On the Calculation of the Minimum Determinable Concentration with a Specified Reliability by the APN and PPN Methods
Abstract Generated abstract
This paper develops semiempirical formulas for estimating the minimum concentration determinable with a specified reliability in amalgam and film polarography with accumulation. Treating the problem as one of statistical hypothesis testing between blank and signal concentrations, the authors use the Neyman Pearson framework, assume normally distributed independent anodic peak measurements based on experimental checks, and derive expressions relating the determinable concentration to error probabilities, number of measurements, signal slope, variance, and experimental parameters. The resulting relations indicate how sensitivity depends on signal and noise characteristics, accumulation and anodic oxidation conditions, sample and solution parameters, and especially the number of measurements. The authors suggest that factorial experimental design can be used to determine variance dependencies and optimize the detection conditions.
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ON THE CALCULATION OF THE MINIMUM CONCENTRATION DETERMINABLE WITH A GIVEN RELIABILITY BY THE APN AND PPN METHODS
A. A. Kaplin, G. N. Yudin, Yu. A. Ivanov, A. N. Pokrovskaya
(Presented by the scientific-methodological seminar of the KhTF)
Increasing the sensitivity of determining impurities in especially pure objects from \(10^{-5}\text{–}10^{-6}\%\) to \(10^{-10}\text{–}10^{-12}\%\), primarily by increasing the absolute sensitivity of methods, is a principal task of analytical chemistry. This presupposes the existence of sufficiently well-founded mathematical criteria for estimating the minimum determinable concentration, the detection limit, accuracy, and resolving power. At present, such criteria are practically absent in the literature on the method of amalgam and film polarography with accumulation. In the present work, a derivation is given of relations for the magnitude of the minimum concentration determinable with a specified guarantee, \(C_{\min}\), for the case of identical nature of the signal \(C_1\) and the interference \(C_0\). The calculations are performed on the basis of the theory of testing statistical hypotheses.
The problem is formulated as follows. We have two possible concentrations \(C_1\) and \(C_0\) and a set of experimental quantities of anodic peaks \(h_1, h_2 \ldots h_n\). The conditional distributions of peak-height values corresponding to the concentrations \(C_0\) and \(C_1\) will be denoted by \(P_n(h_1 \ldots h_n / C_0)\) and \(P_n(h_1 \ldots h_n / C_1)\). Let \(H_0\) and \(H_1\) denote the hypotheses that the selected values \(h_1 \ldots h_n\) belong to the distributions \(P_n(h_1 \ldots h_n / C_0)\) or \(P_n(h_1 \ldots h_n / C_1)\). The finiteness of the sample on the basis of which the choice between the hypotheses \(H_0\) and \(H_1\) is made can lead to errors of the first and second kind: the “false alarm” error \(\alpha\)—an incorrect hypothesis of absence of a signal is accepted—and the “missed signal” error \(\beta\)—the true hypothesis is rejected. In the absence of a priori information about the loss, the rule for choosing a decision between the hypotheses \(H_0\) and \(H_1\) is given by the Neyman–Pearson criterion, according to which the number of experiments \(n\) and the type-I error are fixed, and the type-II error is minimized in the decision.
In our problem the errors of the first and second kind are fixed, and the quantity calculated is
\[ \Delta C = C_{\min} = C_1 - C_0 = f(n,\alpha,\beta). \tag{1} \]
This is achieved by specifying the threshold ratio \(\ln K_0\), which is a function of \(\alpha\) and \(\beta\). In this case the sample space is divided into two nonintersecting subsets. Depending on which subset the sample belongs to, the corresponding decision is made regarding the presence of concentration \(C_0\) or \(C_1\).
