Assumptions about levels for the undetected compounds and their corresponding weightings were made earlier in order to implement the weighted least-squares technique. A simplex optimization procedure was applied to avoid the use of assumed values. An optimum for the t-value as well as for the levels and the weightings of the undetected compounds was established. Analytical results for 378 pure oils were used to develop the models. The optimized technique is slightly better for assessing the composition of unknown blends than the original technique, when tested on 93 samples which were blends of known amounts of sunflower-seed oil, groundnut oil, cottonseed oil, maize oil, olive oil and palm oil.     

Limits of detection and decision. Part 3

It has been shown that the MARLAP (Multi-Agency Radiological Laboratory Analytical Protocols) for estimating the Currie detection limit, which is based on ‘critical values of the non-centrality parameter of the non-central t distribution’, is intrinsically biased, even if no calibration curve or regression is used. This completed the refutation of the method, begun in Part 2. With the field cleared of obstructions, the true theory underlying Currie's limits of decision, detection and quantification, as they apply in a simple linear chemical measurement system (CMS) having heteroscedastic, Gaussian measurement noise and using weighted least squares (WLS) processing, was then derived. Extensive Monte Carlo simulations were performed, on 900 million independent calibration curves, for linear, “hockey stick” and quadratic noise precision models (NPMs). With errorless NPM parameters, all the simulation results were found to be in excellent agreement with the derived theoretical expressions. Even with as much as 30% noise on all of the relevant NPM parameters, the worst absolute errors in rates of false positives and false negatives, was only 0.3%.     

Room-temperature nickel-catalyzed amination of heteroaryl/aryl chlorides with Ni(II)-(σ-Aryl) complex as precatalyst

The room-temperature cross-coupling of heteroaryl and aryl chlorides with secondary cyclic amines can be effected using Ni(II)-(σ-aryl) complex as pre-catalyst. Some useful aromatic and heteroaromatic amine derivatives were readily synthesized in moderate to good yields in the presence of the Ni(II)-(σ-aryl) complex/NHC/KO^tBu/toluene system.     

Distribution of polymer chain lengths inside a spherical volume: Solution in terms of a diffusion problem

The probability W(t) that a given number t of segments of an infinite chain lie within a given sphere can be expressed in terms of the single-pass length probability and the probability of reentrance into the sphere. The problem of calculating these two probabilities is equivalent to that of a diffusing particle exiting or entering the sphere after a given time, when the surface of the sphere is an absorbing wall. It is shown that the boundary condition, c = 0, usually applied to an absorbing surface cannot be used for the present purpose. The boundary condition used instead is the so-called radiation condition ?c/?z = kc; it is shown that when k approaches infinity the final answer for W(t), which is given in the form of an infinite series, approaches the correct limit. In this same limit the ratio 〈t〉²/〈t〉² has the value 2.4     

Study of peak shape and efficiency in butyl acrylate-based monolithic columns for capillary electrochromatography

The study and modelling peak shape of in capillary electrochromatography (CEC), obtained using butyl acrylate (BA)-based monolithic columns, is described in this article. A modified-Gaussian model, which is a function of conventional experimental parameters: retention time (t_R), peak height (H₀) and standard deviation (σ₀) at the peak maximum, and left (A) and right (B) halfwidths, was used to describe the peaks of thiourea and several polyaromatic hydrocarbons compounds, which were eluted under several applied voltages. A mean relative error below 2% in the prediction of peak shape is obtained. Based on peak parameters, an easy and reliable estimation of global chromatographic performance, efficiency and peak capacity of BA-based monoliths was also considered. A comparison in terms of chromatographic performance of BA-based monoliths in CEC with CE mode and RPLC microparticulate columns was also performed.