2-Way Mixed Analysis of Variance

Women’s PBA - 2009

Data Description• Women’s Professional Bowling Association –

Qualifying rounds at Alan Park, Michigan (2009).• Factors:

A: Oil Pattern (Fixed) with a=4 levels: • 1=Viper, 2=Chameleon, 3=Scorpion, 4=Shark

B: Bowler (Random) with b=15 levels:• 1=Diandra Abaty, 2=Shalin Zulkiffi, 3=Liz Johnson, 4=Kelly Kulick,

5=Clara Guerrero, 6=Jennifer Petrick, 7=Wendy MacPherson, 8=Shannon Pluhowski, 9=Stephanie Nation, 10=Tammy Boomershine, 11=Amanda Fagan, 12=Aumi Guerra, 13=Michelle Feldman, 14=Shannon O'Keefe, 15=Jodie Woessner

• Replicates: Each bowler rolled 2 sets of 7 games on each pattern (Y = Total Pins in a game, n=14)

Statistical Model

4

1

1,..., 4; 1,..., 15; 1,..., 14

Overall Population mean across 4 oil patterns and population of bowlers

Effect of Oil Pattern: 0

Effect of Bowler:

ijk i j ijkij

thi i

i

thj

Y b ab e i a j b k n

i

b j

2

2

2

~ 0,

Interaction Effect of Oil Pattern and Bowler: ~ 0,

Random Error term: ~ 0,

Note that this is an Unrestricted Model (wrt interaction effect).

j b

th thabij ij

ijk ijk

j ijkij

b NID

ab i j ab NID

e e NID

b ab e

**

* * * *2 2

'1

Restricted Model (interaction effects sum to zero over oil patterns for all bowlers):

1 10 ~ 0, , '

ijk i j ijk i j ijkj ij j ij

a

ab abij ij ij i ji

Y b ab ab ab e b ab e

aab ab N COV ab ab i i

a a

Covariance Structure / ANOVA (Unrestricted Model)

2 2 2 2 2 2

2 2' '

' ' '

0 0 0

0 0

' : COV , COV ,

' , ' : COV , COV ,

ijk i j ijk i iij

ijk i j ijk b ab b abij

ijk ijk j ijk j ijk b abij ij

ijk i jk j ijk jij i j

E Y E b ab e

V Y V b ab e

k k Y Y b ab e b ab e

i i k k Y Y b ab e b ab e

2' '

' ' ' ' ' '' '

2 2 2

1 1

2 2 2

1 1

2

1 1

' , ', , ' : COV , COV , 0

i jk b

ijk i jk j ijk j i j kij i j

a a

i i

i i

b b

j j

j j

a b

ij i j

i j

j j i i k k Y Y b ab e b ab e

SSA bn Y Y bn Y abnY

SSB an Y Y an Y abnY

SSAB n Y Y Y Y n Y

2 2 2 2

1 1 1 1

2 22

1 1 1 1 1 1 1 1

a b a b

ij i j

i j i j

a b n a b n a b

ij ijijk ijki j k i j k i j

bn Y an Y abnY

SSE Y Y Y n Y

Expectations and Variances of Means - I

2 2 2 2 2 2 2 2 2

22 2 2 2 2 2 2

1

'21 1 '

2 2

1 1

1 12 COV ,

ijk i ijk b ab ijk b ab i

n

ij ijk i ik

n n

ij ijk ijk ijk ijkk k k k

V Y E Y E Y Y E Y E Y E Y E Y V Y

E Y V Y E Y

E Y E Y nn n

V Y V Y V Y Y Yn n

2 2 2 2 22

2 2 22 22 2 2 2 2 2 2 2 2 2

2 2

'

1

12

2

1 1( 1)

1 1 Note: COV , C

b ab b ab

b abijb ab b ab b ab i

b

i ij ij iji ij

nn

n

n nn n n n n n E Y

n n n

E Y E Y b Y Yb b

'1 1

22 2

'2 21 1 '

2 2 2 2 2 22 2

1 1OV , 0

1 1 12 COV , ( 1)(0)

n n

ijk ij kk k

b b

i ij ij ij ij b abj j j j

b ab b abi i

Y Yn n

V Y V Y V Y Y Y b b bb b b n

n n n nE Y

bn bn

Expectations and Variances of Means - II

2 2 2' ' 2

1 1

1 1 1

'21 1 '

22

1 1 1Note: COV , COV ,

1 1 1 10

1 12 COV ,

1

n n

ij i j ijk i jk b bk k

a a n

j ij i ii i i

b a

j ij ij ij i j

j i i i

b a

Y Y Y Y nn n n

E Y E Y aa a a a

V Y V Y V Y Y Ya a

aa

2 2 222 2

2 2 22 2

1

2 2 2 2 2 2

21

2 2 22 2

( 1)

