Approximated Interval Estimation in the Staggered Nested Designs for Precision Experiments

Summary Staggered nested experimental designs are the most popular class of unbalanced nested designs in practical fields. Reproducibility is one of the important precision measures. In our study, interval estimation of reproducibility is proposed and evaluated by precision experiments, which we call the staggered nested experimental design. In this design, the reproducibility estimator is expressed as linear combination of the variance components from ANOVA (analysis of variance). By using a gamma approximation, the shape parameter that is needed for the approximation is introduced in our study. Additionally, general formulae for the shape parameter are proposed. Applying the formulae, we constructed the confidence interval for the reproducibility of three-factor and four-factor staggered nested designs. The performance of the proposed gamma approximations is evaluated with the goodness of fit and compared with each other. The interval estimation of reproducibility is evaluated with the coverage probability through a Monte-Carlo simulation experiment. We also compared it to the method ignoring the covariance terms. As a result, the proposed approximations were better than the method without the covariance terms. The performance of the proposed interval estimation was also superior to that without covariance terms. Additionally, some practical recommendations were obtained for designing precision experiments, including the number of participating laboratories.