Repeatability and Reproducibility - Components of Variance, Percent of Tolerance, and Total Variation

If the ANOVA method is used, the variance components will be estimated assuming a three-way (operators, parts, trials) random-effects model, where all interactions involving the trials factor are equal to zero (see ASQC/AIAG, 1990, pages 65-71; Duncan, 1974, pages 716-734).

Described below are the formulas for the computation of the three major variance components repeatability, reproducibility, and part-to-part variation. The formulas for the computation of the sigma confidence intervals (computed in the ANOVA method only) are described in detail in ASQC/AIAG, 1990).

Variance Components and Mixed Model ANOVA/ANCOVA

A designated module for analyzing experiments with random effects (mixed model ANOVA) is available in Statistica .

The Variance Component and Mixed Model ANOVA/ANCOVA module contains specialized options for estimating variance components for random effects and for analyzing large main effect designs (e.g., with factors with more than 100 levels) with or without random effects or large designs with many factors when you do not need to estimate all interactions.

Repeatability

Repeatability refers to the variability of measurements of the same parts by the same operators across trials.

Thus, repeatability refers to the variation (measurement errors) in the measurement equipment. This component of the measurement variability is also referred to as equipment variation (or EV).

Ranges

If the variance components are estimated from ranges (the customary default method), the standard deviation for the repeatability or

σ_repeat. = R-bar_trials /d2

where R-bar_trials is the average range of measurements within parts and operators, across trials and d2 is the value of the average relative range, as tabulated in Duncan (1974), Table D3; d2 depends on the number of trials (sample size m, using Duncan's Table D3) and the number of operators times parts (number of samples g using Table D3). Note that some textbooks report tables of 1/d2, which may introduce round-off errors, resulting in small differences between the results reported by Statistica and some textbook examples.

ANOVA mean squares

 If the variance components are computed from the ANOVA mean squares, then the repeatability component is computed from the residual mean square error (which will include the operator by parts interaction if you select the No 2-way (Operator-Part) interaction check box on the Advanced tab of the Gage Repeatability & Reproducibility Results dialog).

Reproducibility

Reproducibility refers to the variability of measurements of the same parts across operators; thus, reproducibility is also sometimes referred to as appraiser variation (or AV).

Ranges

If the variance components are estimated from ranges (the customary default method), the standard deviation for the reproducibility or appraiser error (σ_{reproducibility}) is estimated as:

σ_reprod. = sqr. root[(X-bar_diff / d2)² - σ_repeat² / (n*r)]

where X-bar_diff is the range of mean measurements across operators and d2 is the value of the average relative range, as tabulated in Duncan (1974), Table D3; d2 depends on the number of operators (sample size m, using Duncan's Table D3), and the number of samples g (using Table D3) is equal to 1.

Some text books report tables of 1/d2, which may introduce round-off errors, resulting in possible small differences between the results reported in Statistica and some textbook examples. In the above equation, n and r are the number of parts and trials respectively

Some textbooks report a simplified formula for the reproducibility Sigma, without the part past the minus sign in the formula above. The simplified computations will be performed if you cleared the Adjust appraiser variability (AIAG) check box on the Advanced tab of the Gage Repeatability & Reproducibility Results dialog).

ANOVA mean squares

If the variance components are computed from the ANOVA mean squares, then the reproducibility component consists of two parts:

The operator variance and the operator by part interaction variance (the latter component will be pooled into the repeatability component if you select the No 2-way (Operator-Part) interaction check box on the Advanced tab of the Gage Repeatability & Reproducibility Results dialog).

If the No 2-way (Operator-Part) interaction check box is cleared, then the operator and operator by part interaction components are computed as:

σ_Oper = sqr. root[(MS_Oper - MS_Oper*Parts )/(N_Parts * N_Trials)]

σ_Oper*Parts = sqr. root[(MS_Oper*Parts - MS_Error )/( N_Trials)]

where MS refers to the respective ANOVA mean squares, and N_Operat, N_Parts, and N_Trials are the number of operators, parts, and trials, respectively.

If the No 2-way (Operator-Part) interaction check box is selected, then the operator sigma is computed as:

σ_Oper = sqr. root[(MS_Oper - MS_Pool )/(N_Parts * N_Trials)]

where MS_Pool refers to the pooled (1) error and (2) operator by part interaction mean square.

Part-to-Part Variability

The part-to-part variability refers to the variability of measurements across parts, that is, it is an estimate of the variability in measurements due to differences in parts.

Ranges

If the variance components are estimated from ranges (the customary default method), the standard deviation for the part-to-part variability ( σ_parts) is estimated as:

σ_parts = R_parts /d2

where

R_parts is the range of mean measurements across parts and d2 is the value of the average relative range, as tabulated in Duncan (1974), Table D3; d2 depends on the number of parts (sample size m, using Duncan's Table D3), and the number of samples g (using Table D3) is equal to 1.

ANOVA mean squares

If the variance components are computed from the ANOVA mean squares, and the No 2-way (Operator-Part) interaction check box on the Advanced tab of the Gage Repeatability & Reproducibility Results dialog is cleared, then the part-to-part component is computed as:

σ_Part = sqr. root[(MS_Parts - MS_Oper*Parts )/( N_Oper * N_Trials)]

where MS refers to the respective ANOVA mean squares, and N_Oper and N_Trials are the number of operators and trials, respectively.

If the No 2-way (Operator-Part) interaction check box is selected, then the part-to-part component is computed as:

σ_Parts = sqr. root[(MS_Parts - MS_Pool)/(N_Oper.*N_Trials)]

where MS_Pool refers to the pooled 1) error and 2) operator by part interaction mean square.

See also, Unbiasing Constants c4, c5, d2, d3, d4.