1 1

1 1

b abb b

b abj

b

j

j

bb ab b ab

j

j

b ab

an na a

n an

an nE Y

an

E Y E Y bb b

an n an nV Y V Y b

b b an abn

an nE Y

abn

Expected Mean Squares - I

2 2

1

2 2 2 2 2 22 2

1

2 2 2 2 2 2

1 1

2 2 2

1

1 2

1 1

a

i

i

ab ab b ab

ii

n n

b ab i ii i

n

ab ii

SSA bn Y abnY

n n an nE SSA abn bn abn

bn abn

an an an n a bn a abn

Sa n a bn E MSA E

2

2 2 1

2 2

1

2 2 2 2 2 22 2

2 2 2

2 2 2 2 2

1 1

1

1 1 11

n

ii

ab

b

j

j

b ab b ab

b ab

ab b ab

bnSA

na a

SSB an Y abnY

an n an nE SSB abn abn

an abn

abn an bn n b

SSBb n b an b E MSB E n

b

2ban

Expected Mean Squares - II

2 2 2 2

1 1 1 1

2 2 2 2 2 22 2

1 1

2 2 2 2 2 22 2

a b a b

ij i j

i j i j

n ab ab b ab

i ii i

b ab b ab

SSAB n Y bn Y an Y abnY

E SSAB

n n n nabn bn abn bn

n bn

an n an nabn abn

an abn

2 2 2

2 2 2 2

1 1

2 2 2 2

2 2

1

1 1 1 1 1 1

1 1

b ab

n n

i ii i

ab ab

ab

abn an abn an abn an bn n ab a b

bn bn abn abn

ab a b n ab a b a b n a b

SSABE MSAB E n

a b

Expected Mean Squares III & F-Tests

22

1 1 1 1 1

2 2 22 22 2 2

1 1

2 22 2 2 2

1 1

1

1

a b n a b

ijijki j k i j

a ab ab

b ab i ii i

a a

b ab i ii i

SSE Y n Y

n nE SSE abn bn abn bn

n

abn abn abn abn abn ab bn bn ab n

SSEE MSE E

ab n

0

0

0

2

2 20 1 1 , 1

2 20 1 , 1 1

0 1 1 , 1 1

: 0 : 0 : ~

: 0 : 0 : ~

: 0 : Not all 0 : ~

H

ab A ab AB a b ab n

H

b A b B b a b

H

a A i A a a b

MSABH H TS F F

MSE

MSBH H TS F F

MSAB

MSAH H TS F F

MSAB

Bowling Results (a=4, b=15, n=14)Mean OilPatt1 OilPatt2 OilPatt3 OilPatt4 SD OilPatt1 OilPatt2 OilPatt3 OilPatt4Bowler1 223.29 208.79 195.57 211.00 Bowler1 42.44 22.98 27.27 28.85Bowler2 211.79 195.71 196.43 191.57 Bowler2 28.77 27.06 24.31 19.77Bowler3 218.14 208.50 209.64 215.50 Bowler3 24.51 26.23 27.78 14.09Bowler4 219.43 216.64 212.43 216.21 Bowler4 18.72 21.74 27.52 35.62Bowler5 210.57 198.43 204.71 219.07 Bowler5 20.77 18.80 28.70 24.82Bowler6 211.00 203.29 193.07 187.14 Bowler6 30.91 16.90 23.72 30.82Bowler7 223.36 199.29 194.43 221.07 Bowler7 34.26 30.29 20.24 22.84Bowler8 209.57 214.21 208.64 201.29 Bowler8 25.17 31.64 20.76 25.94Bowler9 199.57 198.57 193.29 204.43 Bowler9 27.98 21.67 18.57 26.78Bowler10 205.86 213.71 198.36 219.29 Bowler10 33.02 20.73 31.23 27.84Bowler11 202.50 205.29 194.36 207.57 Bowler11 26.88 14.49 21.40 17.67Bowler12 206.21 182.64 196.14 194.00 Bowler12 31.98 25.80 16.73 26.59Bowler13 198.50 207.86 210.71 193.86 Bowler13 15.51 22.39 25.88 33.64Bowler14 212.00 205.86 208.29 220.21 Bowler14 30.03 25.49 22.43 13.01Bowler15 199.57 209.79 204.86 208.64 Bowler15 27.98 23.93 20.27 13.62

ANOVASource df SS MS MSError F* F(0.05) P-valueOil Pattern 3 8785.1 2928.4 819.6 3.573 2.827 0.0217Bowler 14 29964.7 2140.3 819.6 2.611 1.935 0.0082OilPattxBowler 42 34423.2 819.6 649.6 1.262 1.400 0.1271Error 780 506679.0 649.6Total 839 579852.0

Estimating Population Mean Score

2 2 2

2 2 2

^

/2, 1

.025,14

1 100% CI for :

Bowling Data:

