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.

The paper reviews gauge R&R studies by two-factor mixed models including random and fixed factors. The two-factor mixed models include restricted models and unrestricted models considering the interaction of two factors. This study also classifies the models according to the number of factors, and the combination of various factors such as random factor, fixed factor, block factor and repetition type.

The research develops measurement processes for estimating and evaluating the gauge R&R(Reproducibility & Repeatability) using ANOVA(Analysis of Variance) of experimental design tools. The ten-step processes developed include experimental goal setting, the selection of characteristics(factors, levels), data model, ANOVA, EMS(Expected Mean Square), estimation of gauge precisions, and evaluation indexes. The three-factor combined measurement models are presented to show the processes developed in this paper.

A development in method of evaluating the measurement precisions using approximate F tests and variance components from expected mean square (EMS) is investigated. The research proposes three-factor mixed measurement models with the fixed and random factors. Unrestricted and unconstrained design work was rarely studied, while restricted and constrained designs have been significantly discussed. The unrestricted and unconstrained designs assume to be an independence of interaction. The proposed evaluation method about the measurement precisions can be extended to four-factor random measurement model or mixed measurement model. The study also presents the three evaluation indexes of precisions such as R&RTR (Reproducibility & Repeatability-To-Total Precision Ratio), PTR (Precision-To-Tolerance Ratio), and SNR (Signal-To-Noise Ratio). Numerical examples are proposed to evaluate the approximate F tests with Satterthwaite degrees of freedom and three indexes using the measurement precisions from EMS.