In eCerto: Statistical Tests for the Production of Reference Materials

Homogeneity uncertainty calculation

We first conduct an analysis of variance (ANOVA) on a one factor linear model for $N$ bottles (or containers) with $n$ replicate measurements each. The ANOVA is preformed for each analyte and H_type (if specified during upload) independently. The P-value of each ANOVA is provided in Tab.H1 together with the variance within bottles $s_w$ (M_within) and the variance between bottles $s_a$ (M_between).

A significant ANOVA P-value indicates non homogeneous specimen and is highlighted in red color in Tab.H1. P-values are adjusted for multiple testing in case of several analytes being under investigation per specimen using bonferroni correction. Adjustment can be switched off in the options panel next to Tab.H1.

Note! To account for the case of different number of replicates over all $N$ bottles, $n$ is calculated according to ISO GUIDE 35 as:

$$n=\frac{1}{N-1} \times \left[\sum_{i=1}^N{n_i}-\frac{\sum_{i=1}^N{n_i^2}}{\sum_{i=1}^N{n_i}}\right]$$

where $n_{i}$ is the vector of replicates per group. Together with the overall mean vlaue $\mu$ (mean of all $N$ bottle means), we now can compute two relative uncertainties between bottles:

$$s_{bb}=\frac{\sqrt{\frac{s_a-s_w}{n}}}{\mu}$$

and

$$s_{bb, min}=\frac{ \sqrt{ \frac{s_w}{n} } \times \sqrt[4]{ \frac{2}{N \times (n-1)} }}{\mu}$$

Note! When $s_{a} < s{w}$ we set $s{bb}=0$.

The larger of both values, $s_{bb}$ and $s_{bb,min}$ (rendered in bold font), is selected as uncertainty contribution when the user decides to transfer an uncertainty value to the material table.

A transfer is only possible for analytes which have been found in the table of certified values (Tab.C3 in the certification module). Analytes not present there will be rendered in red.

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