Robust and non-parametric statistics in the evaluation of figures of merit of analytical methods. Practices for students

A set of laboratory practices is proposed in which evaluation of the quality of the analytical measurements is incorporated explicitly by applying systematically suitable methodology for extracting the useful information contained in chemical data. Non-parametric and robust techniques useful for detecting outliers have been used to evaluate different figures of merit in the validation and optimization of analytical methods. In particular, they are used for determination of the capability of detection according to ISO 11843 and IUPAC and for determination of linear range, for assessment of the response surface fitted using an experimental design to optimize an instrumental technique, and for analysis of a proficiency test carried out by different groups of students. The tools used are robust regression, least median of squares (LMS) regression, and some robust estimators as median absolute deviation (m.a.d.) or Huber estimator, which are very useful as an alternatives to the usual centralization and dispersion estimators.     

On the construction of experimental designs for a given task by jointly optimizing several quality criteria: Pareto-optimal experimental designs

Experimental designs for a given task should be selected on the base of the problem being solved and of some criteria that measure their quality. There are several such criteria because there are several aspects to be taken into account when making a choice. The most used criteria are probably the so-called alphabetical optimality criteria (for example, the A-, E-, and D-criteria related to the joint estimation of the coefficients, or the I- and G-criteria related to the prediction variance). Selecting a proper design to solve a problem implies finding a balance among these several criteria that measure the performance of the design in different aspects. Technically this is a problem of multi-criteria optimization, which can be tackled from different views.     

An assisted approach of the global optimization for the experimental conditions in capillary electrophoresis

Experimental conditions have effect on the separation of capillary electrophoresis (CE) directly. In this work, a set of index to describe the separation in CE was established properly. Based on a combination of genetic algorithm and least square support vector machine, an assisted approach of global optimization for experimental conditions was proposed for the first time, and it was applied to the separation of four synthetic compounds by CE in nonaqueous system. Under the optimum conditions obtained by this approach, the result of the experiment was satisfactory and proved that this novel approach was effective. Furthermore, we investigated the most important conditions that mainly affect the separation effectiveness of CE by partial least squares regression analysis. Because of the generalization of this new approach proposed, it can be applied to the optimization of other experimental processes.     

Improvement of multivariate calibration techniques applied to 1-to-N component mixtures through an optimized experimental design

This work describes a novel experimental design aimed at building a calibration set constituted by samples containing a different number of components. The algorithm performs a reiteration process to maintain the number of samples at the lower value as possible and to ensure an homogeneous presence of all the concentration levels. The mixture design was applied to a drug system composed by one-to-four components in different combination. The resolution of the system was performed by three multivariate UV spectrophotometric methods utilizing principal component regression (PCR) and partial last squares (PLS1 and PLS2) algorithms. The calibration set was composed by 61 references on four concentration levels, including 15 samples for each quaternary, ternary and binary composition and 16 one-component samples. The calibration models were optimized through a careful selection of number of factors and wavelength zones, in such a way as to remove interferences from instrumental noise and excipients present in the pharmaceutical formulations. The prediction power of the regression models were verified and compared by analysis of an external prediction set. The models were finally used to assay pharmaceutical specialities containing the studied drugs in one-to-four formulations.     

Sum of ranking differences (SRD) to ensemble multivariate calibration model merits for tuning parameter selection and comparing calibration methods

