Gauge & R&R [Repeatability & Reproducibility] Analysis

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.

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.

Overall Variation

Pat-to-Part Variation

Measurement System Variation

Variation due to Gage

Variation due to Operators

Repeatability

Reproducibility

Operators

Operator by Part

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

1 1 496 497 497

2 1 496 499 498

3 1 499 497 496

4 2 498 498 497

5 2 497 496 499

6 2 499 499 500

7 3 498 497 496

8 3 498 498 498

9 3 498 497 497

10 4 497 496 498

11 4 497 496 497

12 4 498 499 497

13 5 499 499 499

14 5 501 499 499

15 5 500 499 500

Row Punch OperA OperB OperC

16 6 499 500 498

17 6 498 499 498

18 6 499 497 498

19 7 503 498 500

20 7 499 499 499

21 7 502 499 502

22 8 500 501 500

23 8 499 498 501

24 8 499 499 499

25 9 499 500 500

26 9 500 500 499

27 9 499 498 500

28 10 497 500 496

29 10 496 494 498

30 10 496 496 496

A statistical Model for Describing the Gage R&R Study

ijk The observation , y is the kth measurement made by operator j on part i.

It can be expressed by a two-way random effect model:

( ) , i=1,2,...,I; j = 1,2,..., J; k = 1,2,. ..,r.ijk i j ij ijky e

where is the unknow grand mean,

~ (0, ), which are the ranom effects of different parts.

~ (0, ), which are the random effects of different operators.

~ (0, ), which are th erandom effec

i

j

ij

N

N

N

ijk

2 2 2 2

ts of the parts-operator interaction

e ~ (0, ), which is the random error due to replications.

And these components are independent.

Note we have four variance components: , , , .

N

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, On the other hand, the reproducibility is the uncertainty among operators for measuring the same part. Therefore, the variance component for the reproducibility is

2 2

2 2

2 2 2 2 2

2 2 2 2

,

Further more, the overall uncertaity of a mesurement is

NOTE :

T

repeatability reprodicibilty

overall reproducibility repeatability

overall

he proportion of uncertainty due to Repeatability and Reproducibility can be obtain.

How to estimate these variance components?

Two ways to determine these variance components:

1. X-bar & R-chart approach.: can be done by hand, but it less accurate.

2. ANOVA approach : Computer will be handy.

We will discuss both approaches

Our goal is to estimate

2

R =

( )

.

repeatability d r

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) . Therefore,

and repeatability reporducibility

The following is the Rij’s for the Measured Punch Heights

Punch Operator 1 Operator 2 Operator 3

1 3 2

2 2 3

3 0 1

4 1 3

5 2 0

6 1 3

7 4 1

8 1 3

9 1 2

10 1 0

The average range is __________ And d2(r=3) = _________

Therefore, ˆ ___________repeatability

1.9, 1.693, 1.12

Determine the reproducibility

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

ij

2 2 2ij

max min from the model,

y

Now, for a given fixed part i, y has the variance /

(since part is fixed. No variation among parts among these means).

Therefore

i ij ijjj

i j ij ij

y y

e

r

2 2 22

2 2

2

, using the range to estimate s.d., we have:

ˆ ˆ ˆ E( ) ( ) /

By some simple algebra, the variance component for reporducibility

is estimated by ( )

i d J r

d J

2 2

2

2 2

2 2

1

( )

Since this difference may be negative, we take reproducibility to be:

1ˆ max(0,

( ) ( )reporducibility

R

r d r

R

d J r d r

For the Punch Height case study, the following table gives the measurement means of ith part by the jth operator:

Punch (i)

1 497.00 497.67 497.00 .67

2 498.00 497.67 498.67 1.00

3 498.00 497.33 497.00

4 497.33 497.00 497.33

5 500.00 499.00 499.33

6 498.67 498.67 498.00

7 501.33 498.67 500.33

8 499.33 499.33 500.00

9 499.33 499.33 499.67

10 496.33 496.67 496.67

1iy 3iy2iy i

2 2_________, d ( ) (___) _______

ˆ max(0, _________) _________reproducibility

J d

0

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

2 2

2

2

2

ˆ ˆ

1ˆ max(0, ( ))

1ˆ max(0, ( ))

1 1ˆ max{0, ( ( 1) )) }

repeatability

reproducibility

MSE

MSB MSABIr

MSAB MSEr

MSB I MSAB MSEIr r

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).

How to determine the capability of a gage?For most gages (instruments), it usually comes with an engineer specification limit.

The specification limits speficies the range of the measurement the instruct will produce.

It is usually stated as th standarde form of x .

When the measurement is out of the limit, the instrucment is no longer function properly,

and some adjusment or calibration will be needed.

One of the final goal of a Gage R&R ana

xU

lysis is to evaluate how cacapble of the

instrument be measuring the Gage Capability Ratio:

ˆ ˆ6 6GCR =

2U Upper Spec - Lower Specoverall overall

x

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 ˆbest overallx

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.

