In eCerto: Statistical Tests for the Production of Reference Materials

Calculation of possible storage time

Besides Recovery and RSD for each temperature level $T_i$, the slope k_eff and the log-transformation of it's negative value log(-k_eff) are provided in Tab.S2.

In the case that at least 3 finite values for $log(-k_{\mathit{eff}})$ exist, a linear model over 1/K is calculated (depicted in Fig.S3). Based on slope $m$ and intercept $n$ of this model values $K_i$ (provided in log(k)_calc) are established using $K_i = m \times x_i + n$.

To estimate the confidence interval of the model (CI_upper and CI_lower) we need to estimate some intermediate values which are shown below Tab.S2. First, eCerto calculates $a=\sum{x_i}$, $b=\sum{x_i^2}$ and $n=length(x_i)$ where $x_i$ are the inverse temperature values, $1/K$, as well as the standard error $err$ of the model:

$$err=\sqrt{\frac{1}{n-2} \times \sum{(y_i-\overline{y})^2} - \frac{\sum{(x_i-\overline{x}) \times (y_i-\overline{y})}^2}{\sum{(x_i-\overline{x})^2}}}$$

Next, these four calculated values $a$, $b$, $n$ and $err$ allow to compute the dependent variables:

$$u(i) = \sqrt{\frac{err^2 \times b}{(b \times n-a^2)}}$$

$$u(s) = \sqrt{\frac{err^2 \times n}{(b \times n-a^2)}}$$

$$cov = -1 \times \frac{err^2 \times a}{(b \times n-a^2)}$$

which, finally, can be used to estimate the confidence interval $\mathit{CI}$ as:

$$\mathit{CI}_{(upper,~lower)} = K_i \pm \sqrt{ u(i)^2 + u(s)^2 \times x_i^2 + 2x_i \times cov }$$

When a certified value $\mu_c$ and corresponding uncertainty $U_{abs}$ (cert_val and U_abs) are available for an analyte in Tab.C3 of the certification module of eCerto, these can be used together with $\mathit{CI}{upper}$ to calculate the storage time $S{Month}$ for each evaluated temperature level using:

$$S_{\mathit{Month}}(T) = \frac{ \log( \frac{\mu_c - U_{abs}}{\mu_c})}{e^{\mathit{CI}_{upper}(T)} }$$

Note! Extrapolation beyond the range of storage conditions tested (for example, predicting degradation rates at −20°C from an experiment involving only temperatures above 0°C) can be unreliable and is not recommended.

eCerto documentation built on April 12, 2025, 9:13 a.m.