1.2.4. Iterative determination of Decision threshold and Detection limit¶
The calculation of detection limits is based on ISO 11929:2019 which is derived from Bayesian methods (see also Weise et al., 2006). It utilizes complete uncertainty propagation taking all individual uncertainties and covariances into account where the numerical calculations are based on the routines RESULT and UncPropa. The values of Decision threshold and Detection limit for the output quantity \(y\) are calculated by using an iterative procedure. In this procedure, the value of \(y\) is varied, now designated as “assumed value” \(\tilde{y}\). From this, the iterated value \(\tilde{\mathbf{R}}_{\mathbf{b}}\) of the gross counting rate is obtained via calculating the net counting rate from \(\tilde{y}\). For each iteration step the combined standard uncertainty of \(\tilde{y}\), now called uncertainty function \(\tilde{\mathbf{u}}\left(\tilde{\mathbf{y}} \right)\) is in turn derived from the easily calculated \(\tilde{u}(\tilde{R_{b}}).\)
The \(y^{*}\) for the output quantity \(y\) is calculated according to ISO 11929 as follows:
where \(k_{1 - \alpha}\) is the normal quantile belonging to the error of first kind, \(\alpha\). UncPropa is used for calculating the combined standard uncertainty \(u_{c}(y)\) of the output quantity under the constraint that the net counting rate is set equal to zero. This is easily done.
The Detection limit \(y^{\#}\) for the output quantity \(y\) is calculated as follows, where \(y^{*}\) is the value of the Decision limit taking from the preceding step and \(k_{1 - \beta}\) is the normal quantile belonging to the error of second kind, \(\beta\):
This represents an implicit equation for \(y^{\#}\), because on the right-hand side of the equation the uncertainty \(u\) is to be calculated for a value of \(R_n\), which corresponds to the value \(y^{\#}\) on the left-hand side; the latter is obtained by the inverse function \(f^{-1}\) which is easily established as \(R_n = (y -F_C)/F_L\) from the simple linear relationship between \(y\) and \(R_n\), \(y = F_L \cdot R_n + F_C\).
The solution of the implicit equation (1.2.9) is obtained by a simple iterative procedure which is demonstrated for the detection limit case. The value of Factor is determined in the subprogram RESULT, while the uncertainty in Eq. (2) is calculated with the function subprogram UncPropa. In order to use UncPropa correctly, for each iteration step the corresponding value \(R_{n}^{i}\) is obtained from the associated \(\mathbf{y}^{\mathbf{\# i}}\), which then is transformed to the gross counting rate, which in this example is stored in the array element \({p(8)}^{i}\). The uncertainty of the latter is calculated from the “uncertainty function (standard uncertainty) of the gross counting rate” which has been supplied to the program by the user. In the example the gross counting rate is calculated as if \({p(8)}^{i}\) had been obtained by simple single-channel counting, which applies to most cases: \(u(8)^{i} = \sqrt{p(8)^{i}/t}\).
With the version 2.2.02 (2017/12) the iteration procedure described above has been replaced by the Ridder’ method (subroutine zriddr from the Numerical Recipes, Press et al., 1992). It works more effectively than the secant method. Since version 2.2.11 (2018/11) the method by Brent is applied.
Special cases
In the case of linear unfolding by using linear least squares analysis, e.g. in the evaluation of a decay curve, the fitting parameter for the desired net counting rate is that quantity which is varied by iteration for estimating Decision threshold and Detection limit (c.f. Note on Decision threshold and Detection limit with linear unfolding).
For determining the activity of a radionuclide from several gamma lines the quantity associated with this activity is the one of which the value is iterated (c.f. Method for calculating Decision threshold and Detection limit with Gamspk1).