ˆ ˆ6 6%GCR for Repeatability = 100 100

2U Upper Spec - Lower Specrepeatability repeatability

x

ˆ ˆ6 6%GCR for Reproducibility = 100 100

2U Upper Spec - Lower Specreporducibility reproducibility

x

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.

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

Two Types of Gage R&R Experimental Designs

( ) , i=1,2,...,I; j = 1,2,..., J; k = 1,2,. ..,r.


~ (0, ), which are the ranom effects of different parts.

~ (0, ), which are the random e

ijk i j ij ijk

i

j

y e

N

N

ijk

ffects of different operators.

~ (0, ), which are th erandom effects of the parts-operator interaction


And these components are independ

ij N

N

ent.

NOTE: The model has an interaction term between Operator abd Part.

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 :

( ) ( )

( ) ( )

, i=1,2,...,I; j = 1,2,..., J; k = 1,2,...,r.


~ (0, ), which are the random effects of different operators.

~ (0, ), which are the

ijk j i j k ij

j

i j

y e

N

N

ijk

ranom effects of different parts within operator.


NOTE: The the Part is nested within Operator.

N

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:

1. Go to Manip, choose ‘Satck’, select ‘Stack Columns’.

2. In the dialog box, enter column # where the new stacked column will be, and the corresponding index column as the subscripts.

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):

1. Go to Stat, go to Quality Tools, choose Gage R&R Study (Crossed).

2. In the dialog box, enter the Columns for Part Operator and Measurement. Choose the Method – Either ANOVA or Xbar-R.

3. 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.

Gage name:Date of study:Reported by:Tolerance:

Misc:

494495496497498499500501502503

1 2 3 4 5Part

Heig

ht

OperA

OperB

OperC

494495496497498499500501502503

6 7 8 9 10Part

Heig

ht

Runchart of Height by Part, Operator

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.

StdDev Study Var %Study Var

Source (SD) (5.15*SD) (%SV)

Total Gage R&R 1.12310 5.78398 70.87

Repeatability 1.12227 5.77968 70.82

Reproducibility 0.04331 0.22304 2.73

Part-to-Part 1.11810 5.75821 70.55

Total Variation 1.58477 8.16158 100.00

Number of distinct categories = 1

Gage R&R Study - XBar/R Method for Height

%Contribution

Source Variance (of Variance) Total Gage R&R 1.26136 50.22

Repeatability 1.25949 50.15

Reproducibility 0.00188 0.07

Part-to-Part 1.25015 49.78

Total Variation 2.51151 100.00

ˆ

ˆ

repeatability

reproducibility

496

497

498

499

500

501

502 OperA OperB OperC

Xbar Chart by Operator

Sam

ple

Mea

n

Mean=498.4

UCL=500.3

LCL=496.4

0

1

2

3

4

5

6OperA OperB OperC

R Chart by Operator

Sam

ple

Ran

ge

R=1.9

UCL=4.891

LCL=0

1 2 3 4 5 6 7 8 9 10

496

497

498

499

500

501

Part

OperatorOperator*Part Interaction

Ave

rage

OperA OperB OperC

OperA OperB OperC

494495496497498499500501502503

Operator

Response by Operator

1 2 3 4 5 6 7 8 9 10

494495496497498499500501502503

Part

Response by Part

%Contribution %Study Var

Gage R&R Repeat Reprod Part-to-Part

0

50

100

Components of Variation

Perc

ent

Gage R&R (Xbar/R) for Punch Height Study

Gage R& R Analysis – ANOVA Method

Two-Way ANOVA Table With Interaction

Source DF SS MS F P

Part 9 122.400 13.6000 14.9268 0.00000

Operator 2 2.489 1.2444 1.3659 0.28037

Operator*Part 18 16.400 0.9111 0.6260 0.86484

Repeatability 60 87.333 1.4556

Total 89 228.622

Two-Way ANOVA Table Without Interaction

Source DF SS MS F P

Part 9 122.400 13.6000 10.2262 0.00000

Operator 2 2.489 1.2444 0.9357 0.39666


Total 89 228.622

Gage R&R (BY ANOVA Method – Crossed Design)

%Contribution

Source VarComp (of VarComp)

Total Gage R&R 1.3299 49.38



Operator 0.0000 0.00

Part-To-Part 1.3633 50.62




Total Gage R&R 1.15322 5.93908 70.27

Repeatability 1.15322 5.93908 70.27


Operator 0.00000 0.00000 0.00

Part-To-Part 1.16762 6.01326 71.15

Total Variation 1.64111 8.45174 100.00

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?

If the specs for the gage is

What is the GCR? What is the %GCR for Repeatability? Does the gage need calibration?

