Statistical Method for Calculating the Minimum Detectable Concentration
Abstract Generated abstract
This paper proposes a unified statistical approach for estimating the minimum detectable concentration in analytical chemistry, addressing methods such as polarography where detection depends on distinguishing a sample signal from a blank or residual current. It formulates the problem as recognition between two hypotheses about the difference of means for two random samples, derives minimax threshold rules under normal assumptions, and gives expressions for the minimum recognizable signal when the variance is known, unknown, or only one sample is available. The resulting detection limit is related to concentration through the linear current, concentration relationship used in amalgam and film polarography, and the same principle is suggested as a basis for a statistical theory of resolving power.
Full Text
STATISTICAL METHOD FOR CALCULATING THE MINIMUM DETECTABLE CONCENTRATION
A. A. Kaplin, A. I. Ruban, A. N. Pokrovskaya
One of the principal metrological criteria in analytical chemistry is sensitivity and accuracy. In various analytical methods (spectral, activation, photometric, etc.) diverse methods are used for their quantitative assessment, but there is no unified approach. In the present work a general statistical approach is considered for estimating the detection limit of an analytical signal.
Let there be two random samples: 1. \(x_1,\ldots,x_{n_1}\), with mathematical expectation \(m_x\) and variance \(\sigma_x^2\); 2. \(y_1,\ldots,y_{n_2}\), with mathematical expectation \(m_y\) and variance \(\sigma_y^2\). In particular, in amalgam and film polarography, the sample \(\{x_i\}\) should be understood as measurements of the current caused, for example, by the presence of a “blank experiment” (or residual current), and \(\{y_i\}\) as measurements after introducing the analyzed sample into the solution. In what follows we shall assume that \(\sigma_x^2=\sigma_y^2=\sigma^2\), while \(m_x\) and \(m_y\) are unknown. It is necessary, from the available information, to determine whether \(m_x\) and \(m_y\) are equal, and, if they are not equal, to estimate the minimum value of their difference \(\Delta m\) that can be recognized with a prescribed probability. We calculate the sample means \(\bar{x}\) and \(\bar{y}\) and form the new quantity \(z=\bar{x}-\bar{y}\), whose mathematical expectation is \(m_z=m_x-m_y\), and whose variance is \(\sigma_z^2=\sigma^2(n_1^{-1}+n_2^{-1})\). To solve the problem posed, we shall use the results of the statistical theory [1, 2] of recognizing the following two hypotheses:
\(H_1\) — the hypothesis that \(m_z=0\); \(H_2\) — the hypothesis that \(m_z\), with a prescribed probability, is equal to the minimum recognizable quantity \(\Delta m\).
Denote by \(f_1(z)\) the density of the distribution of the random variable \(z\) when hypothesis \(H_1\) is true, and by \(f_2(z)\) when …
hypothesis \(H_2\). Then, for distribution laws possessing the property
\[ f_2(z)=f_1(z), \tag{1} \]
the minimax rule for recognizing hypotheses ultimately takes the form: if \(f_2(z)>f_1(z)\), then hypothesis \(H_1\) is accepted; if \(f_2(z)<f_1(z)\), then hypothesis \(H_2\) is accepted; or, if \(z>\Pi\), then \(H_1\) is accepted, and if \(z<\Pi\), then \(H_2\) is accepted. Here \(\Pi\) is the threshold, determined either from the condition \(f_2(z)=f_1(z)\), or from the equation \(\alpha=\int_{\Pi}^{\infty} f_1(z)\,dz\) for a specified error of the first kind \(\alpha\). The minimally detectable nonzero signal \(\Delta m\) is then recognized with probability \(p=1-\alpha\), and \(\Delta m=2\Pi\). Let us calculate the magnitude of this signal, assuming that the sample values \(\{x_i\}\), \(\{y_i\}\) are independent and have a normal distribution law.
- Suppose that \(\sigma^2\) is known; then
\[ \Delta m=2U_{1-\alpha}\sigma_z=2U_{1-\alpha}\sigma\sqrt{n_1^{-1}+n_2^{-1}}, \tag{2} \]
where \(U_{1-\alpha}\) is the quantile of the normal distribution law.
- If \(\sigma^2\) is unknown, then its estimate is calculated as
\[ S^2=\frac{\sum_{i=1}^{n_1}(x_i-\bar{x})^2+\sum_{i=1}^{n_2}(y_i-\bar{y})^2}{n_1+n_2-2} \tag{3} \]
and then \(f_1(z)\) has the Student distribution law with \((n_1+n_2-2)\) degrees of freedom. Since this law satisfies condition (1), applying the recognition rule described, we shall have
\[ \Delta m=2t_{1-\alpha}S\sqrt{n_1^{-1}+n_2^{-1}}, \tag{4} \]
where \(t_{1-\alpha}\) is the quantile of the Student distribution.
- If there is one sample \(x_1,\ldots,x_n\) and it is necessary to determine the minimally recognizable signal \(m_x\), then in the preceding calculations it is necessary to put \(\Delta m=m_x\), \(z=\bar{x}\), \(\sigma_z^2=\sigma^2 n^{-1}\). As a result we obtain
\[ \Delta m=m_x=2t_{1-\alpha}\frac{S}{\sqrt{n}};\quad S=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2. \tag{5} \]
In the method of amalgam and film polarography, there is a functional dependence between the recorded current and the concentration,
\[ J=AC, \]
where the sensitivity coefficient \(A=K_a S\gamma V\nu^{-1}\) is determined by the parameters of all stages of the process. Taking this into account, the value of the minimum ion concentration in solution \(C\), detectable with a specified guarantee, is calculated from the corresponding relations (2, 4, 5), in which \(\Delta m=\Delta J=AC^*\).
The principle set forth above forms the basis for developing a statistical theory of resolving power; in this case the sample \(x_1, \ldots, x_n\) is determined by measurements of the anode-current values of the interfering element at the peak potential of the element being determined.
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