Uncertainty of variance estimators in analytical and process variability studies

Abstract

This thesis demonstrates that the half-t distribution is the prior of choice for estimating uncertainty of variance estimators in routine analysis of analytical and process variance components studies.

Industrial studies are often performed to estimate sources of variation e.g. to improve and quantify measurement or process capability. Understanding the uncertainty of those estimators is important, especially for small studies. A Bayesian analysis is proposed – providing a flexible methodology which easily copes with the complex and varied nature of the studies and the varied quantities of interest.

The prior is a fundamental component of a Bayesian analysis. The choice of prior is appraised and the coverage of the credible intervals obtained using six families of priors is assessed. A half-t prior (with several degrees of freedom) on the standard deviation is recommended in preference to a uniform or half-Cauchy prior, when some information exists on the magnitude of variability ‘core’ to the process or analytical method. Whilst a half-t prior has been previously proposed, through extensive simulation it is demonstrated that it is the prior of choice for estimating uncertainty of variance estimators in routine analysis of analytical and process variation studies. The coverage of 95% credible intervals for variance components and total variance is 93% (approximately) or above across a range of realistic scenarios. Other priors investigated, including Jeffreys’, a FLAT prior and inverse gamma distributions on stratum variances available in PROC MIXED1 in the SAS/STAT® software, are less satisfactory. This evaluation is novel: for one-way variance component designs there is very limited evaluation of the half-t prior when estimating the uncertainty of the variance component estimators; for the two-way or more complex none has been found. Since the coverage issues were primarily for the mid-level variance component, evaluation of designs more complex than one-way is important. Highest posterior density intervals are recommended with the metric of the parameter being important. Additionally, a scale based on the intra-class correlation coefficient is proposed for plotting the credible intervals.

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