Setting Alarm Thresholds in Measurements with Systematic and Random Errors

Abstract

For statistical evaluations that involve within-group and between-group variance components (denoted σ W 2 and σ B 2 , respectively), there is sometimes a need to monitor for a shift in the mean of time-ordered data. Uncertainty in the estimates σ ^ W 2 and σ ^ B 2 should be accounted for when setting alarm thresholds to check for a mean shift as both σ W 2 and σ B 2 must be estimated. One-way random effects analysis of variance (ANOVA) is the main tool for analysing such grouped data. Nearly all of the ANOVA applications assume that both the within-group and between-group components are normally distributed. However, depending on the application, the within-group and/or between-group probability distributions might not be well approximated by a normal distribution. This review paper uses the same example throughout to illustrate the possible approaches to setting alarm limits in grouped data, depending on what is assumed about the within-group and between-group probability distributions. The example involves measurement data, for which systematic errors are assumed to remain constant within a group, and to change between groups. The false alarm probability depends on the assumed measurement error model and its within-group and between-group error variances, which are estimated while using historical data, usually with ample within-group data, but with a small number of groups (three to 10 typically). This paper illustrates the parametric, semi-parametric, and non-parametric options to setting alarm thresholds in such grouped data.

Introducing Robust Statistics in the Uncertainty Quantification of Nuclear Safeguards Measurements

“…Ref. [ 12 ] shows that (i) assuming a truncated normal distribution for that guarantees finite moments for the ratio , and (ii) considering that typically and , we have that: the distribution of the ratio is extremely close to a normal distribution; it provides an accurate approximation of the target variable. …”

“… Effect of the introduction of one single outlier (highlighted in red) on the simulated case study of [ 12 ] (dashed lines represent the average of the 10 observations in each group). ( a ) Clean data; ( b ) One outlier.…”

“…In random effect models, even one single anomalous observation may greatly influence the properties of the statistics [ 11 ]. The bias introduced by a single anomalous value will also affect the result of the subsequent analyses based on the between and within group standard deviations, as the setting of alarm thresholds in measurements (see, for example, [ 12 ]) or the material balance evaluation [ 1 ]. Consider, for example, the simulated case study in [ 12 ] (see Figure 1 a).…”

“…The bias introduced by a single anomalous value will also affect the result of the subsequent analyses based on the between and within group standard deviations, as the setting of alarm thresholds in measurements (see, for example, [ 12 ]) or the material balance evaluation [ 1 ]. Consider, for example, the simulated case study in [ 12 ] (see Figure 1 a). The 30 observations, equally split in 3 groups, are simulated using a random effect model with between and within group standard deviations both equal to 0.010.…”