An Advanced Statistical Approach Using Weighted Linear Regression in Electroanalytical Method Development for Epinephrine, Uric Acid and Ascorbic Acid Determination

In this study, the use of weighted linear regression in the development of electrochemical methods for the determination of epinephrine (EP), ascorbic acid (AA), and uric acid (UA) is presented. The measurements were performed using a glassy carbon electrode and square-wave voltammetry (SWV). All electroanalytical methods were validated by determination of the limit of detection, limit of quantification, linear concentration range, accuracy, and precision. The normal distribution of all data sets was checked using the quantile-quantile plot and Kolmogorov-Smirnov statistical tests. The heteroscedasticity of the data was tested using Hartley’s test, Bartlett’s test, Cochran’s C test, and the analysis of residuals. The heteroscedastic behavior was observed with all analytes, justifying the use of weighted linear regression. Six different weighting factors were tested, and the best weighted model was determined using relative percentage error. Such statistical approach improved the regression models by giving greater weight on the values with the smallest error and vice versa. Consequently, accuracy of the analytical results (especially in the lower concentration range) was improved. All methods were successfully used for the determination of these analytes in real samples: EP in an epinephrine auto-injector, AA in a dietary supplement, and UA in human urine. The accuracy and precision of real sample analysis using best weighted model gave satisfactory results with recoveries between 95.21–113.23% and relative standard deviations between 0.85–7.98%. The SWV measurement takes about 40 s, which makes the presented methods for the determination of EP, AA, and UA a promising alternative to chromatographic techniques in terms of speed, analysis, and equipment costs, as the analysis is performed without organic solvents.


The reversibility of the potassium hexacyanoferrate system
The reversibility of the diffusion-controlled potassium hexacyanoferrate (K3[Fe(CN)6]) system was checked one time per day before the electroanalytical measurements in order to determine the appropriateness of the glassy carbon working electrode. In this work, four criteria were investigated to check if the K3[Fe(CN)6] oxidation/reduction reaction is reversible and diffusion-controlled: i) the difference between the anodic peak potential (Epa) and the cathodic peak potential (Epc), ∆Ep, should be 59.00 mV (for one-electron transfer redox reaction), ii) Epa and Epc should not change with increasing potential sweep, iii) the absolute value of the anodic peak current (ipa) and cathodic peak current (ipc) ratio should be equal to 1, and iv) ipa and ipc should change linearly with √ [1]. For the K3[Fe(CN)6] diffusion-controlled reversible system, the theoretical value of ∆Ep is 59.00 mV, as the number of exchanged electrons in the redox reaction is 1. At a potential sweep rate ( ) of 10.00 mV/s, the experimentally obtained values ∆Ep were between 72.00 and 76.00 mV. The potential difference then increased with increasing . Figure S1 shows a slight shift of Epa towards more positive potentials and of Epc towards more negative potentials with increasing . The reason for the Epa and Epc shifts and consequently the larger potential difference than the theoretical can be explained with the iR-ohmic drop (where i is current and R is the solution resistance). Thus, the potential resulting from the iR-drop must be added to the applied potential on the working electrode (in the case of an anodic ). This effect was minimized by installing working and reference electrodes at the closest distance possible. Considering the above mentioned, the potential shift is not significant. The third criterion was met as the ratios of ipa and ipc were close to 1. Finally, ipa and ipc changed linearly with increasing √ as the R 2 were higher than 0.9950 and therefore the fourth criterion was met. Furthermore, the diffusion coefficients were calculated using the Randles-Sevčink equation and compared with values reported in the literature. The diffusion coefficients were in the same order of magnitude as the reported value, which is a good indicator of a properly working electrode. We can conclude that despite small deviations from the ideal required conditions, the glassy carbon electrode fulfils all four criteria and can be used for further electroanalysis.

LOD and LOQ determination
The LOD and LOQ values were determined experimentally based on the signal to noise (S/N) ratio, where S represents the analyte's peak height and N represents the background noise (determined as the difference between the highest and lowest points in the background contribution at the more positive or more negative E side of the analyte's peak). The S/N ratio was obtained by performing SWV measurements by successively injecting diluted solutions of the analyte's standard and measuring the current response. The criterion for LOD was S/N ≥ 3.00 (but close to 3.00 and lower than 10.00), and the criterion for LOQ was S/N ≥ 10.00 (but close to 10.00) [2].

Linearity
Where (Δ measured ) i is the measured current at the i th calibration point (peak height) and (Δ model ) i is the corresponding interpolated signal from the obtained regression equation.

Weighted Linear Regression
Where i −2 is the square of the inverse of the variance response at the i th calibration point and is the number of calibration points.
Where () i is the g at a certain calibration point given by the weighted regression model upon response measurement, and  theoretical is the theoretical g of the solution of the diluted analyte standard at a certain calibration point i.
Where j is the g and j is the signal (the response of the analytical method) at a certain concentration point j. The weighted R 2 , which describes the statistical relationship between two variables, is calculated by Equation S6.

Accuracy and Precision of the Method
To test precison and accuracy, new solutions were prepared every time before electroanalytical determination. The recovery was calculated as recovery % = 100.00 · determined/theoretical. The determined concentration (determined) is obtained using the weighted regression model. The measurement is deemed to be accurate if the recovery is between 80.00% and 120.00%. For the method to be precise, an RSD value of < 20.00% was considered [3,4]. At least three replicate measurements were performed, and the presence of possible outliers were checked using Dixon's and Grubbs' statistical tests. If an outlier was present, this particular value was discarded and not used for the calculation of the average recoveries and RSD values. In that case, the electrochemical cell was spiked again, and the recovery and RSD values were determined until three measurements were obtained at every tested level without outliers present. Figure S2: a, c, e, g) Q-Q plots and b, d, f, h) K-S statistical tests confirming the normal distribution of the data for the first set of calibration curves (the average response out of three replicate measurements at every calibration point) for a,b) EP (anodic sweep), c,d) EP (cathodic sweep), e,f) AA, and g,h) UA determination. The parameter ztheoretical represents the z-value of the standard normal distribution, zactual is the actual z-value calculated based on the experimental data. FO and FE stand for the observed and expected frequency, respectively. FO k and FO k-1 represent FO for the k/n and k-1/n (k = 1, 2, ..., n) calibration points, respectively [5]. Figure S3: a, c, e, g) Q-Q plots and b, d, f, h) K-S statistical tests confirming the normal distribution of the data for the second obtained set of calibration curves (one measurement for every calibration point) that were used for the weighted linear regression; for a,b) EP (anodic sweep), c,d) EP (cathodic sweep), e,f) AA, and g,h) UA determination. Figure S4: Voltammograms for the real samples measured using SWV in 0.15 M PBS; a) EP from epinephrine autoinjector (anodic sweep), b) EP from epinephrine autoinjector (cathodic sweep), c) AA from a nutrition supplement, and d) UA from a human urine sample.