Gage Repeatability and Reproducibility Experiment for Measuring Drill Holes
Explore a 2-way random effects ANOVA study on the measured diameter of drill holes in wood parts, addressing gage repeatability and reproducibility. The experiment examines factors A (Drill Hole) and B (Measurer) with 10 and 3 levels, respectively, using 3 replicates per measurer/hole. The statistical model, variance analysis, and expected mean squares are detailed in the study.
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2-Way Random Effects ANOVA Gage Repeatability and Reproducibility Experiment for Measuring Drill Holes in Wood Parts Source: M-H. C. Li and A. Al-Refaie (2008). Improving Wooden Parts Quality by Adopting DMAIC Procedure, Quality and Reliability Engineering International, Vol. 24, pp. 351-360
Data Description Gage Repeatability and Reproducibility Experiment Response: Measured Diameter of Drill Hole (10s mm) Factors: A: Drill Hole (Random Factor, a = 10 Levels) B: Measurer (Random Factor, b = 3 Levels) Replications (3 replicates per Measurer/Hole, in random order) Measurer Hole\Rep 1 2 3 4 5 6 7 8 9 10 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 256.0 255.8 256.1 255.7 255.7 257.2 256.6 257.2 255.1 256.1 256.1 256.2 255.8 256.3 255.6 256.7 256.3 256.6 254.6 255.5 255.7 256.3 256.1 256.2 255.9 256.8 255.9 256.9 254.6 255.7 255.6 256.5 256.2 255.7 256.4 256.8 256.1 256.8 254.7 256.2 256.2 256.4 255.9 256.3 256.2 256.7 256.3 256.7 254.4 255.9 255.9 256.3 256.1 255.9 256.7 256.4 256.6 257.3 254.9 256.4 255.7 256.7 256.2 255.9 256.8 256.5 256.9 257.2 254.7 256.1 255.9 256.6 256.4 256.2 256.2 256.3 256.4 256.7 255.4 255.7 256.2 256.4 256.7 256.3 256.5 256.7 256.8 256.9 254.9 256.2
Statistical Model ( ) = + + + + = = = = = = = = 1,..., 10; 1,..., 3; 1,..., 3 90 Y i a j b k n N abn ijk i j ijk ij ( ) ( ) ( ( ) ( ) i ) ( ) 2 a 2 b 2 ab 2 ~ 0, ~ 0, ~ 0, ~ 0, NID NID NID NID i j ijk j ijk ij ij ( ) = + + + + = 0 0 0 0 + + + + = E Y E ijk i j ijk ij ( ) = + + + + = + + + + = + + + 2 a 2 b 2 ab 2 2 a 2 b 2 ab 2 0 V Y V ijk i j ijk ij + + + + + = = = = = = = 2 a 2 b 2 ab 2 ', ', ', ', ', ', ', ', ', ', ' ' i i i j i j i j i j i j j k j k j j j k k 2 a 2 b 2 ab = , COV Y Y 2 a , ' , ' k k k k i i i k k ' ' ' i j k ijk 2 b 0 , ' Analysis of Variance: ( ) ( ) ( ) a b n a 2 2 2 = = = 1 SS Y Y bn Y abn Y df a i i A A = = = = 1 1 1 1 i j k i ( ) ( ) ( ) a b n b 2 2 2 = = = 1 SS Y Y an Y abn Y df b j j B B = = = = 1 1 1 1 i j k j ( ) ( ) ( ) ( ) ( ) a b n a b a b 2 2 2 2 2 ( )( ) = + = + = 1 1 SS Y Y Y Y n Y bn Y an Y abn Y df a b ij i j ij i j AB AB = = = = = = = 1 1 1 1 1 1 1 i j k i j i j ( ) ( ) a b n a b n a b 2 2 ( ) = = = 2 1 SS Y Y Y n Y df ab n ij ij ERR ERR ijk ijk = = = = = = = = 1 1 1 1 1 1 1 1 i j k i j k i j
