ON THE QUESTION OF STATISTICAL CRITERIA FOR SENSITIVITY AND RESOLUTION

A. A. Kaplin, A. A. Zheltonozhko, A. N. Pokrovskaya, B. A. Kubrak

(Presented by the scientific-methodological seminar of the Department of Physical Chemistry)

1. Statistical Criteria of Sensitivity

Earlier [1], on the basis of the theory of statistical hypotheses, the following relation was proposed as a criterion for the minimally determinable, with a specified guarantee, concentration of an element in solution—the detection limit:

\[ C_0=(U_\alpha+U_\beta)\delta_x A^{-1}n^{-1/2}, \tag{1} \]

where \(\sigma\) is the root-mean-square deviation of the blank experiment for a sample size \(n\); \(U_\alpha\) and \(U_\beta\) are the \(P\)-quantiles of the normal distribution for the specified errors of the first and second kind, \(\alpha\) and \(\beta\); \(A\) is the coefficient of proportionality between current and concentration. For \(\alpha=\beta\),

\[ C_0=2U_\alpha x A^{-1}n^{-1/2}. \tag{1a} \]

Relation (1) was obtained under the assumption of a normal law of distribution of signal and noise; the noise was taken to be the anodic peaks of the element being determined, caused by contamination of the system by this same element or by an element with the same value of \(\varphi_{\mathrm{p}}\) relative to the reference electrode and the same value of \(A\). Another substantially important assumption consisted in the equality of the variances of the “blank experiment” and the “blank experiment” plus sample, with equal sample sizes.

Under the condition \(U_\alpha=U_\beta\), we have

\[ I_0=(U_\alpha+U_\beta)\sigma_x n^{-1/2}, \tag{1б} \]

and for \(U_\alpha=U_\beta;\ \sigma_1^2=\sigma_2^2\)

\[ I_0=2U_\alpha x^{-1/2}. \tag{2} \]

In the present work, studies are presented to refine the magnitude of the minimally determinable signal with a specified guarantee: quantitative accounting for the variances of noise and of the total anodic current—the signal after the noise—without imposing the condition of their equality, accounting for the condition of small samples in the APN method, etc. On the basis of an approximate procedure for recognizing fluctuating signals against a background of fluctuating no...

we obtained a relation for the minimal statistically resolvable, with a specified guarantee against a background of interference, signal

\[ I_0=g, \tag{3a} \]

\[ g=2(\sigma_1^2 t_1 n_1^{1/2}+\sigma_2^2 t_2 n_2^{-1/2})(\sigma_1^2 n_1^{-1/2}+\sigma_2^2 n_2^{-1/2}), \tag{3b} \]

\(\sigma_1^2\) and \(\sigma_2^2\) are estimates of the variances of the interference and of the total current of the determined element and interference at the potential of the anodic current of the determined element, and in samples \(n_1\) and \(n_2\), respectively;

\(t_1\) and \(t_2\) are Student coefficients for samples \(n_1\) and \(n_2\) and the probability of correct detection \((1-\alpha)\). Relation (3) was derived under the assumptions of a normal distribution law and the additive character of the signal and interference, and equality of type I and type II errors \((\alpha=\beta)\), where \(\alpha=\psi(U'_\alpha)\) and \((U_\alpha)\) is the distribution function of a normal random variable with zero mean and unit variance.

For the coefficient \(g\), which characterizes the statistical properties of signals and interference, we propose the term “interference-immunity coefficient.”

In deriving relation (3), no restrictions were imposed on the nature of the signal and the interference; therefore relation (3b) is of a general character and is valid, under the conditions indicated above, for the recording of any signals (and not only polarographic ones) and in the presence of interferences of any nature. In the APN method these may be the anodic currents of the “blank experiment,” interfering elements¹, the residual current, including the capacitive current, electrical interferences, etc.

The minimally determined, with a specified guarantee, concentration of ions in solution corresponding to the recording of the minimal statistically resolvable, with a specified guarantee, signal \(I_0\) is

\[ C_0=gA^{-1}, \]

where \(A\), as in relations (1, 2), is the coefficient of proportionality between the current and the concentration of the determined element in solution; consequently, \(A\) characterizes the change in the signal with a change in concentration. For the coefficient \(A\) we propose the term “sensitivity coefficient.”

