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

The Process Analysis module uses the standard formulas for estimating the variance components from ranges (e.g., see ASQC/AIAG, 1991, Montgomery, 1991, DataMyte, 1992). 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. Note that 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

srepeat. = R-bartrials /d2

where

R-bartrials 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 (sreproducibility) is estimated as:

sreprod. = sqr. root[(X-bardiff / d2)2 - srepeat2 / (n*r)]

where X-bardiff 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. Note that 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

Note that 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:

sOper = sqr. root[(MSOper - MSOper*Parts )/(NParts *NTrials)]

sOper*Parts = sqr. root[(MSOper*Parts - MSError )/(NTrials)]

where MS refers to the respective ANOVA mean squares, and NOperat, NParts, and NTrials 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:

sOper = sqr. root[(MSOper - MSPool )/(NParts *NTrials)]

where MSPool 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 (sparts) is estimated as:

sparts = Rparts /d2

where

Rparts 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:

sPart = sqr. root[(MSParts - MSOper*Parts )/(NOper *NTrials)]

where MS refers to the respective ANOVA mean squares, and NOper and NTrials 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:

sParts = sqr. root[(MSParts - MSPool)/(NOper.*NTrials)]

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

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