Robust estimation of between and within laboratory standard deviation with measurement results below the detection limit

Appendix

The computation of reproducibility and repeatability parameters according to the Q method can be broken down into several steps. The first step is the determination of the jump discontinuities of the function

$$ H_{1} \left( y \right) = \frac{1}{{\left( {\begin{array}{*{20}c} J \\ 2 \\ \end{array} } \right)}}\mathop \sum \limits_{{1 \le j_{1} < j_{2} \le J}} \frac{1}{{n_{{j_{1} }} \cdot n_{{j_{2} }} }}\mathop \sum \limits_{{i_{1} = 1}}^{{n_{{j_{1} }} }} \mathop \sum \limits_{{i_{2} = 1}}^{{n_{{j_{2} }} }} 1_{{\left\{ {\left| {x_{{j_{1} i_{1} }} - x_{{j_{2} i_{2} }} } \right| \le y} \right\}}} , $$

(1)

where x _ji denotes the ith result of laboratory j, n _j denotes the number of replicates for laboratory j and 1 _A denotes the indicator function for the set A. J denotes the number of laboratories. The function H ₁ maps a (positive) real number y to the percentage of pairwise absolute differences between results from different laboratories which are less than or equal to y, i.e. it is an empirical cumulative distribution function. Let $y_{1} , \ldots , y_{k}$ denote the jump discontinuities of H ₁ (in ascending order). A linear interpolation G ₁ is then computed on the basis of values defined at the jump discontinuities:

$$ G_{1} \left( {y_{i} } \right) = \left\{ {\begin{array}{ll} {0.5 \cdot \left( {H_{1} \left( {y_{i} } \right) + H_{1} \left( {y_{i - 1} } \right)} \right)} & \text{for}\quad i \ge 2 \\ {0.5 \cdot H_{1} \left( {y_{1} } \right) } & \text{for}\quad i = 1 \quad {\rm and} \quad y_{1} > 0 \\ 0 & \text{for}\quad i = 1\quad {\rm and} \quad y_{1} = 0. \\ \end{array} } \right.$$

(2)

Having defined

$$ q = 0.25 + 0.75 \cdot H_{1} \left( 0 \right), $$

(3)

the reproducibility sd s _R is obtained as

$$ s_{R} = \frac{{G_{1}^{ - 1} (q)}}{{\sqrt 2 \cdot \Phi^{ - 1} (0.5 + 0.5 \cdot q)}}, $$

(4)

where Φ denotes the cumulative distribution function of the standard normal distribution. It should be noted that, if no rounding takes place, the values H ₁(0) = 0 and q = 0.25 are obtained. The denominator in Eq. 4 is defined in such a way that, in the case of a standard normal distribution, s _R  = 1 will be obtained on average.

For the estimation of the repeatability sd, absolute differences between results within each laboratory constitute the basis for the definition of the empirical cumulative distribution function:

$$ H_{2} \left( y \right) = \frac{1}{J}\mathop \sum \limits_{j = 1}^{J} \frac{2}{{n_{j} (n_{j} - 1)}}\mathop \sum \limits_{{1 \le i_{1} < i_{2} \le n_{j} }} 1_{{\left\{ {\left| {x_{{ji_{1} }} - x_{{ji_{2} }} } \right| \le y} \right\}}} $$

(5)

Just as in the case of the reproducibility sd, a linear interpolation is then performed to obtain the function G ₂. The repeatability sd s _r is then obtained in the same way as s _R (see Eq. 4), the only difference being the definition

$$ q = 0.5 + 0.5 \cdot H_{2} \left( 0 \right). $$

(6)