Most multivariate calibration methods require selection of tuning parameters, such as partial least squares (PLS) or the Tikhonov regularization variant ridge regression (RR). Tuning parameter values determine the direction and magnitude of respective model vectors thereby setting the resultant predication abilities of the model vectors. Simultaneously, tuning parameter values establish the corresponding bias/variance and the underlying selectivity/sensitivity tradeoffs. Selection of the final tuning parameter is often accomplished through some form of cross-validation and the resultant root mean square error of cross-validation (RMSECV) values are evaluated. However, selection of a “good” tuning parameter with this one model evaluation merit is almost impossible. Including additional model merits assists tuning parameter selection to provide better balanced models as well as allowing for a reasonable comparison between calibration methods. Using multiple merits requires decisions to be made on how to combine and weight the merits into an information criterion. An abundance of options are possible. Presented in this paper is the sum of ranking differences (SRD) to ensemble a collection of model evaluation merits varying across tuning parameters. It is shown that the SRD consensus ranking of model tuning parameters allows automatic selection of the final model, or a collection of models if so desired. Essentially, the user’s preference for the degree of balance between bias and variance ultimately decides the merits used in SRD and hence, the tuning parameter values ranked lowest by SRD for automatic selection. The SRD process is also shown to allow simultaneous comparison of different calibration methods for a particular data set in conjunction with tuning parameter selection. Because SRD evaluates consistency across multiple merits, decisions on how to combine and weight merits are avoided. To demonstrate the utility of SRD, a near infrared spectral data set and a quantitative structure activity relationship (QSAR) data set are evaluated using PLS and RR.     

Multivariate analytical sensitivity in the determination of selenium, copper, lead and cadmium by stripping voltammetry when using soft calibration

A new approach to the multivariate sensitivity concept based on the determination of the capability of discrimination of a method of analysis is shown. Thus the analytical sensitivity is defined in this work by the analyte concentration that a analytical method is able to discriminate, which implies the estimation of the ‘false noncompliance’ and ‘false compliance’. In this approach the estimation of the multivariate analytical sensitivity is independent of scale factors and calibration models, and allows one to study the behavior of a analytical method for several concentrations and matrix. The estimation of this parameter in the simultaneous determination of selenium, copper, lead and cadmium by stripping voltammetry when using soft calibration is carried out, showing that different multivariate analytical sensitivities are obtained for each metal.     

A computer program for searching the best model for describing different experimental systems

An algorithm for searching the best polynomial analytical function for describing different experimental systems is presented. It is based

1. (1)on the generation of all possible analytical functions of a given order, with a given number of terms and with a given number of independent variables, and

2. (2)on the calculation of the parameters of all selected functions using the linear regression method.

To show the ability of the program two different examples are given:

1. (1) searching the best univariate polynomial model, and

2. (2) modelling of the stability of a two-component mixture as a function of three factors.

Robust regression and outlier detection in the evaluation of robustness tests with different experimental designs

Robustness tests are usually based on an experimental design approach. As designed experiments generally lead to a large variability among the results, erroneous results are often not readily detected. As a consequence, the ordinary least squares (OLS) estimates of the effects of the robustness test can be biased. Here, two robustness tests are studied, which both contain a suspicious result. Moreover, simulated datasets are considered to examine the influence of the extent of the outlier as well as the influence of multiple outliers. On the one hand, different methods are applied to inspect the results of the experiments for outliers: the half-normal plot of the OLS residuals, the normal probability plot of the effects and a method, which is based on experimental design reconstruction. On the other hand, two robust regression methods are applied to calculate the effects with a minimum influence of possible outliers. The different methods are compared and it is evaluated under which circumstances they can be applied.     

Comparison between ordinary least squares regression and weighted least squares regression in the calibration of metals present in human milk determined by ICP-OES

In this study we compared the use of ordinary least squares and weighted least squares in the calibration of the method for analyzing essential and toxic metals present in human milk by ICP-OES, in order to avoid systematic errors in the measurements used. Human milk samples were provided by maternity clinic Odete Valadares and digested by means of a high-performance microwave (MW) oven. Evaluation of plasma short and long-term stability was made using a solution of digested milk (1:50) with 2.0 mg L⁻¹ Mg in HNO₃ 2% (v/v). The detection power resulted to be at or below the μg L⁻¹ level, whilst the precision expressed as relative standard deviation R.S.D. was almost always equal to or better than 3.3%. Certified reference material Infant Formula (NIST SRM 1846) was used to assess the accuracy of the proposed method, which proved to be accurate and precise. Recovery rates were in the range of 83-117%. Aqueous calibration was carried out for each element under study.     

