Hasse Diagrams for Linear Models
2006 Professional Bowlers Association
Qualifying Scores
Description
• 2006-7 Pro Bowlers Association Tournaments
• Bowlers: 37 Bowlers Competing in all Tournaments
• Oil Patterns: 5 Patterns Used (Chameleon, Cheetah,
Scorpion, Shark, Viper)
• Tournaments: 15 Tournaments at Different Venues Across
U.S. (3 Tournaments per Oil Pattern)
• Replications: 2 Sets of 7 Games/set at each tournament for
each bowler
• Fixed: Oil Pattern Random: Tournament, Bowler
• Nested: Tournament(Oil Pattern)
• Crossed: Bowler x Oil, Bowler x Tourney(Oil)
• Response: Y = 7 Game Score for each Replication (in 100s)
Basic Hasse Diagram
M 11
B 
37
36
5
4
O
T 
15
10
(OB )185
144
(TB)555
360
E 
1110
555
Obtaining Test Denominators
1.
2.
3.
4.
5.
Denominator for Factor U is “leading” random term below U
No Random terms between eligible V and U
2 or more leading eligible terms  approximate F-test
Unrestricted Model  All Random Terms below U are eligible
Restricted Model  All Random terms below U are eligible,
EXCEPT those containing a Fixed term not in U
•
Unrestricted Model  Interaction Effect between Fixed and
Random factors changes across repetitions of experiment
Restricted Model  Interaction Effect between Fixed and
Random factors Remains constant across repetitions
•
Unrestricted (Oil x Bowler) Interaction
• Suppose Interaction between Bowler and Oil Pattern is not
consistent across repetitions of experiment (controlling for
alley, etc.). That is, bowlers do not have “consistent
preferences” among Oil Patterns
• Eligible Random Terms for Oil are Tourney(Oil),(Oil x
Bowler),(Bowler x Tourney) since all are directly below Oil.
• Eligible Random Terms for Bowler are (Oil x Bowler) and
(Bowler x Tourney) since Unrestricted Model allows
interaction with Fixed effect (Oil) not included in Random
Effect (Bowler)
• Eligible Random Term for Tourney is (Tourney x Bowler)
Restricted (Oil x Bowler) Interaction
• Suppose Interaction between Bowler and Oil Pattern is
consistent across repetitions of experiment (controlling for
alley, etc.). That is, bowlers do have “consistent
preferences” among Oil Patterns
• Eligible Random Terms for Oil are Tourney(Oil) & (Oil x
Bowler),(Bowler x Tourney) since all are directly below Oil.
• Eligible Random Term for Bowler is (Tourney x Bowler)
since Restricted Model does not allow for interaction with
Fixed effect (Oil) not included in Random Effect (Bowler)
• Eligible Random Term for Tourney is (Tourney x Bowler)
Obtaining Expected Mean Squares
1. Representative element for each random term is its
Variance Component
2. Representative element for fixed terms is Q=effects2/df
3. Contribution of term = (N/#effects)*Rep element
where #effects is the superscript for that term
4. E(MS) for U = sum of contributions for U and all eligible
random terms below U
5. Unrestricted Model  All Random Terms below U are
eligible
6. Restricted Model  All Random terms below U are
eligible, EXCEPT those containing a Fixed term not in U
Representative Elements and E(MS) Terms
Model : Yijkl     i   j (i )   k  ik   kj (i )   l (ijk ) i  1,...,5; j  1,2,3; k  1,...,37; l  1,2
5
Representa tive Elements : Oil : Q 
o
i 1
2
i
4
Bowlers :  2 Tourneys :  2
Oil  Bowler :  2
Bowler  Tourney :  2
Expected Mean Squares :
Unre stricted
Restricted
Oil :
222Qo  74 2  6 2  2 2   2
222Qo  74 2  6 2  2 2   2
Bowler :
30 2  6 2  2 2   2
30 2  2 2   2
Tourney :
74 2  2 2   2
74 2  2 2   2
Oil  Bowler :
6 2  2 2   2
6 2  2 2   2
Bowler  Tourney : 2 2   2
2 2   2
F-Tests
Oil (Unrestric ted and Restricted ) :
MS O  MS BT 
MST  MS OB 
MS O  MS BT
FO 
1 


2
MST  MS OB
 MS O 2 MS BT 2 
 MST 2 MS OB 2 






 df


df BT 
df OB 
O

 dfT
Bowler :
MS B
MS B
Unrestrict ed : FB 
Restricted : FB 
MS OB
MS BT
2
Tourney (Unrestric ted and Restricted ) :
MST
FT 
MS BT
Oil x Bowler (Unrestric ted and Restricted ) :
FOB 
MS OB
MS BT
Bowler x Tourney (Unrestric ted and Restricted ) :
MS BT
FBT 
MS E
2
Analysis of Variance (Scores Divided by 100)
Yijkl     i   j (i )   k  ik   kj ( i )   l (ijk )
5
3
37