Expected Mean Squares - I ijk E Y = = = = = E Y E Y E Y E Y ij i j n + 1 n n n n ( ) ( ) ij V Y ijk V Y = = + = + + + + + + 2 a 2 b 2 ab 2 2 a 2 b 2 ab 2 , 2 V Y COV Y Y n ' ijk ijk ijk 2 = = = = + 1 1 + 1 ' 1 k + k k k k ( ) )( ) ( = + + + = + + + 2 a 2 b 2 ab 2 2 a 2 b 2 ab 2 2 a 2 2 b 2 2 ab ) 2 1 n n n n n n n + = ) 2 2 ( 1 n 1 n n ij V Y 2 = = + + + = + + + + 2 a 2 b 2 ab 2 a 2 b 2 ab 2 V Y V Y E Y ij ij ijk 2 n n = 1 k 1 1 b n b n b n n b b n n ijk V Y = = + + 2 , 2 , V Y V Y COV Y Y COV Y Y ' ' ' i ijk ijk ijk ijk ij k = = = = = = = + = = + = ' 1 = 1 1 1 1 1 1 ' ) 1 1 ' 1 1 j k j k j k k k j j j k k ( V Y )( + ( ) ( ( ) ) + + + + + + = + + + 2 a 2 b 2 2 2 a 2 b 2 ab 2 2 a 2 2 2 a 2 2 b 2 ab 2 1 1 bn bn n b b n b n bn bn ab + 2 b 2 ab 2 b 2 ab 2 2 ( 1 2 = = + + = + + + 2 a 2 a 2 V Y E Y i i i 2 2 b n b bn + b bn 2 a 2 ab 2 ( ) 2 = + + + 2 b 2 By direct analogy: E Y j a an ( ) ( ) ( ) ( ) ( ) = + + + + + + + + + 2 a 2 b 2 ab 2 2 a 2 b 2 ab 2 2 a 2 2 b 1 1 1 0 V Y abn abn n ab b n a a bn 2 a 2 b 2 ab 2 = + + + = + + + 2 2 2 a 2 2 2 b 2 2 ab 2 ab n a bn abn abn V Y a b ab abn 2 a 2 b 2 ab 2 ( ) 2 = + + + + 2 E Y a b ab abn
Expected Mean Squares - II = + a b n ( ) + + + + = 2 2 a 2 b 2 ab 2 2 1) E Y abn ijk = = = 1 1 1 i j k = + + + 2 a 2 b 2 ab 2 2 abn abn abn abn abn 2 ( ) a b 2 = + + + + + = 2 a 2 b 2 ab 2 2) E n Y abn ij n = = 1 + 1 i j = + + 2 a 2 b 2 ab 2 2 abn abn abn ab abn + 2 b 2 ab = an 2 ( ) a 2 + + + = 2 a 2 3) E bn abn Y a bn i b bn = 1 i + = + + + 2 a 2 b 2 ab 2 2 an a abn + 2 a 2 ab 2 ( ) b 2 = + + + = 2 b 2 4) E an Y abn j a an = 1 j = + + + + 2 a 2 b 2 ab 2 2 bn abn bn b abn 2 a 2 b 2 ab ( ) 2 = + + + + = 2 2 5) E abn Y abn a + b ab = + + + 2 a 2 2 ab 2 2 bn an n abn b
Expected Mean Squares - III Analysis of Variance: ( = ( ) ( ) a 2 2 ) ( ) ( ) = = + + 2 a 2 ab 2 1 1 1 E SS E bn Y E abn Y a bn a n a i A = 1 i = + + 2 a 2 ab 2 1 df a E MS bn n A A = ( ) ( ) b 2 2 ( ) ( ) ( ) = = + + 2 b 2 ab 2 1 1 1 E SS E an Y E abn Y a b n b n b j B = 1 j = + + 2 b 2 ab 2 1 df b E MS an n B B ( ) ( ) ( ) ( ) a b a b 2 2 2 2 = + = E SS E n Y E bn Y E an Y E abn Y ij i j AB = = + = = 1 1 1 1 i j i j ( )( ( a ) b ( )( E MS ) = 2 ab 2 1 1 1 1 a b n a b )( ) = = + 2 ab 2 1 1 df n AB AB ( ) a b n a b 2 ( ) ( ) E MS = = = = 2 2 2 1 1 E SS E Y E n Y ab n df ab n ij ERR ERR E RR ijk = = = = = 1 1 1 1 1 i j k i j
Estimating and Testing Variance Components ^ = = 2 2 E MSE MSE ^ MSAB MSE = + = 2 2 ab 2 ab E MSAB n n ^ MSA MSAB bn MSB an = + + = 2 2 ab 2 a 2 a E MSA n bn ^ MSAB = + + = 2 2 ab 2 b 2 b E MSB n an 2 k g MS i i ^ k = = = 1 i 2 * Satterthwaite Approximation for Degrees of Freedom: where df g MS ( ) * i i 2 g MS df k = 1 i i i = 1 i i ^ ^ 2 * 2 * df df ( ) 2 * Approximate 1 100% CI for : , * * 2 2 1 /2; /2; df df MSAB MSE MSA MSAB MSB MSAB = = 2 ab 2 ab AB AB A : 0 : 0 Test Stat: Rejection Region: H H F F F ( )( ) ( ) 1 , 0 AB AB ; 1 1 a b ab n = = 2 a 2 a A A A : 0 : 0 Test Stat: Rejection Region: H H F F F ( )( ) 0 A A ; 1, 1 1 a a b = = 2 b 2 b B B A : 0 : 0 Test Stat: Rejection Region: H H F F F ( )( ) 0 B B ; 1, 1 1 b a b