Robust regression techniques A useful alternative for the detection of outlier data in chemical analysis

The validation of an analytical procedure means the evaluation of some performance criteria such as accuracy, sensitivity, linear range, capability of detection, selectivity, calibration curve, etc. This implies the use of different statistical methodologies, some of them related with statistical regression techniques, which may be robust or not. The presence of outlier data has a significant effect on the determination of sensitivity, linear range or capability of detection amongst others, when these figures of merit are evaluated with non-robust methodologies.In this paper some of the robust methods used for calibration in analytical chemistry are reviewed: the Huber M-estimator; the Andrews, Tukey and Welsh GM-estimators; the fuzzy estimators; the constrained M-estimators, CM; the least trimmed squares, LTS. The paper also shows that the mathematical properties of the least median squares (LMS) regression can be of great interest in the detection of outlier data in chemical analysis. A comparative analysis is made of the results obtained by applying these regression methods to synthetic and real data. There is also a review of some applications where this robust regression works in a suitable and simple way that proves very useful to secure an objective detection of outliers. The use of a robust regression is recommended in ISO 5725-5.     

Fusion strategies for selecting multiple tuning parameters for multivariate calibration and other penalty based processes: A model updating application for pharmaceutical analysis

New multivariate calibration methods and other processes are being developed that require selection of multiple tuning parameter (penalty) values to form the final model. With one or more tuning parameters, using only one measure of model quality to select final tuning parameter values is not sufficient. Optimization of several model quality measures is challenging. Thus, three fusion ranking methods are investigated for simultaneous assessment of multiple measures of model quality for selecting tuning parameter values. One is a supervised learning fusion rule named sum of ranking differences (SRD). The other two are non-supervised learning processes based on the sum and median operations. The effect of the number of models evaluated on the three fusion rules are also evaluated using three procedures. One procedure uses all models from all possible combinations of the tuning parameters. To reduce the number of models evaluated, an iterative process (only applicable to SRD) is applied and thresholding a model quality measure before applying the fusion rules is also used. A near infrared pharmaceutical data set requiring model updating is used to evaluate the three fusion rules. In this case, calibration of the primary conditions is for the active pharmaceutical ingredient (API) of tablets produced in a laboratory. The secondary conditions for calibration updating is for tablets produced in the full batch setting. Two model updating processes requiring selection of two unique tuning parameter values are studied. One is based on Tikhonov regularization (TR) and the other is a variation of partial least squares (PLS). The three fusion methods are shown to provide equivalent and acceptable results allowing automatic selection of the tuning parameter values. Best tuning parameter values are selected when model quality measures used with the fusion rules are for the small secondary sample set used to form the updated models. In this model updating situation, evaluation of all possible models, thresholding, and iterative SRD performed equivalently for the three fusion rules with TR and PLS performed worse. While the application is model updating, the fusion processes are applicable to other situations requiring selection of multiple tuning parameter values.     

Randomization tests to identify significant effects in experimental designs for robustness testing

The screening designs applied in robustness tests are usually fractional factorial or Plackett-Burman designs. Different methods to identify significant factor effects estimated from experimental designs for robustness testing are described. In this paper, the use of randomization tests as a statistical interpretation method is examined and compared with both graphical (half-normal probability plot) and statistical methods, such as the estimation of error based on a priori considered negligible effects and the algorithm of Dong. It was found that all statistical methods usually gave similar results, i.e. the same effects are found to be significant. However, sometimes randomization tests indicate either less or more significant factor effects compared to the other methods, regardless the design size. Both randomization tests and the algorithm of Dong become unreliable when about 50% of the examined factors are significant. In such situation, it is advisable to perform an experimental design from which enough negligible effects can be estimated. The graphical interpretation method did not always succeed in indicating the correct number of significant effects.     

