The effects of errors in measuring the independent variable in least-squares regression analysis

It is well known that linear regression analysis gives unbiased estimates of the slope and intercept of a straight line if the dependent variable y is subject to random errors of measurement while the independent variable x is not. It is much less well known that, if x is also subject to random errors of measurement, linear regression analysis yields an underestimate of the slope and a correspondingly biased estimate of the intercept. These errors cannot be removed by weighted regression. Similar errors arise in non-linear regression when the independent variable is afflicted by random errors of measurement.     

The quality coefficient as performance assessment parameter of straight line calibration curves in relationship with the number of calibration points

Knowledge of the response function (y?=?f(x)) is essential in the validation of quantitative analysis methods as it describes the mathematical relationship between measurable responses and the concentrations or quantities of the analyte in the sample within a suitable range. The most common response function used is a straight line obtained by ordinary least squares (OLS) regression. Suitability of calibration lines obtained by OLS regression might be verified by calculation of a quality coefficient (QC_mean). Mathematical modelling performed previously showed that with respect to critical limit values for g, which controls the symmetry of the prediction interval of the abscissa value obtained from the confidence intervals around the OLS calibration curve, a corresponding quality coefficient value exists as a quality performance parameter which is related to the spread of the abscissa values around their mean. In this paper, new mathematical models are developed to demonstrate to which extend also the number n of calibration points (x _i ,y _i ) defines the required value for the quality coefficient (QC_mean) for different values of g. From these models, it could be established that the attribution of a critical limit value to QC_mean as a performance parameter for straight line calibration cannot be arbitrary chosen but has to rely on the mathematical model relating QC_mean, the g-value, the number n of calibration points and the spread of the x _i -values around their mean. Practical measures for analysts are provided which tend to lower the g-value of straight calibration lines beneath critical values and enable to improve the quality of the calibration line applied for analysis, as demonstrated in an elaborated example.     

Quantitative analysis of the three-dimensional trap stiffness of a dielectrophoretic corral trap

Dielectrophoresis is a robust approach for the manipulation and separation of (bio)particles using microfluidic platforms. We developed a dielectrophoretic corral trap in a microfluidic device that utilizes negative dielectrophoresis to capture single spherical polystyrene particles. Circular-shaped micron-size traps were employed inside the device and the three-dimensional trap stiffness (restoring trapping force from equilibrium trapping location) was analyzed using 4.42 μm particles and 1 MHz of an alternating electric field from 6 V_P-P to 10 V_P-P. The trap stiffness increased exponentially in the x- and y-direction, and linearly in the z-direction. Image analysis of the trapped particle movements revealed that the trap stiffness is increased 608.4, 539.3, and 79.7% by increasing the voltage from 6 V_P-P to 10 V_P-P in the x-, y-, and z-direction, respectively. The trap stiffness calculated from a finite element simulation of the device confirmed the experimental results. This analysis provides important insights to predict the trapping location, strength of the trapping, and optimum geometry for single particle trapping and its applications such as single-molecule analysis and drug discovery.     

Numerical observation of preferred directionality in ion ejection from stretched rectilinear ion traps

We report on numerical investigations of directionality of ion ejection in stretched rectilinear ion traps (RIT). Three 4-electrode trap geometries have been investigated. In all cases, one pair of electrodes has slits at their center and the other pair has no slits. The studied traps include the RIT-S, in which the mass analyzer electrodes are symmetrically positioned around the central axis; the RIT-X, in which the mass analyzer has a stretch in the direction of the electrodes which have slits (labeled as x-direction); and the RIT-Y, in which the mass analyzer has a stretch in the direction of the electrodes which have no slits (labeled as y-direction).Our analysis has been carried out on two-dimensional (2D) fields at the centre of an infinitely long mass analyzer. The boundary element method (BEM) has been used for field computations. The trajectory of ion motion has been generated using Runge Kutta fourth order integration.Three sets of simulations have been carried out on each of the RIT-S, the RIT-X and the RIT-Y to check for directionality of ion ejection. In the first, we numerically obtain the stability region on the potential (U_dc– V_rf) axes. In the second we generate an escape velocity plot with U_dc=0 for different values of V_rf. In the third, we simulate the mass selective boundary ejection experiment on a single ion.In the symmetric RIT-S, as expected, all three simulations show that there is an equal probability of ion reaching the trap boundary in either of the x- or y-directions. For the stretched traps, however, the results are dramatically different. For the RIT-X, all three simulations suggest that ion destabilization at the stability boundary occurs in the x-direction. Similarly, for the RIT-Y, ions preferentially get destabilized in the y-direction. That is, ions reaching the trap boundary overwhelmingly prefer the stretch direction.     

