1. Module 13: Gage R&R Analysis – Analysis of Repeatability and Reproducibility
This is a technique to measure the precision of gages and other measurement systems. The name of this technique originated from the operation of a gage by different operators for measuring a collection of parts. The precision of the measurements using this gage involves at least two major components: the systematic difference among operators and the differences among parts. The Gage R&R analysis is a technique to quantify each component of the variation so that we will be able to determine what proportion of variability is due to operators, and what is due to the parts.
A typical gage R&R study is conducted as the following:
A quality characteristic of an object of interest (could be parts, or any well defined experimental units for the study) is selected for the study. A gage or a certain instrument is chosen as the measuring device. J operators are randomly selected. I parts are randomly chosen and prepared for the study. Each of the J operators is asked to measure the characteristic of each of the I parts for r times (repeatedly measure the same part r times). The variation among the m replications of the given parts measured by the same operation is the Repeatability of the gage. The variability among operators is the Reproducibility.

2. Gage repeatability and reproducibility studies determine how much of your observed process variation is due to measurement system variation. The overall variation is broken down into three categories: part-to-part, repeatability, and reproducibility. The reproducibility component can be further broken down into its operator, and operator by part, components.
Pat-to-Part Variation
Variation due to Gage
Overall Variation
Repeatability
Measurement System Variation
Operators
Variation due to Operators
Reproducibility
Operator by Part

3. Case Study (Data are from Vardeman & Jobe (1999):
A study was conducted to investigate the precision of measuring the heights of 10 steel punches (in 10-3 inches) using a certain micrometer caliper. Three operators were randomly selected for the the study. Ten parts with steel punches are randomly selected. Each operator measured each punch three time. Here are the data.
Row Punch OperA OperB OperC
Row Punch OperA OperB OperC

4. A statistical Model for Describing the Gage R&R Study

5. The Repeatability is the uncertainty among replications of m measurements of a given part made by the an operator. Since part and operator are fixed, the variance component for the repeatability is the random error, s2. On the other hand, the reproducibility is the uncertainty among operators for measuring the same part. Therefore, the variance component for the reproducibility is
How to estimate these variance components?

6. We will discuss both approaches Our goal is to estimate
Two ways to determine these variance components:
X-bar & R-chart approach.: can be done by hand, but it less accurate.
ANOVA approach : Computer will be handy.
We will discuss both approaches
Our goal is to estimate
Repeatability is the variation due to random error introduced by the differences of the repeated measurements of the same part by the same operator. That is this the variation among:
Yij1, yij2, …yijr
The range Rij can be easily obtained and applied to estimate this variability (recall the X-bar & R-charts):
E(Rij) = d2(r)s . Therefore,

7. The following is the Rij’s for the Measured Punch Heights
Operator 1
Operator 2
Operator 3 1 3 2 4 5 6 7 8 9 10
The average range is __________ And d2(r=3) = _________
Therefore,
1.9, 1.693, 1.12

8. Determine the reproducibility
Reproducibility measures the uncertainty among operators for a given part. This uncertainty is closely related to the range:

9. For the Punch Height case study, the following table gives the measurement means of ith part by the jth operator:
Punch (i) 1
497.00
497.67
.67 2
498.00
498.67
1.00 3
497.33 4 5
500.00
499.00
499.33 6 7
501.33
500.33 8 9
499.67 10
496.33
496.67

10. Using ANOVA approach, we have the following results:
(to be shown by computer output)
NOTE: Recall from Module 12, This is two random effect factor model with applications to repeatability and reproducibility

11. Overall Uncertainty of the Gage under evaluation when it is used to produce measurements
To sum up the discussion of this Gage R&R Analysis, the final goal is to determine the uncertainty of the gage, and determine the capability of the gage. This information will be taken for further study of possible calibration study or for setting up quality control procedure to ensure the gage to be ‘capable’, that is , to ensure the expanded uncertainty is within a certain range with a high level of confidence (95%, 99% or higher).