Use of factorial experimental design for the rapid evaluation of main and interactive factors affecting linearity in calibration curves for sulfate analysis by ion chromatography

Analytical results of anion determination by suppressed ion chromatography are significantly affected by calibration curve calculation. In this paper, as expected, eluent pK_a is shown to influence calibration linearity in the range 1–20 mg/l sulfate, with A carbonate-hydrogencarbonate mixture producing a larger non-linearity than NaOH. Evidence is given for very large errors (about 30–40%) in estimating sample sulfate concentration when linear regression is used instead of a quadratic calibration curve. This study was performed following a 2⁴ run full factorial experimental design, including eluent pK_a, counterion type, solution composition and current level for background suppression as main variables.     

A combination of statistical and analytical evaluation methods as a new optimization strategy for the quantification of pharmaceutical residues in sewage effluent

In this study, a new solid-phase microextraction (SPME) method for simultaneous extraction of pharmaceutical compounds with acidic and basic characteristics (ibuprofen, fenoprofen, diclofenac, diazepam and loratadine) from residual water samples is proposed. In this procedure, the extraction is processed using two distinct sample pH values. The extraction is begun at pH 2.5 to promote the sorption of acidic pharmaceuticals and after 35 min the sample pH is changed to 7.0 by adding 0.4 mol L⁻¹ disodium hydrogenphosphate, so that the basic compounds can be sorbed by the fiber (20 min). The pH change is performed without interruption of the extraction process. A comparison between the proposed method and the SPME method applied to each group of the target compounds was performed. Gas chromatography coupled to mass spectrometry was used for separation and detection of analytes. The extraction conditions for the three methods were optimized using full factorial experimental design, response surface through a Doehlert matrix and central composite design. Limits of detection (0.02-0.43 μg L⁻¹) and correlation coefficients (0.9970-0.9998) were determined for the three methods. The proposed extraction procedure was applied to samples of sewage treatment plant effluent and untreated wastewater. Recovery and relative standard deviation values ranged from 67 to 116% and 4.6 to 14.5%, respectively, for all compounds studied. Modification of sample pH during the extraction procedure was shown to be an excellent option for all of the compounds and may be extended to the simultaneous extraction of other compounds with different acid-base characteristics.     

Development and validation of an HPLC method for the determination of process-related impurities in pridinol mesylate, employing experimental designs

A simple high performance liquid chromatographic method for the determination of process-related impurities in bulk drug of the central anticholinergic compound pridinol mesylate, has been developed and validated. Spectroscopically characterized synthetic impurities were used as standards. The chromatographic separation was optimized employing an experimental design strategy, and was achieved on a C₁₈ column with a mobile phase containing 50 mM potassium phosphate buffer (pH 6.4), MeOH and 2-propanol (20:69:11, v/v/v), delivered at a flow rate of 1.0 mL min⁻¹. UV detection was performed at 245 nm. The optimized method was thoroughly validated, demonstrating to be selective, when the chromatogram was recorded with a diode-array detector and peak purities were evaluated (>0.9995). The method is robust and linear (r² > 0.99) over the range 0.05-2.5% (5-250% with regards to the 1% specification limit for both process-related impurities); it is also precise, regarding repeatability (RSD ≤ 1.5% for all of the analytes) and intermediate precision aspects and LOQ values for the impurities are below 0.01%. Method accuracy, evidenced by low bias of the results and analyte recoveries in the range of 99.1-102.7%, was assessed at five analyte concentration levels. The usefulness of the determination was also demonstrated through the analysis of different lots of pridinol mesylate bulk substance. The results indicate that the method is suitable for the quality control of the bulk manufacturing of pridinol mesylate drug substance.     

Hierarchical classification designs for the estimation of different sources of variability in proficiency testing experiments

An approach to the estimation of possible different sources of variation found in proficiency testing experiments is described. Four errors namely, technique, analyst, laboratory and geographical location are considered and calculated by using a rational experimental design based on hierarchical classification. The treatment of the confidence of the design over different experimental arrangements is explored and visualised by calculating a function that depends only on the design and not on the experimental response. An illustrative example based on simulated data is used to show how the theory could be applied in practice.     