2.0

5.0

Gage name:Date of study:Reported by:Tolerance:

Misc:

0

496

497

498

499

500

501



Sam

ple

Mea

n

Mean=498.4

UCL=500.3

LCL=496.4

0

0

1

2

3

4

5

6OperA OperB OperC

R Chart by Operator

Sam

ple

Ran

ge

R=1.9

UCL=4.891

LCL=0

1 2 3 4 5 6 7 8 9 10

496

497

498

499

500

501

Part

OperatorOperator*Part Interaction

Ave

rage

OperA OperB OperC

OperA OperB OperC

494495496497498499500501502503

Operator

By Operator

1 2 3 4 5 6 7 8 9 10

494495496497498499500501502503

Part

By Part



0

50

100


Perc

ent

Gage R&R (ANOVA) for Height

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 2 2.489 1.24444 0.24207 0.78668

Part (Operator) 27 138.800 5.14074 3.53181 0.00002


Total 89 228.622

Gage R&R (Nested Model)

%Contribution

Source VarComp (of VarComp) Total Gage R&R 1.45556 54.23



Part-To-Part 1.22840 45.77




Total Gage R&R 1.20646 6.21329 73.64

Repeatability 1.20646 6.21329 73.64


Part-To-Part 1.10833 5.70790 67.65

Total Variation 1.63828 8.43713 100.00

NOTE: No Operator by Part Interaction

Gage name:Date of study:Reported by:Tolerance:Misc:

496

497

498

499

500

501



Sam

ple

Mea

n

Mean=498.4

UCL=500.3

LCL=496.4

0

1

2

3

4

5

6OperA OperB OperC

R Chart by Operator

Sam

ple

Ran

ge

R=1.9

UCL=4.891

LCL=0

OperA OperB OperC

494495496497498499500501502503

Operator

By Operator

1 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 9101 2 3 4 5 6 7 8 910OperA OperB OperC

494495496497498499500501502503

PartOperator

By Part (Operator)



0

50

100


Perc

ent

Gage R&R (Nested) for Height

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 3 OperA OperB OperC

Part random 10 1 2 3 4 5 6 7 8 9 10

Analysis of Variance for Height, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P

Operator 2 2.489 2.489 1.244 1.37 0.280

Part 9 122.400 122.400 13.600 14.93 0.000

Operator*Part 18 16.400 16.400 0.911 0.63 0.865

Error 60 87.333 87.333 1.456

Total 89 228.622

Unusual Observations for Height

Obs Height Fit SE Fit Residual St Resid

3 499.000 497.000 0.697 2.000 2.03R

20 499.000 501.333 0.697 -2.333 -2.37R

42 499.000 497.000 0.697 2.000 2.03R

58 500.000 496.667 0.697 3.333 3.38R

59 494.000 496.667 0.697 -2.667 -2.71R

Expected Mean Squares, using Adjusted SS

Source Expected Mean Square for Each Term

1 Operator (4) + 3.0000(3) + 30.0000(1)

2 Part (4) + 3.0000(3) + 9.0000(2)

3 Operator*Part (4) + 3.0000(3)

4 Error (4)

Error Terms for Tests, using Adjusted SS

Source Error DF Error MS Synthesis of Error MS

1 Operator 18.00 0.911 (3)

2 Part 18.00 0.911 (3)

3 Operator*Part 60.00 1.456 (4)

Variance Components, using Adjusted SS

Source Estimated Value

Operator 0.01111

Part 1.40988

Operator*Part -0.18148

Error 1.45556

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.

Variance Components, using Adjusted SS

Source Estimated Value

Operator 0.01111

Part 1.40988

Operator*Part -0.18148

Error 1.45556

Least Squares Means for Height

Operator Mean

OperA 498.5

OperB 498.1

OperC 498.4

Part

1 497.2

2 498.1

3 497.4

4 497.2

5 499.4

6 498.4

7 500.1

8 499.6

9 499.4

10 496.6

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.

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.

04

Hands-on Project for Gage R&R Analysis

(Data Source: Vardeman & Jobe, 1999)

Row Anal Speci Angle

1 1 1 23

2 1 1 20

3 1 1 20

4 1 2 17

5 1 2 20

6 1 2 20

7 1 3 23

8 1 3 20

9 1 3 22

10 1 4 16

11 1 4 22

12 1 4 15

13 1 5 20

14 1 5 20

15 1 5 22

16 2 1 20

17 2 1 25

18 2 1 17

19 2 2 15

20 2 2 13

21 2 2 16

22 2 3 20

23 2 3 23

24 2 3 20

25 2 4 20

26 2 4 22

27 2 4 18

28 2 5 18

29 2 5 20

30 2 5 18

31 3 1 20

32 3 1 19

33 3 1 16

34 3 2 15

35 3 2 20

36 3 2 16

37 3 3 15

38 3 3 20

39 3 3 22

40 3 4 17

41 3 4 18

42 3 4 15

43 3 5 20

44 3 5 16

45 3 5 17

46 4 1 18

47 4 1 18

48 4 1 21

49 4 2 21

50 4 2 17

51 4 2 16

52 4 3 17

53 4 3 20

54 4 3 18

55 4 4 16

56 4 4 15

57 4 4 17

58 4 5 18

59 4 5 18

60 4 5 20

Gauge & R&R [Repeatability & Reproducibility] Analysis

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Transcript of Gauge & R&R [Repeatability & Reproducibility] Analysis