# Methods and formulas for Expanded Gage R&R Study

Select the method or formula of your choice.

## Gage R&R expanded study methods

Minitab uses the general linear model approach with three types of ANOVA models to perform Gage R&R studies: the random-effects model, the mixed-effects model, and the nested designs model. The random-effects model is the default. The mixed-effects or the nested design model is used if any factors are fixed or nested.

The final selected model only includes the main effects terms, the significant highest order interactions, and the relevant interactions in between. Minitab calculates the ANOVA table for the appropriate model. That table is then used to calculate the variance components, which appear in the Gage R&R tables.

### References

Burdick, R. K., Borror, C. M., and Montgomery, D.C. (2003). "A Review of Methods for Measurement Systems Capability Analysis", Journal of Quality Technology, 35(4) 342–354.

Adamec, E. and Burdick, R.K. (2003). "Confidence Intervals for a Discrimination Ratio in a Gauge R&R Study with Three Random Factors", Quality Engineering, 15(3) 383–389.

## Random effects model

The default model used in this command is the random-effect model. If you specify a full model for the three factors, then:

Yijkl = μ + Pi + Oj + Ak + (PO)ij + (PA)jk + (OA)jk + (POA)ijk + εijkl

where:
TermDescription
μconstant
Pithe ith part
TermDescription
Oj the jth operator
TermDescription
Akthe kth level of the additional factor

Pi, Oj , Ak, (PO)ij, (PA)jk, (OA)jk, (POA)ijk, and εijkl are independently normally distributed with mean of zero and, respectively, variances of .

Minitab uses Fit General Linear Model to estimate the variance components. For more details about estimating variance components, go to Methods and Formulas for Fit General Linear Model.

If the Part term is the only term that is used to calculate the Part-to-Part variation:
Total Gage R&R
Repeatability
Reproducibility
Operator
A
Part * Operator
Part * A
Part-to-Part
Part
Total variation
###### Note

When you specify to the historical standard deviation to estimate the process variation, Minitab does the following:

• If the historical standard deviation is greater than the total gage standard deviation calculated from the data, then the total standard deviation is σ and the part-to-part standard deviation is .
• Otherwise Minitab uses the data to estimate the total standard deviation and the part-to-part variation.

If more terms are specified for the Part-to-Part variation the table changes accordingly. For instance, if both the Part factor and Factor A represent process variation, then Part, A, and their interactions are specified to estimate Part-to-Part variation:
Total Gage R&R
Repeatability
Reproducibility
Operator
Part * Operator
Part-to-Part
Part
A
Part * A
Total variation
###### Note

When you specify to the historical standard deviation to estimate the process variation, Minitab does the following:

• If the historical standard deviation is greater than the total gage standard deviation calculated from the data, then the total standard deviation is σ and the part-to-part standard deviation is .
• Otherwise Minitab uses the data to estimate the total standard deviation and the part-to-part variation.

For more than 3 factors, the variance components for Gage Repeatability, Reproducibility, and Part-to-Part are similarly defined as in the 3 factor case. Usually:
• Gage Repeatability = the variance component for the error term
• Part-to-Part variation = Variance component for Part or the Sum of the Variance components for the Part-to-Part terms
• Gage reproducibility = the Sum of the Variance components for the rest of the terms

## Mixed-effects model

If some terms in the linear model are fixed, it is a mixed effect model. The variance components for the random terms are obtained using the results from Fit General Linear Model.

For more details about estimating variance components, go to Methods and Formulas for Fit General Linear Model.

For the fixed terms, there are no variance components. The variability across the levels of a fixed term is estimated as follows:
1. By fitting the linear model, Minitab estimates the coefficients for the first J-1 levels of the factor.
2. The coefficient for level J = –(Sum of coefficients across the first J-1 levels).
3. Estimated variability = Sum of (coefficient)2 across all levels / number of levels.

In the calculation of the Gage reproducibility for mixed effects, the variance components for the fixed terms will be replaced by φ , but the random effects model definitions hold.

## Nested design model

If some factors are nested under other factors, Minitab fits the model using Fit General Linear Model. For more details about estimating variance components, go to Methods and Formulas for Fit General Linear Model.

Gage repeatability, reproducibility, and part-to-part variation are defined the same way as in the random and the fixed factor cases.

## Gage R&R expanded calculations

Minitab displays two tables for Expanded Gage R&R Study. The first table contains the VarComp column and the %Contribution (of VarComp) column. For more details about estimating variance components, go to Methods and Formulas for Fit General Linear Model.

%Contribution = value of VarComp / Total Variation.

The second table contains:
• StdDev (SD) = sqrt (VarComp)
• Study Var = number of standard deviation * StdDev
• %Study Var (%SV) = Study Var / Study Var for Total Variation
• %Tolerance = Study Var / Process Tolerance
• %Process = StdDev / historical standard deviation

## Number of distinct categories

The number of distinct categories represents the number of non-overlapping confidence intervals that will span the range of product variation. You can also think of it as the number of groups within your process data that your measurement system can discern.

Minitab then truncates this value, except when the value is less than 1. In that case, Minitab sets the number of distinct categories equal to 1.

### Confidence interval

Suppose L and U are the lower and upper bounds of the ratio of the gage variance and the total variance, then the lower and the upper bounds for the number of distinct categories are:

###### Note

L and U must be in (0, 1). If L and U are outside the range, the lower and upper bounds for the number of distinct categories are missing.

## Probabilities of misclassification

Minitab calculates the probabilities of misclassifications as both joint probabilities and conditional probabilities when you enter at least one specification limit.

### Joint probabilities

Probability that part is bad and you accept it:

Probability that part is good and you reject it:

### Conditional probabilities

Probability that given a part is bad, you accept it (false accept):

Probability that given a part is good, you reject it (false reject):

### Notation

F(X,Y) is the cumulative distribution function (CDF) of the bivariate normal random vector (X,Y)T with:

mean, μ = (θ,θ)T

F(X) and F(Y) are the corresponding marginal CDFs.

That is,

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