Experimental Model/Results Source Part Operator Part:Operator Residuals Df Sum Sq 25.9721 0.7716 2.0129 4.0067 Mean Sq 2.88579 0.38578 0.11183 0.06678 9 2 18 60 Here we treat the Drill Hole as the "Product" and Measurer as the "Operator" = + + + = 2 Product 2 Operator 2 Product + = 2 Reproducibility + 2 Repeatability 2 Total 2 PxO 2 2 Gage 2 Gage 2 Reproducibility = 2 Operator + 2 Repeatability = 2 PxO 2 2 ^ ^ = = = Repeatability = = 2 2 2 0.06678 E MS MS ERR ERR 2 0.11183 0.06678 3 MS an MS MS ^ = + = = = 2 2 PxO 0.015017 PxO ERR E MS n PxO PxO n MS 2 0.38578 0.11183 10(3) ^ Operator PxO = + + 2 Operator Operator = = = 2 2 PxO 0.009132 E MS n an Operator 2 2.88579 0.11183 3(3) MS MS ^ = + + 2 Product Product = = = 2 2 PxO 0.308218 Product PxO E MS n bn Product bn 2 2 2 ^ ^ ^ Reproducibility = Operator + = 0.009132 0.015017 + = 0.024149 PxO 2 2 2 ^ ^ ^ = + Repeatability = 0.024149 0.06678 + = 0.090929 Gage Reprod ucibility
Tests Concerning Variance Components 0.11183 0.06678 MS MS = = = = PxO 0 2 PxO PxO A 2 PxO : 0 : 0 : 1.6746 PxO H H TS F PxO ERR = ( ) = : 1.7784 1.6746 .0703 RR F .05;18,60 F P F PxO 18,60 0.38578 0.11183 = MS MS = = = = O 0 2 O O A 2 O : 0 : 0 : 3.4497 O H H TS F O PxO ( ) = : 3.5546 3.4497 .0539 RR F .05;2,18 F P F O 2,18 2.88579 0.11183 = MS MS = = = = 2 P 2 P P P A : 0 : 0 : 25.8052 H H TS F P 0 P PxO ( ) = : 2.4563 25.8052 .0000 RR F .05;9,18 F P F 9,18 P
Approximate Confidence Intervals ( ) 2 2 0.015017 MS MS 1 3 1 3 ^ = = + = = = 0.015017 2.64 PxO ERR MS MS df PxO PxO ERR PxO 2 n 2 1 3 1 3 ( ) ( ) 0.06678 0.11183 + 18 60 ( ) 2 2 0.009132 MS MS an 1 30 1 ^ = = + = = = 0.009132 1.00 O PxO MS MS df O O PxO O 2 30 2 1 1 30 ( ) ( ) 0.11183 0.38578 30 + 2 18 ( ) 2 2 0.308218 MS MS bn 1 9 1 9 ^ = = + = = = 0.308218 8.31 PxO P MS MS df P PxO P P 2 2 1 9 1 9 ( ) ( ) 0.11183 2.88579 + 9 18 ( 8.6651 ) ( 0.1430 ) 2.64 0.015016 2.64 0.015016 2 .975;2.64 = .025;2 = 2 2 PxO 0.1430 8.6651 Approximate 95% CI for : , 0.004573, 0.277108 .64 ( 5.0239 ( 17.9992 ) ( 0 ( 2.3381 ) 1.00 0.009132 1.00 0.009132 .0010 2 .975;1.00 = 2 .025;1.00 = 2 O 0.0010 5.0239 Approximate 95% CI for : , 0.001816,9.124630 ) ) 8.31 0.308218 8.31 0.308218 2 .975;8.31 = 2 .025;8.31 = 2 P 2.3381 17.9992 Approximate 95% CI for : , 0.142296,1.095427 Approximate CIs for Standard Deviations: : 0.067624, 0.526410 : 0.0426146,3.020700 : 0.37722,1.046627 PxO O P