A possible term for the coefficient \(A\) is apparently “sensitivity of the method”; however, in the APN method this term denotes the minimal concentration.

For calculation of the sensitivity coefficient \(A\) one may use:

1. Single measurements of the anodic current at a specified concentration of ions in solution and fixed values of the other parameters:

\[ A=I\cdot C^{-1} \]

1. Calibration graphs in the coordinates: magnitude of the anodic current—concentration of ions in solution

\[ A=\frac{\Delta I}{\Delta C} \]

1. Regression equations for the dependence of the anodic current on concentration, obtained in the form of polynomials using experimental design under specified restrictions on the parameters:

\[ A=\frac{dI}{dC};\quad I=f(x_i);\quad i=1;\ 2;\ 3;\ \ldots\ x_i-x_a. \]

1. Equations of parametric theory:

\[ A = K_a S \cdot \gamma; \]

\[ \gamma = 1 - e^{-\theta}, \quad \theta = \frac{K_{\mathrm{e}}}{ZF} S \frac{\overline{t}}{V}. \tag{5} \]

The quantities \(K_a, K_{\mathrm{e}}\) can be determined experimentally and theoretically. The theoretical calculation of \(K_a\) is based on the solution of Fick’s second law for given initial and boundary conditions; consequently, the theoretical calculation of \(K_a\) is based on an analysis of the mathematical equations of the anodic current. The quantity \(K_{\mathrm{e}}\), in principle, can be calculated from hydrodynamic theory.

Thus, the value of the sensitivity coefficient determines a determinate relation between the minimum determinable concentration and the “true”—theoretical (item 4) or mean statistical values of the parameters (items 1–3) of the stages of accumulation, anodic oxidation, and peak recording.

The value of the interference-resistance coefficient, characterizing the error of an experiment \(\sigma_i\), as follows from relation (3), is determined only by statistical characteristics—the variances of the signal and the interference, i.e., by the precision of the method.

Therefore

\[ g = f(\sigma_1^2;\ \tau_2^2;\ C_1;\ n_1;\ n_2;\ \alpha); \]

\[ \sigma_1;\ \sigma_2 = f(A,\ C_1,\ i); \]

\[ \sigma_1^2;\ \sigma_2^2 = f(Ka,\ S,\ \sigma,\ C,\ i). \]

The determination of the coefficient \(g\) is based on the processing of experimental data.

For calculating \(g\) one may use:

a) samples of \(n_1\) and \(n_2\) repeated measurements of the interference and of the total anodic current of the signal and interference at fixed values of the analysis parameters;

b) regression equations for the dependence of the anodic current of the interference and of the total signal–interference current on various parameters; methods of experimental design are applicable for finding regression dependences. Obtaining regression dependences and processing them for the purpose of finding them are applicable if the interference is, for example, the anodic

\[ \sigma_1^2,\ \sigma_2^2 = f(x_i), \quad \text{where } x \text{—} (\gamma,\ S,\ K_{\mathrm{e}},\ \tau\ldots) \tag{6} \]

current of the interfering element; and, for the interference, the determining parameters and the intervals of their variation may be selected.

It follows from the foregoing that the minimum determinable concentration of ions in solution with a prescribed guarantee depends only on two generalized parameters: the sensitivity coefficient \(A\) and the interference-resistance coefficient \(g\), determined by the parameters of all stages of the amalgam-polarographic determination process and by their statistical characteristics.

2. Statistical Resolution

The question of resolving power arises in polarography in the presence of a mutual interfering influence of two elements (or only of one upon another), expressed in partial or complete mutual superposition of waves or peaks. As the basis of the statistical theory of the resolving power of the methods of APA and PPA, we have taken the same premises as in considering the theory of statistical sensitivity. The quantity \(I_0\), determined by relation (3), is a general criterion

minimum, statistically resolvable polarographic signal with a specified guarantee against a background of interferences. A particular case of interference in the SPA method is the anodic current of an interfering element at the potential of the anodic peak of the element being determined.