205.9 2.145 2140.3 4 15 14

2140.3 2140.32.55 1

4(15)(14) 840

b ab

b ab

b

E Y

an nV Y

abn

E MSB an n

MSBV Y

abn

MSBY t

abn

Y t MSB a b n

MSB

abn

/2, 1

.60

1 100% CI for : 205.9 2.145(1.60) 205.9 3.43 (202.47,209.33)b

MSBY t

abn

Simple Effects – Comparing Oil Patterns Within Bowlers

'

22 2 2 2 2 2 2

'2

2 2 2' ' 2

1 1

' '

Differences in pattern means for a given bowler:

Estimator:

1

1 1 1COV , COV ,

ij i j

ij i jb ab b ab

n n

ij i j ijk i jk b bk k

ij i j ij i

Y Y

V Y n n n V Yn n

Y Y Y Y nn n n

V Y Y V Y V Y

2 222 2 2

'

^2 2

'

' /2,( 1)( 1)

.025,42

22COV , 2 2

2

21 100% CI for Simple Effect:

2 2(819.6)For Bowling Data: =2.018

14

abj ij i j b ab b

ij i jab

ij i j a b

nY Y

n n

MSABE MSAB n V Y Y

n

MSABY Y t

n

MSABt

n

/2,( 1)( 1)

10.82

22.018(10.82) 21.83a b

MSABt

n

Marginal Effects – Comparing Oil Patterns Across Bowlers

'

2 2 2

'

' ' ' ' '21 1 1 '

22

Differences in pattern means across bowlers:

Estimator:

1 1 1COV , COV , COV , 2 COV ,

1

i i

b abi i

b b b

i i ij i j ij i j ij i j

j j j j j

b

Y Y

n nV Y V Y

bn

Y Y Y Y Y Y Y Yb b b

b b bb

2

2 22 2 2 2

' ' '

^2 2

'

' /2,( 1)( 1)

1 0

22COV , 2 2

2

21 100% CI for Marginal Effect:

For Bo

b

abb ab bi i i i i i

i iab

i i a b

b

nn nV Y Y V Y V Y Y Y

bn b bn

MSABE MSAB n V Y Y

bn

MSABY Y t

bn

.025,42 /2,( 1)( 1)

2 2(819.6) 2wling Data: =2.018 2.80 2.018(2.80) 5.65

15 14 a b

MSAB MSABt t

bn bn

Pairwise Comparisons Among Oil PatternsFamily: All Comparisons among Oil Patterns (Within Bowler and Across Bowlers)

= 4 Oil Patterns = 15 Bowlers = 14 Replicates = 819.6

# of Oil Patterns = 4, # Pairs of Oil Patte

a b n MSAB

' '

rns = 4(3)/2 = 6 ( ) = ( -1)( -1) = 42

2 210.82 2.80

Critical Values: Tukey: .05,4,42 3.784 Bonferroni: .025 / 6,42 2.764

Comparing Simple (Within Bowler)

ij i j i i

df MSAB a b

MSAB MSABSE Y Y SE Y Y

n bn

q t

Means:

3.784Tukey: 10.82 = 28.95 Bonferroni: 2.764(10.82) = 29.91

2

Comparing Marginal (Across Bowler) Means:

3.784Tukey: 2.80 = 7.49 Bonferroni: 2.764(2.80) = 7.74

2

Oil Pattern i MarginalScorpion 3 201.40Chameleon 2 204.57Shark 4 207.39Viper 1 210.09

Oil Pattern i Bowler 1Scorpion 3 195.57Chameleon 2 208.79Shark 4 211.00Viper 1 223.29

Estimating Variance Components

~2 2

~2 2 2

~2 2 2 2

~ ~2

~ ~2

~2

649.6 819.6 2140.3 14 4

649.6 25.5

819.6 649.6 170.012.14 3.5

14 14

ab ab

ab b b

ab ab

b

E MSE MSE

MSAB MSEE MSAB n

nMSB MSAB

E MSB n anan

MSE MSAB MSB n a

MSE

MSAB MSE

n

MSB MS

~2140.3 819.6 1320.723.58 4.9

4(14) 56 b

AB

an

Output from SAS PROC MIXEDdata wpba2009; infile 'wpba2009.dat'; input bowler 7-8 pattern 16 set 24 game 32 score 38-40; run; proc mixed covtest cl; class bowler pattern; model score = pattern; random bowler bowler*pattern; run; quit; The Mixed Procedure Covariance Parameter Estimates Standard Z Cov Parm Estimate Error Value Pr > Z Alpha Lower Upper bowler 23.5846 14.7947 1.59 0.0555 0.05 9.2431 138.95 bowler*pattern 12.1437 12.9894 0.93 0.1749 0.05 3.1052 757.62 Residual 649.59 32.8932 19.75 <.0001 0.05 589.65 719.20 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F pattern 3 42 3.57 0.0217