Preliminarily, on the basis of sufficiently large samples (up to 400 measurements), we for the first time for the APN method estimated the type of distribution of the ve-
of the values of the anodic currents using Pearson’s \(\chi^2\) criterion. The results in the range of the concentrations studied \((10^{-7}—10^{-8}\ \text{g/ml})\) showed satisfactory agreement with the normal distribution law\(^1\). The sample values \(h_1,\ldots,h_n\) are independent. We may write
\[ n=\frac{\left(U_{p,\alpha}\sqrt{D_{C_0}M}+U_{p,\beta}\sqrt{D_{C_0}M}\right)^2} {\left(E_{C_1}M-E_{C_0}M\right)^2}, \tag{2} \]
where
\[ M=\ln \frac{P(h/C_1)}{P(h/C_0)}; \tag{3} \]
\[ D_C=\int_{-\infty}^{+\infty} \left[\ln \frac{P(h/c_1)^2}{P(h/c_0)^2}\right]P(h/c)\,dh -\left(E_CM\right)^2; \tag{4} \]
\[ E_CM=\int_{-\infty}^{+\infty} \left[\ln \frac{P(h/c_1)^2}{P(h/c_0)^2}\right]P(h/c)\,dh; \tag{5} \]
\(U_p\) is the \(p\)-quantile of the normal distribution. The quantities \(D_CM\) and \(E_CM\) can be expressed in terms of the variance \(\sigma^2\) and the mean values \(h\) of measurements of the anodic peak of the blank experiment and of the signal. We take \(\alpha=\beta\) and \(U_{p,\alpha}=U_{p,\beta}=U_p\). This assumption is justified for analytical measurements, since the conclusion that an impurity is present in a given concentration range when it is in fact absent (an error of the first kind), and the conclusion that impurities are absent when they are present in the sample, are equally undesirable.
We approximate the experimental dependence of the mean peak value \(h\) on the concentration of ions in the solution and the dependence of the variance on the concentration by the relations
\[ h=KC \quad \text{and} \quad s^2=ac^2+bc+d. \tag{6} \]
After substituting the values of \(h\) and \(\sigma^2\) into relation (2), taking (3—5) into account and after a number of transformations, we obtain
\[ C_{\min}= \frac{2U_p} {\sqrt{n}\left[ \frac{2(ac+b)^2+2K^2(ac^2+bc+d)} {2(ac^2+bc+d)^2} \right]^{1/2}}, \tag{7} \]
for
\[ 2(ac+b)^2 \ll 2K^2(ac^2+bc+d), \]
\[ C_{\min}=\frac{2U_p\sigma_c}{\sqrt{n}K}. \tag{8} \]
Taking into account the theoretical dependence of the anodic-current value on the parameters of a series of stages,
\[ J=hi=KC=K_aS\cdot C\gamma, \tag{9} \]
we have
\[ C_{\min}=\frac{2U_p\sigma}{\sqrt{n}K_aS\gamma}\,i. \tag{10} \]
If the ion concentration is expressed in \(\text{g/ml}\), then the minimum determinable amount of the element, \(a,\ \%\) (percent), in a solution volume \(V\) with sample weight \(m\) is determined as
\[ a,\ \%=\frac{2U_pV\sigma}{\sqrt{n}\,K_a\cdot S\cdot \gamma\cdot m}\,i. \tag{11} \]
\(^1\) This conclusion may not be confirmed in the region of extremely dilute solutions.
In the general case (taking into account the dependence of \(K_a\) and \(\gamma\) on the parameters of the accumulation and anodic-oxidation stage),
\[ \sigma = f(C, h, n, S, m, Pa, Du, \tau, r, W, K_s, K_e, Z \text{ and others}). \]
From relations (10) and (11) it follows that the minimum concentration determinable with a guarantee is a function of the characteristics of the signal and noise (\(K_a\)), the accumulation conditions (\(\gamma\)), the conditions of anodic oxidation and recording (\(K_a, i, n\)), and the statistical characteristics of the signal and noise, determined by the variance \(\sigma^2\). In the presence of a blank experiment, the sensitivity of determination can be increased by increasing \(K_a, S, \gamma\) and by decreasing \(V, i\), provided that, when these parameters are changed, \(\sigma\) decreases (\(n = \mathrm{const}\)). The principal condition for increasing sensitivity (and accuracy) in the presence of a blank experiment is an increase in the number of measurements \(n\). In the absence of a blank experiment, increased sensitivity is achieved by increasing \(n, K_a, \gamma, S, m\), by decreasing \(V, i\), and by reducing the value of \(\sigma\).
The dependence of the variance on various factors can be determined by carrying out an experiment using factorial design and by solving equations (9) and (10) for the minimum.
Conclusions
Semiempirical relations have been obtained for calculating the value of the minimum concentration determinable with a specified reliability.