The problem of calculating the limiting statistical resolving power reduces to estimating the minimum statistically resolvable signal \(I_2\) of element (2) against the background of interference—the anodic current \(i_1\) of element (1) present in solution at concentration \(C_1\); the potentials of the anodic peaks of elements (1) and (2) are, respectively, \(\varphi_1\) and \(\varphi_2\). As follows from relation (3), the minimum value of the signal of element (2) is

\[ I_2^0 = I_0 = g. \tag{7} \]

The limiting ratio of the anodic currents of the signal and the interference at which separation of the minimum signal \(I_2^0\) with a specified guarantee \((1-\alpha)\) is possible will be denoted by \(P\).

Relation (7) characterizes the limiting resolving power with respect to current. In polarography, to characterize resolving power it is customary to use the ratio of the currents (and concentrations) of the interfering element to those of the element being determined. Consequently, the limiting resolving power with respect to current is

\[ P_I = \frac{i_1(\varphi_2)}{I_2^0(\varphi_2)} = g^{-1} A_1(\varphi_2) C_1, \tag{8} \]

\[ P_I = K_1(\varphi_2) S\nu S_1 g^{-1}, \tag{8a} \]

where \(K_1\) is the coefficient of the anodic current of the interfering element (1) at the potential of the limiting element (2). From the theory of SPA it follows that the value \(K_1(\varphi_2)\) is related to the anodic-current constant and depends on a number of parameters:

\[ \Delta \varphi = \varphi_2 - \varphi_1; \ \varphi_0; \ Z; \ I; \ D \ldots \]

In a particular case one may write

\[ K_1(\varphi_2) = K_1(\varphi_1)\cdot B \exp \,[f(\varphi_2-\varphi_1)]. \tag{8b} \]

Understanding the resolving power with respect to potential as the minimum \(\Delta \varphi=\varphi_2-\varphi_1\) at which registration of the limiting signal \(I_2^0\) is possible, i.e., separation with respect to current \(P_I\) is possible, we have

\[ P_{\varphi} = (\varphi_2-\varphi_1)_{\min} = \frac{1}{f}\left[\ln \frac{I_1(\varphi_1)}{I_2(\varphi_2)} + \ln \frac{1}{B}\right]. \tag{9} \]

The resolving power with respect to concentration is

\[ P_c = \frac{C_1}{C_0} = C_1 A_2(\varphi_2)\, g^{-1} \quad \text{or} \tag{10} \]

\[ P_c = K_2(\varphi_2) S \gamma C_1 g^{-1}. \]

It follows from relations (8) that the resolving power with respect to current is determined by the sensitivity coefficient of the interfering element and decreases as \(A\) or \(K_1(\varphi_2)\) decreases. The resolving power with respect to concentration is determined by the “sensitivity” of determination of element (2) and increases with increasing \(A_2(\varphi_2)\). Hence follows, for example, the physical meaning (at \(g=\mathrm{const}\)) of the conclusion that, for instance, the current of irreversible oxidation of an element may either increase or decrease the resolving power. The difference is connected with whether the element under consideration is interfering or being determined.

It follows from relations (8)—(10) that the statistical resolving power depends on the parameters of all stages of the process, and the sensitivity criterion is determined by the sensitivity coefficients and by-

stability. The conclusions drawn above concerning the dependence of \(A\) and \(g\) on various parameters, and the methods for estimating them, are applicable in calculating \(P_1\) and \(P_c\). A unified approach to the consideration of criteria of sensitivity, resolving power, and accuracy makes it possible to establish a clear quantitative relationship among these criteria.

Conclusions

1. Criteria are proposed for the concentration of an element in a solution that is minimally determinable with a specified guarantee, and for the limiting resolving power with respect to current, amplitude, and concentration.

2. Variants are proposed for the quantitative calculation of the criteria of limiting sensitivity and resolving power of the method.

LITERATURE

1. A. A. Kaplin, B. A. Kubrak, Yu. A. Ivanov, “Electronic Engineering,” series 12, issue 5 (11), 1971.

1. This has been made by us the basis of the statistical theory of resolving power. ↩