Reliability of measurements of pesticide residues in food

This paper accounts for the major sources of errors associated with pesticide residue analysis and illustrates their magnitude based on the currently available information. The sampling, sample processing and analysis may significantly influence the uncertainty and accuracy of analytical data. Their combined effects should be considered in deciding on the reliability of the results. In the case of plant material, the average random sampling (coefficient of variation, CV=28–40%) and sample processing (CV up to 100%) errors are significant components of the combined uncertainty of the results. The average relative uncertainty of the analytical phase alone is about 17–25% in the usual 0.01–10 mg/kg concentration range. The major contributor to this error can be the gas-liquid chromatography (GLC) or high-performance liquid chromatography (HPLC) analysis especially close to the lowest calibrated level. The expectable minimum of the combined relative standard uncertainty of the pesticide residue analytical results is in the range of 33–49% depending on the sample size.The gross and systematic errors may be much larger than the random error. Special attention is required to obtain representative random samples and to eliminate the loss of residues during sample preparation and processing.     

Supplement on applicability of the Kissinger equation in thermal analysis

The relative errors (e%) in the determination of the activation energy from the slope of the Kissinger straight line ln(β/βT _p²) vs. 1/T _p (β is the heating rate) are in-depth discussion. Our work shows that the relative errors is a function containing the factors of x _p and Δx _p, not only x _p (x _p = E/RT _p, E is the activation energy, T _p is the temperature corresponding to maximum process rate, R is the gas constant). The relative error between E _k and E _p will be smaller with the increase of the value of x and/or with the decrease of the value of Δx. For a set of different heating rates in thermal analysis experiments, the low and close heating rates are proposed from the kinetic theory.     

Assessment of quality performance parameters for straight line calibration curves related to the spread of the abscissa values around their mean

In validation of quantitative analysis methods, knowledge of the response function is essential as it describes, within the range of application, the existing relationship between the response (the measurement signal) and the concentration or quantity of the analyte in the sample. The most common response function used is obtained by simple linear regression, estimating the regression parameters slope and intercept by the least squares method as general fitting method. The assumption in this fitting is that the response variance is a constant, whatever the concentrations within the range examined.The straight calibration line may perform unacceptably due to the presence of outliers or unexpected curvature of the line. Checking the suitability of calibration lines might be performed by calculation of a well-defined quality coefficient based on a constant standard deviation.The concentration value for a test sample calculated by interpolation from the least squares line is of little value unless it is accompanied by an estimate of its random variation expressed by a confidence interval. This confidence interval results from the uncertainty in the measurement signal, combined with the confidence interval for the regression line at that measurement signal and is characterized by a standard deviation s_x0 calculated by an approximate equation. This approximate equation is only valid when the mathematical function, calculating a characteristic value g from specific regression line parameters as the slope, the standard error of the estimate and the spread of the abscissa values around their mean, is below a critical value as described in literature.It is mathematically demonstrated that with respect to this critical limit value for g, the proposed value for the quality coefficient applied as a suitability check for the linear regression line as calibration function, depends only on the number of calibration points and the spread of the abscissa values around their mean.     

Background elimination in three-dimensional spectroscopy using the intensian method

The method to eliminate background in the case of quantitative multidimensional spectroscopy, chromatography or any analytical 3-dimensional technique is shown. The 3-dimensional signal is required to be proportional to the concentration of determined substance and the additivity of signals should be obeyed. Eliminated background is assumed to be a low-order polynomial of two variables. The intensian method [1] is a generalization of the Beer-Lambert law, where a certain determinant called intensian replaces absorption and absorptivity. In practice there will be no need to use determinants, since usually they are replaced by expressions of few terms. Some details on the practical use of the method are given.Index of used symbols x, y UV, IR, GC, NMR or other scale. - A (x, y) intensity (absorption) of the 3-dimensional band of interest. - a (x, y) standard intensity (absorption) of the 3-dimensional band of interest. - B(x, y) intensity (absorption) of 3-dimensional background. - S(tx, y) intensity (absorption) of 3-dimensional multicomponent spectrum. - f (x, y) auxiliary function:f (x, y) =A (x, y), B(x, y), a(x, y), S(x, y). - (x _i, Yi) selected point,i positive integer number. - f(xi, yi) value off (x, y) in point (xi, yi). - S _i value ofS(x, y) in point (x _i, y_i). - b pathlength, measurement coefficient,c concentration. - , , , real numbers. - ij, _ij real coefficients of power expansions. - x ⁱy^j monomial of degree (i +j). - F(·) linear functional acting on 3-dimensional spectral functions. - J^3-dim(·) 3-dimensional intensian acting on 3-dimensional spectral functions. - J _n ^3-dim (·) 3-dimensionaln-points intensian. - d _i ith intensian coefficient, cofactor of expansion of J _n ^3-dim (f(x, y)) according to its first row (eq. (10)). - (.) absolute error. - r _i,, R random variables: eqs. (13) and (14). - G(.) (normal) distribution function. - z ordinate axis. - a - f , h abreviations for some arguments. - d _ijk,D _mnp abreviations defined in eq. (22).     