12. How to determine the capability of a gage?
The multiple, 6, is chosen based on normal distribution so that 99.5% of chance the measurements will covered. The estimate overall uncertainty is the most current condition of the instrument. Therefore,
If GCR > 1, it indicates that the current condition is out of specs, and some adjustment or calibration of the instrument may be necessary.
The current condition can also be presented as

13. In many cases, one may be interested in repeatability and reproducibility components separately, and present each in terms of %GCR for Repeatability, %GCR for Reproducibility.
A simple guideline can then be determined to monitor the instrument (gage or system). One common guideline used in industry is:
If %GCR Repeatability > 5%, a yellow warning sign is flagged.
If %GCR Repeatability > 10%, a red sign is flagged, and an immediate attention is needed for instrument adjustment or calibration.
Similarly, one can set up a flag system for reproducibility.
If If %GCR Repeatability > 20%, a yellow warning sign is flagged.
If %GCR Repeatability > 30%, a red sign is flagged, and an immediate action is needed for operator retraining or monitoring the process produces the parts.

14. Two Types of Gage R&R Experimental Designs
Gage R&R (Crossed): When the same parts are used cross the entire study. That is every operator measures the same parts.In experimental design language, this is a two-factor factorial design with both factors are random effect factors.
The two factors are Operators and Parts.
The statistical model is

15. Gage R&R (Nested): In cases where one part can only be measured once
Gage R&R (Nested): In cases where one part can only be measured once. Once it is used, it can no longer be used. In this case, parts are nested within operator. Each operator measures different sets of parts. It is important to choose the parts (the experimental units) as homogeneous as possible, so that the variability due to operator reflects the uncertainty of operators not because of the different parts being used by different operators.
The statistical model describes this design is a two nested random effects model :

16. Use Minitab to perform the analysis for the Punch Height Case Study
Minitab provides two methods for Gage R&R(Crossed Design) : Xbar and R, or ANOVA, one method for the Gage R&R (Nested Design): ANOVA method. In addition, there is a Gage R&R Run Chart tool to show the measurements for each operator from part to part.
Data Preparation: Gage R& R data must be arranged in three columns: One for Operator ID, one for Part number and one for the measurement. If the data originally are created as each operator’s measurement is entered in one column (eg., C1 for Parts Number, C2 for the measurement of Operator 1, C3 for Operator 2 and C4 for Operator 3), then data have to be ‘STACKED’ together. Steps of using Minitab to ‘Stack’ several columns into one:
Go to Manip, choose ‘Satck’, select ‘Stack Columns’.
In the dialog box, enter column # where the new stacked column will be, and the corresponding index column as the subscripts.

17. Gage R&R (Crossed Design):
Before running the Gage R&R analysis, you need to decide if the design is a ‘crossed factorial design’, or ‘a nested design’ by checking if the same parts are used by each operator (Crossed) or different parts are use by each operator (Nested).
Gage R&R (Crossed Design):
Go to Stat, go to Quality Tools, choose Gage R&R Study (Crossed).
In the dialog box, enter the Columns for Part Operator and Measurement. Choose the Method – Either ANOVA or Xbar-R.
There are two selections: Gage Infor is for keeping track of the Gage information. Options is for Tolerance Analysis. In the uncertainty study, this is irrelevant.
However, if one is conducting a quality control monitoring, this provides the information about the process tolerance and how it is compared with a given tolerance range. In the box: “study Variation is 5.15 by default, 5.15x(s.d.), which is designed to cover 99% of all possible values (NOTE 5.15 = 2(z(.005)). Under normal curve, 5.15(s.d.) covers approximately 99% of the data values). One can choose different multiple for difference level of confidence.

18. The following graph is from the Gage Run Chart Procedure
The following graph is from the Gage Run Chart Procedure. In Minitab, go to Stat, choose Quality Tools, selcet Gage Run Chart, then enter columns numbers for Part, Operator, and Measurement.

