# HOW TO USE ANOVA IN GAGE R&R

Firstly gage R&R studies are used to determine if a measurement system (includes measuring instruments, calibration standards, human operator, part to be measured and environmental influence) is capable for its intended purpose. If the measurement system variation is small compared to the process variation then the measurement system is considered capable, since the parts and appraisers are randomly selected there is a measure of variability associated with each.

## Gage Variability is a function of following variance components:

**σ² part to part**

**σ² reproducibility**

** σ² repeatability**

**σ² part to part**

The actual variance component among the parts which is referred to as part to part variation.

**σ² reproducibility**

Reproducibility is the variance component among the individual appraisers.

** σ² repeatability**

Repeatability is the variance component within the appraiser himself and also includes the random error of the experiment.

## Two commonly used methods for estimating the repeatability and reproducibility metrics are

- Range method
- Analysis of Variance method (ANOVA)

ANOVA method has a become a common choice for practitioners conducting gage R&R studies since the computations can be easily carried out using modern statistical software and ANOVA method have the advantage of quantifying the interaction between the factors whereas range method cant estimate

Suppose the response of interest in a gage R&R study can be expressed by a random two-factor model (part & operator) then the total variability can be expressed as

**= +**

**σ²gage = σ²reproducibility + σ²repeatability**

** σ²reproducibility = σ²O + σ²PO**

**σ²O- Variance associated with different appraisers**

**σ²PO- Variance associated with the interaction between the part and appraiser.**

**σ² repeatability= σ²E which is associated with the error in the experiment also called within the variation**

** **Necessary quantities for the analysis of variance can be shown in a table that can be called an ANOVA table.

**p** – No of parts considered for the experiment

**o** – No of the appraiser involved in the experiment

**n** – No of replicates considered for the experiment

**General notations of the study**

The first factor is the operator ** o** and the second factor is no of parts

**and which is measured**

*p***times will end up like**

*n*The sum of squares **SS** can be calculated by statistical software as it is difficult by hand and the means squares can be obtained by

Mean squares **MS **is the sum of squares divided by the appropriate degrees of freedom provides a measure of variability adjusted to the sample size and no of treatments taken for an experiment.

**So the estimates of variance components are**

**If the variance components result in a negative value it can be set to be zero as a thumb rule**

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**Conclusion**

We can estimate the measurement systems capability using metrics like discrimination percent agreement and PTR etc, the common choice will always be using PTR.

Precision-to-tolerance ratio which compares the square root of the gage variability to the tolerance of the characteristic to be measured.

If the PTR values are

Less than 0.10 the measurement system can be considered acceptable

Between 0.10 and 0.30 measurement system can be considered marginal and should be examined for improvement.

More than 0.30 measurement systems should be considered inadequate.

The ANOVA method can be more computationally intensive than other methods but with modern statistical software, this is less of an issue. ANOVA method is more flexible in case of adding another factor and interaction between the factors whereas other methods cannot handle the same.

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