Errors in isotope ratios measured on a laser mass spectrometer with photographic recording

The random and systematic measurement errors were determined for tin isotope ratios measured by laser mass spectrometry with photographic recording. The analytical isotope signals were treated by the Hull equation using the widths of mass-spectrometric lines. This method significantly reduced the systematic measurement error in the isotope ratios and extended the working range of signal intensities (0.1 <T< 10). Within this range, the relative standard deviation (s _r ) of the measured isotope ratios wass _r < 0.15, and the relative systematic error was S < 0.15. For isotope lines with close signal intensities in the region of normal blackening of the photographic emulsion, the valuess _r = 0.04-0.08 and δ = 0.01 were obtained. A discrimination effect was revealed for isotopes with large masses 120, 122, and 124 amu, which increased δ to 0.21     

Miscibility of poly (styrene-co-methylmethacrylate)s differing in composition

The mutual miscibility of random poly (styrene-co-methylmethacrylate)s with different compositions but a constant molecular weight of M_w ? 150,000 was studied at room temperature and at 180°C. Compatibility was analyzed with films cast from solutions with different solvents. The reliability of the analytical technique is discussed. The miscibility windows {x, y_x}, which define the limits of miscibility of the blends of a given copolymer P (S_xMMA_1?x) with other copolymers P (S_yMMA_1?y), were determined for all x. The widths |x ? y_x| of these windows are, contrary to the predictions of the Flory–Huggins model, different for y_x > x and y_x > x, and depend strongly on x. Miscibility is better for blends with a high MMA content. The windows are markedly wider at room temperature. Many blends are, therefore, “semicompatible,” i.e., have a critical point of the LCST type between room temperature and 180°C.     

The accurate determination of major components in GaxSey by means of instrumental neutron activation

The various phases of the major component analysis of Ga_xSe_y by means of instrumental neutron activation were examined. The overall uncertainty for each element was of the order of 0.1% of the final concentration, and no indication of systematic error was found between neutron activation analysis and the comparative method, i.e. wavelength-dispersive X-ray fluorescence analysis. The pronounced reduction in the overall uncertainty was made possible by an improved version of a NaI(T1) detector (scintillation chamber) and by rotation of the samples during irradiation.     

Monitoring the precision of routine analyses by using duplicate determinations

To ensure the reliability of results, analytical laboratories require a continuous qualitycontrol program which must take account of both systematic and random errors. Analyses of reference materials can be used to estimate systematic errors but estimates of random errors (precision) tend to be optimistic, mainly because reference materials cannot be put through the whole analytical process (e.g., primary sampling is often a major source of error). Estimates of precision must be based on routine samples. If duplicate determinations are done on routine samples, the precision can be estimated reliably. Within the optimum concentration range of analytical method (usually starting from 5-10 times the detection limit), the relative standard deviation (s_r can be regarded as being almost constant or independent of concentration. The precision can then be estimated by first calculating the s_r value of each pair of results. Individually, these are not reliable estimates of the true s_r, but they can be regarded as independent measurements of the same s_r and so can be pooled to obtain a more reliable estimate of precision with the number of duplicates as the degrees of freedom. The applicabiilty of the method is tested on soil, rock and ore samples.     

Prediction of wood property in Chinese Fir based on visible/near-infrared spectroscopy and least square-support vector machine

A method for the quantification of density of Chinese Fir samples based on visible/near-infrared (vis–NIR) spectrometry and least squares-support vector machine (LS-SVM) was proposed. Sample set partitioning based on joint x–y distances (SPXY) algorithm was used for dividing calibration and prediction samples, it is of value for prediction of property involving complex matrices. A stepwise procedure is employed to select samples according to their differences in both x (instrumental responses) and y (predicted parameter) spaces. For comparison, the models were also constructed by Kennard–Stone method, as well as by using the duplex and random sampling methods for subset partitioning. The results revealed that the SPXY algorithm may be an advantageous alternative to the other three strategies. To validate the reliability of LS-SVM, comparisons were made among other modeling methods such as support vector machine (SVM) and partial least squares (PLS) regression. Satisfactory models were built using LS-SVM, with lower prediction errors and superior performance in relation to SVM and PLS. These results showed possibility of building robust models to quantify the density of Chinese Fir using near-infrared spectroscopy and LS-SVM combined SPXY algorithm as a nonlinear multivariate calibration procedure.     