19. Gage R&R Study - XBar/R Method for Height
%Contribution
Source Variance (of Variance) Total Gage R&R
Repeatability
Reproducibility
Part-to-Part
Total Variation
StdDev Study Var %Study Var
Source (SD) (5.15*SD) (%SV)
Total Gage R&R
Repeatability
Reproducibility
Part-to-Part
Total Variation
Number of distinct categories = 1

21. Gage R& R Analysis – ANOVA Method
Two-Way ANOVA Table With Interaction
Source DF SS MS F P
Part
Operator
Operator*Part
Repeatability
Total
Two-Way ANOVA Table Without Interaction
Part
Operator
Repeatability

22. If the specs for the gage is
Gage R&R (BY ANOVA Method – Crossed Design)
%Contribution
Source VarComp (of VarComp)
Total Gage R&R
Repeatability
Reproducibility
Operator
Part-To-Part
Total Variation
StdDev Study Var %Study Var
Source (SD) (5.15*SD) (%SV)
Total Gage R&R
Repeatability
Reproducibility
Operator
Part-To-Part
Total Variation
Number of Distinct Categories = 1
If the specs for the gage is
What is the GCR? What is the %GCR for Repeatability? Does the gage need calibration?

24. Gage R&R (Nested Design)
The same Punch Height Case Study is use to demonstrate the Nested Model – Assuming the steel punches can only be measured once by an operator. In this set up, we need to prepare 30 different experimental units, in stead of 10.
Gage R&R (Nested) for Height
Nested ANOVA Table
Source DF SS MS F P
Operator
Part (Operator)
Repeatability
Total

25. NOTE: No Operator by Part Interaction
Gage R&R (Nested Model)
%Contribution
Source VarComp (of VarComp) Total Gage R&R
Repeatability
Reproducibility
Part-To-Part
Total Variation
StdDev Study Var %Study Var
Source (SD) (5.15*SD) (%SV)
Total Gage R&R
Repeatability
Reproducibility
Part-To-Part
Total Variation
NOTE: No Operator by Part Interaction

27. Use of General Linear Model Approach to Analyze the Punch Height Gage R&R data
General Linear Model: Height versus Operator, Part
Factor Type Levels Values
Operator random OperA OperB OperC
Part random
Analysis of Variance for Height, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Operator
Part
Operator*Part
Error
Total
Unusual Observations for Height
Obs Height Fit SE Fit Residual St Resid R R R R R

28. Source(1) Operator is tested by suing Source (3) as Error Term.
Expected Mean Squares, using Adjusted SS
Source Expected Mean Square for Each Term
1 Operator (4) (3) (1)
2 Part (4) (3) (2)
3 Operator*Part (4) (3)
4 Error (4)
Error Terms for Tests, using Adjusted SS
Source Error DF Error MS Synthesis of Error MS
1 Operator (3)
2 Part (3)
3 Operator*Part (4)
Variance Components, using Adjusted SS
Source Estimated Value
Operator
Part
Operator*Part
Error
The EMS provides information for determining how we should conduct the F-test.
Source(1) Operator is tested by suing Source (3) as Error Term.
Source (2) Part uses Source (3) as the Error Term.
Source (3) Operator*Part uses the Source (4) Random Error as the Error Term.

29. The estimated Variance Component for Operator*Part is Negative!
Variance Components, using Adjusted SS
Source Estimated Value
Operator
Part
Operator*Part
Error
Least Squares Means for Height
Operator Mean
OperA
OperB
OperC
Part
NOTE:
The estimated Variance Component for Operator*Part is Negative!
This is not realistic in real world applications. If it is negative, the value zero is usually taken.

30. Hands-on Project for Gage R&R Analysis
(Data Source: Vardeman & Jobe, 1999)
In a system of a sequence of operations, one process is to measure the angle at which fibers are glued to a sheet of base material. The correct angle is extremely important in the later processes. A Gage R&R project is determined to study the uncertainty of the degree of the angle. A team of four members are chosen for the study. Each team member measures five specimens for three times. The same five specimens are used through the the project. The tolerance specification is
The measurements are recorded in the following table.
Conduct a through analysis of Gage R&R study and prepare a summary report of the findings.

31. Row Anal Speci Angle