2
Oil : SSO   Y i  Y 

2
df O  5  1  4
i 1 j 1 k 1 l 1
5
3
37
2

Tourney(Oi l) : SST (O )   Y ij  Y i
i 1 j 1 k 1 l 1
5
3
37

2
Bowler : SS B   Y  k   Y 
i 1 j 1 k 1 l 1
5
3
37
2


2

2
dfT ( O )  53  1  10
df B  37  1  36
Oil  Bowler : SSOB   Y ik   Y i  Y  k   Y 

2
df OB  4(36)  144
i 1 j 1 k 1 l 1
5
3
37
2

Bowler  Tourney : SS BT   Y ijk  Y ij  Y ik   Y i

2
i 1 j 1 k 1 l 1
  Y
5
Error : SS E
3
37
2
ijkl
i 1 j 1 k 1 l 1
5
3
37
2

 Y ijk

2
Total : SSTotal   Yijkl  Y 
i 1 j 1 k 1 l 1
df BT  36(10)  360
df E  5(3)(37)( 2  1)  555

2
dfTotal  5(3)(37)( 2)  1  1109
ANOVA and F-Tests
Source
Oil
Tourney(Oil)
Bowler
Oil*Bowler
Bowler*Tourney
Error
Total
df
4
10
36
144
360
555
1109
SS
337.07
97.65
84.44
129.32
321.05
296.26
1265.80
Unrestricted
MS
F
84.266 7.986
9.765 10.950
2.346
2.612
0.898
1.007
0.892
1.671
0.534
P-value
0.0028
0.0000
0.0000
0.4721
0.0000
Restricted
F
P
2.630
0.0000
Oil (Unrestric ted and Restricted ) :
MSO  MS BT   4.1   MST  MSOB   11.8
MS O  MS BT
FO 
1 
2
MST  MS OB
 MS O 2 MS BT 2 
 MST 2 MS OB 2 






 df


df BT 
df OB 
O

 dfT
2
2
Rules for Variances of Means (Fixed Factors)
1. Only Consider Main Effects and Interactions containing
only Fixed Factors
2. Identify BASE TERMS and FACTORS
a) Main Effects: Base Term=Base Factor
b) Interactions: Base Term=Interaction, Base Factor=Main Effects
3. V(Mean) is sum over all contributing terms T of:
 T2
Product of superscrip ts of all base factors above T(includin g M)
Superscrip t of term T
4. Unrestricted Model  All random terms contribute to
variance of mean of interest
5. Restricted Model  All random terms contribute to
variance of mean of interest except those containing
fixed factor not in main term
Rules for Covariances of Means (Fixed Factors)
1. Identify BASE TERMS and FACTORS
2. Determine whether subscripts agree or disagree for
each base factor
3. COV(Means) is sum over all contributing terms T of:
 T2
Product of superscrip ts of all base factors above T(includin g M)
Superscrip t of term T
4. Unrestricted Model  All random terms contribute to
covariance of means of interest except those below a
base factor with disagreeing subscripts
5. Restricted Model  Same as Unrestricted but also
excludes Random terms containing Fixed Factors not in
the Base factor
Variances and Covariances
• Fixed Factor: Oil Pattern
• Base Factor: O
• Variances: All Random terms contribute since there
are no other fixed factors
• Covariances: All Random Terms are included except
those below a base factor with disagreeing subscripts
(Tourney(Oil), OilxBowler, BowlerxTourney(Oil)).
 
5 
5
 1 
2  5 
2  5 
2
V Y i   2     2     

    
  

 15 
 37 
 185 
 555 
 1110 
 1 
COV Y i , Y i '   2  
 37 
 1
2  1 
2  1 
2  1 
V Y i  Y i '  2V Y i  2COV Y i , Y i '  2  2     
     
  

3
37
111
222
 




  





 

2E MST   E MS OB   E MS BT 
222


Comparing All 10 Pairs of Oil Patterns

2E MST   E MS OB   E MS BT  E MST   E MS OB   E MS BT 

222
111
^
MST  MSOB  MS BT   9.765  0.898  0.892  0.0880  SE

Y i  Y i '  0.2967
111
111
V Y i    Y i '    
^

V Y i    Y i '   



 MST  MS OB  MS BT 


^
111

 10.00
 MST 1112 MS OB 1112 MS BT 1112 




df
df
df BT
T
OB


2
^


Bonferron' s MSD : t.05 2(10),10 SE Y i  Y i '  3.58(0.2967)  1.063
Oil Type
2
4
1
3
5
Row-Col
2
4
1
3
5
Mean
15.9045
14.35261
15.43563
15.80703
15.32086
4
5
2
0
-1.55189
-0.46887
-0.09748
-0.58365
1
3
4
1.551892
0
1.083018
1.454414
0.968243
2
1
0.468873874
-1.08301802
0
0.371396396
-0.11477477
3
0.097477
-1.45441
-0.3714
0
-0.48617
5
0.583648649
-0.96824324
0.114774775
0.486171171
0