Oscillations in Lotka-Volterra systems of chemical reactions

For a chemical reaction system modeled byx =k ₁ Ax –k ₂ x ² –k ₃ xy +k ₄ y ²,y =k ₃ xy –k ₄ y ² –k ₅ y +k ₆ B, it is shown that for each positive choice of parametersk _i A, B there exists a unique stationary state which is globally asymptotically stable in the positive quadrant. A criterion for the non-existence of periodic solutions is given for the generalized Lotka-Volterra system:x = f(x)h(x, y),y.     

The susceptibility of calibration methods to errors in the analytical signal

Calibration methods differ as to the number and concentration of calibration standards, and whether these are added to the samples or separate from them. The four main calibration mehods (single separate or added standard and multiple separate or added standards) and some modifications are desribed mathematically and subjected to error-propagation analysis, to examine the likely effects of errors in the analytical signal on the overall accuracy and precision of the concentration estimate. Comparison of the results throws light on the influence of the number, concentration and nature of the calibration standards, the effects of sample and standard replication, and the costs and the benefits of blank measurements. It is shown that all standard-addition methods are immune to proportional signal error, but more sensitive to nonlinearity. In separate-standard methods, all bias disappears when the true sample concentration (x_s) is equal to the standard concentration or to the mean standard concentration ($\text{x}$). Only the multiple separate standard method is unaffected by constant error common to sample and standards, without blanking. In multiple-standard methods, precision is best at x_s = $\text{x}$. Precision is always improved by increasing the number of sample and standard measurements; standard-addition methods respond best to sample replication. Whichever calibration method is used, recovery correction will eliminate proportional concentration error, at the cost of decreased precision.     

Determination of low concentrations by spectrophotometry: Uranium-thiocyanate system

Errors in the determination of low concentrations by spectrophotometry are investigated with the uranium-thiocyanate system as an example. The reagent blank has significant absorption and measurements are made at 375 nm instead of the λ_max. The error in the intercept of the calibration curve is an important factor in such measurements and the errors involved in the estimation of 1 μg/ml (normal working range 4–40μg/ml) are studied. It is shown that both random and systematic errors associated with the intercept are responsible for observed errors. The two types of errors are resolved by ANOVA (analysis of variance). The error in the measurement of a single value is estimated and compared with measured values for different calibration ranges. It is seen that two factors predominantly influence the error in the measured concentration — the variance from regression and the closeness of the measured value to the mean of the calibration range.     

High accuracy volumetric determination of hydrogen in rare earth hydrides

An improved volumetric determination of the hydrogen content of rare earth hydrides is described and its accuracy is compared with determinations using PVT and gravimetric (hydrogen uptake) methods. Increased experimental accuracy and calibration with pure metal samples give a reproducibility error of only Δx = ±0.008 for CeH₃ and CeH_x. An even smaller error (Δx = ±0.005) is obtained for pure lanthanum and cerium metals during calibration. Compared with this we estimate the error of the PVT method to be Δx ⩽ 0.1 and of the gravimetric method to be Δx ≈ ±0.03 even with extreme purity precautions.     

Effects of combined substitution of Dy and Mo or Si on the crystal structures and magnetic properties of Nd1?yDyyFe11?xTiMx (M = Mo, Si) compounds

A series of Nd_1−y Dy _y Fe_11−x TiM_x (M = Mo, Si) alloys have been prepared by arc melting and studied by X-ray diffraction and neutron diffraction. All samples are found to crystallize in the ThMn₁₂-type structure. The lattice parameters a and c and unit cell volume V of Nd_0.5Dy_0.5Fe_11−x TiMo _x alloys increase linearly with increasing content of Mo (x), while the lattice parameters a and c and unit cell volume V of Nd_0.5Dy_0.5Fe_11−x TiSi _x alloys decrease linearly with increasing content of Si(x). In Nd _y Dy_1−y Fe_11−x TiM _x (M = Mo, Si) compounds, Ti and Mo atoms preferentially occupy the 8i sites and Si atoms preferentially occupy the 8j and 8f sites. Magnetic measurements show that the substitution of Fe by either Mo or Si leads to a decrease in the Curie temperature. Supported by the National Natural Science Foundation of China (Grant No. 20775088) and the Foundation of State Key Laboratory of Environmental Chemistry and Ecotoxicology, Research Center for Eco-Environmental Sciences, Chinese Academy of Sciences (Grant No. KF2008-06)