The research proposes the three-factor random measurement models for estimating the precision about operator, part, tool, and various measurement environments. The combined model with crossed and nested factors is developed to analyze the approximate F test by degrees of freedom given by Satterthwaite and point estimation of precisions from expected mean square. The model developed in this paper can be extended to the three useful models according to the type of nested designs. The study also provides the three-step procedures to evaluate the measurement precisions using three indexes such as SNR(Signal-To-Noise Ratio), R&R TR(Reproducibility&Repeatability-To-Total Precision Ratio), and PTR(Precision-To-Tolerance Ratio), The procedures include the identification of resolution, the improvement of R&R reduction, and the evaluation of precision effect.

We consider a mixed model with covariates considered as fixed effects and a random factor. In this paper, we consider methods for constructing confidence intervals on measures of variability in repeatability and reproducibility study to the mixed model with fixed effects and random effects. Computer simulation is used to determine how well confidence intervals maintain the stated confidence level and compare confidence interval lengths for the methods. A numerical example is considered to illustrate the confidence intervals proposed.