A kernel‐based method to determine optimal sampling times for the simultaneous estimation of the parameters of rival mathematical models

When several models are proposed for one and the same process, experimental design techniques are available to design optimal discriminatory experiments. However, because the experimental design techniques are model‐based, it is important that the required model predictions are not too uncertain. This uncertainty is determined by the quality of the already available data, since low‐quality data will result in poorly estimated parameters, which on their turn result in uncertain model predictions. Therefore, model discrimination may become more efficient and effective if this uncertainty is reduced first. This can be achieved by performing dedicated experiments, designed to increase the accuracy of the parameter estimates. However, performing such an additional experiment for each rival model may undermine the overall goal of optimal experimental design, which is to minimize the experimental effort. In this article, a kernel‐based method is presented to determine optimal sampling times to simultaneously estimate the parameters of rival models in a single experiment. The method is applied in a case study where nine rival models are defined to describe the kinetics of an enzymatic reaction (glucokinase). The results clearly show that the presented method performs well, and that a compromise experiment is found which is sufficiently informative to improve the overall accuracy of the parameters of all rival models, thus allowing subsequent design of an optimal discriminatory experiment. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Dispersive liquid–liquid microextraction coupled with capillary electrophoresis for simultaneous determination of sulfonamides with the aid of experimental design

A novel method for the simultaneous determination of sulfonamides (SAs) in water samples has been developed by using dispersive liquid–liquid microextraction (DLLME) coupled with CE. Orthogonal and Box–Behnken designs were employed together to assist the optimization of DLLME parameters, including volumes of extraction and disperser solvents, ionic strength, extraction time, and centrifugation time and speed as variable factors. Under the optimum extraction and detection conditions, successful separation of the five SAs was achieved within 5 min, and excellent analytical performances were attained, such as good linear relationships (R>0.980) between peak area and concentration for each SA from 0.5 to 50 μg/mL, low limits of detection for the five SAs between 0.020 and 0.570 μg/mL and the intra‐day precisions of migration time below 0.80%. The method recoveries obtained at fortified 10 μg/mL for three water samples ranged from 53.6 to 94.0% with precisions of 1.23–5.60%. The proposed method proved highly sensitive and selective, rapid, convenient and cost‐effective, showing great potential for the simultaneous determination of SAs in water samples.     

Optimization of modified dispersive liquid–liquid microextraction coupled with high‐performance liquid chromatography for the simultaneous preconcentration and determination of nitrazepam and midazolam drugs: An experimental design

A simple, sensitive, and rapid microextraction method, namely, ultrasound‐assisted surfactant‐enhanced emulsification microextraction based on the solidification of floating organic droplet method coupled with high‐performance liquid chromatography was developed for the simultaneous preconcentration and determination of nitrazepam and midazolam. The significant parameters affecting the extraction efficiency were considered using Plackett–Burman design as a screening method. To obtain the optimum conditions with consideration of the selected significant variables, a Box–Behnken design was used. The microextraction procedure was performed using 29.1 μL of 1‐undecanol, 1.36% (w/v) of NaCl, 10.0 μL of sodium dodecyl sulfate (25.0 μg mL^?1), and 1.0 μL of Tween80 (25.0 μg mL^?1) as an emulsifier in an extraction time of 20.0 min at pH 7.88. In order to investigate the validation of the developed method, some validation parameters including the linear dynamic range, repeatability, limit of detection, and recoveries were studied under the optimum conditions. The detection limits of the method were 0.017 and 0.086 ng mL^?1 for nitrazepam and midazolam, respectively. The extraction recovery percentages for the drugs studied were above 91.0 with acceptable relative standard deviation. The proposed methodology was successfully applied for the determination of these drugs in a number of human serum samples.