The Gage R&R (Repeatability and Reproducibility) method is a technique used in statistics to assess the reliability and precision of a measurement process. It is widely used in quality assurance and process improvement.

The aim of the Gage R&R method is to determine to what extent the total variability of a measurement is attributable to variability in the measurement process itself (repeatability) and to variability between different operators or measurement equipment (reproducibility).

The Gage R&R process generally involves the following steps:

**Gage selection (measuring instrument)**Select the measuring instrument to be evaluated.

**Operator selection**Select the operators who will perform the measurements. These operators may represent different members of staff, different teams, or different measuring equipment.

**Defining tests**Select a representative sample of the item to be measured and perform repeated measurements. This allows you to assess repeatability.

**Repeat the tests**Operators measure the same sample several times to assess repeatability.

**Total change**Analyze total measurement variation to determine the proportion due to repeatability (variability of the measurement process), reproducibility (variability between operators) and interaction (interaction between operators and parts).

**Gage R&R calculation**: La variabilité totale est souvent exprimée sous forme de pourcentage, appelé Gage R&R, qui indique la proportion de la variabilité totale attribuable à la répétabilité et à la reproductibilité comparée à l’intervalle de tolérance de la caractéristique.

This method is particularly useful in industries where measurement accuracy is crucial, such as manufacturing. It enables sources of variability to be identified and quantified in order to improve the reliability of the measurement process.

## Gage R&R ANOVA method (Nested=emboited)

The nested Gage R&R method is a variation of the standard Gage R&R method which is used when each operator measures a specific sample of parts, and the operators do not necessarily measure the same parts. It is suited to situations where sample destruction is unavoidable, and where each operator is assigned to specific groups of parts from the same batch (homogeneous parts). This approach is particularly relevant in destructive testing, where sample production is limited.

Pour calculer le GRR et le Cpc en utilisant la ANAVAR method on utilise les analyses statistiques du test de Fisher :

Sources of variability | Sum of squares | Degree of freedom | Average square | F-Statistics |
---|---|---|---|---|

Operator | SSA | a-1 | \text{MSA}=\frac{\text{SSA}}{\text{a-1}} | \text{F}=\frac{\text{MSA}}{\text{MSE}} |

Parts | SSB | b-1 | \text{MSB}=\frac{\text{SSB}}{\text{b-1}} | \text{F}=\frac{\text{MSB}}{\text{MSE}} |

Interaction (Operator/part) | SSAB | (a-1)(b-1) | \text{MSAB}=\frac{\text{SSAB}}{\text{(a-1)(b-1)}} | \text{F}=\frac{\text{MSAB}}{\text{MSE}} |

Instrument | SSE | ab(n-1) | \text{MSE}=\frac{\text{SSE}}{\text{ab(n-1)}} | |

Total | TSS | N-1 |

with:

- a = number of operators
- b = number of pieces
- n = number of repetitions
- N = total number of measurements = abn

\text{SSA}=\sum^{a}{\frac{Y_{i}^{2}}{\text{bn}}}-\frac{Y_{**}^{2}}{N}

\text{SSB}=\sum^{b}{\frac{Y_{i}^{2}}{\text{an}}}-\frac{Y_{**}^{2}}{N}

\text{SSAB}=\sum^{a}\sum^{b}{\frac{Y_{ij}^{2}}{n}}-\frac{Y_{**}^{2}}{N}-\text{SSA}-\text{SSB}

\text{TSS}=\sum^{a}\sum^{b}\sum^{n}Y_{ijk}^{2}-\frac{Y_{**}^{2}}{N}

\text{SSE}=\text{TSS}-\text{SSA}-\text{SSB}-\text{SSAB}

The repeatability of the measurement process is given by :

\text{Répétabilité}=5.15\sqrt{\text{MSE}}

The reproducibility of the measurement process is given by :

\text{Reproductibilité}=5.15\sqrt{\frac{\text{MSA}-\text{MSAB}}{\text{bn}}}

The interaction is given by :

\text{Intéraction}=5.15\sqrt{\frac{\text{MSAB}-\text{MSE}}{\text{n}}}

The variability of the measurement process is given by

\text{Dispersion}=5.15\sqrt{\text{Repeatability}^2+\text{Reproducibility}^2+\text{Interpretation}^2}

Finally, we calculate :

\text{GRR}=frac{Dispersion}{IT}

## Anova Nested - Nested

The nested Gage R&R method is a variation of the standard Gage R&R method which is used when each operator measures a specific sample of parts, and the operators do not necessarily measure the same parts. It is suited to situations where sample destruction is unavoidable, and where each operator is assigned to specific groups of parts from the same batch (homogeneous parts). This approach is particularly relevant in destructive testing, where sample production is limited.

To calculate the GRR and Cpc using the ANAVAR method, we use the Fisher test:

Source | Degree of freedom | Sommes des carrés (SS) | Moyennes des carrés (MS) | F |

Operator | b-1 | SS_{op}=an\sum_{j=1}^{b}(\overline{yj}-\overline{y})^{2} | \frac{SS_{op}}{b-1} | \frac{MS_{op}}{MS_{pièces(opérateur)}} |

Pièces (opérateur) | b(a-1) | SS_{pièce(opérateur)}=n\sum_{i=1}^{a}\sum_{j=1}^{b}(\overline{yij}-\overline{y})^{2} | \frac{SS_{pièces(opérateur))}}{b(a-1)} | \frac{MS_{pièces(opérateur)}}{MS_{répétabilité}} |

Répétabilité | ab(n-1) | SS_{répétabilité}=\sum_{i=1}^{a}\sum_{j=1}^{b}\sum_{k=1}^{n}(\overline{yij}-\overline{y})^{2} | \frac{SS_{répétabilité}}{ab(n-1)} | |

Total | abn-1 | SS_{TOTAL}=SS_{opérateur}+SS_{pièces(opérateur)}+SS_{répétabilité} |

With :

- b : nombre des opérateurs
- a : nombre de pièces
- n : nombre de répétition

On peut donc déduire les variabilités dues aux différentes sources de variabilité :

Source | Variances |

Répétabilité | \sigma_{répétabilité}^{}2 = M S_{répétabilité} |

Reproductibilité | \sigma_{répétabilité}^{}2 = \frac{MS_{opérateur}-MS_{pièce(opérateur)}}{an} |

Pièces à pièce | \sigma_{\text{pièce à pièce}}^{}2 = \frac{MS_{pièce(opérateur))}-MS_{répétabilité}}{n} |

Méthode de mesure | \sigma^2{<em>{\text{méthode de mesure}}}= \sigma</em>{{répétabilité}^{2}}+ \sigma_{{reproductibilité}^{2}} |

Total | \sigma^2{<em>{\text{total}}}= \sigma</em>{{\text{méthode de mesure}}^{2}}+ \sigma_{{\text{pièce à pièce}}^{2}} |