Univariate Split-Plot Analysis
2003 LPGA Data
Background Information
• 6 Golfers (Treated as only 6 of interest Fixed)
• 8 Tournaments (Treated as random sample of all
possible tournaments)
• 4 Rounds per tournament (fixed factor)
Daniel
Kung
Park
Webb
Ochoa
Pak
Data Description and Model
• Tournaments act as blocks. They are each associated
with a particular golf course, region and weather pattern
(they may differ significantly in terms of difficulty)
• Tournaments are made up of Rounds (these tournaments
are all 4 rounds). It is impossible to break up rounds
within blocks, thus they are the whole plot factor (in an
experiment, the treatments would be randomly assigned
to whole plots)
• Golfers all play rounds on the same day (all play round
1, then 2, etc), thus they are the subplot factor (in an
experiment, their positions would be assigned at random
within whole plots)
Data Description and Model
• Let factor A be whole plot factor (round) with a=4 levels
and subscript i be associated with it
• Let factor B be block factor (tournament) with b=8
levels and subscript j be associated with it
• Let factor C be subplot factor (golfer) with c=6 levels
and subscript k be associated with it
• Interaction between round and tournament allows for
climate effects to vary across courses (WP error term)
• Interaction between golfer and round allows golfer skill
to vary across rounds (e.g. pressure effects)
• Model assumes no tournament by golfer interaction (can
be tested) or 3-way interaction (SP error term)
Data Description and Model
Yijk     i  b j  (ab) ij   k  ( ) ik   ijk
 i  effect of round i
a

i 1
0
i

b j  effect of tournamen t j b j ~ N 0,  b2


2
(ab) ij  round i/tourney j interactio n (ab) ij ~ N 0,  ab
 k  effect of golfer k
c

k 1
k

0
( ) ik  round i/golfer k interactio n
a
c
 ( )  ( )
i 1
 ijk  random error term  ijk ~ N 0,  2 
ik
k 1
ik
0
Observed Means
Daniel
Kung
Park
Webb
Ochoa
Pak
70.4375
72.0938
69.5398
70.5000
71.3438
69.5000
Tourney1
Tourney2
Tourney3
Tourney4
Tourney5
Tourney6
Tourney7
Tourney8
68.2917
70.2083
70.9583
70.9583
69.9167
69.8750
70.9583
73.4583
Round1
Round2
Round3
Round4
70.9792
70.5417
69.875
70.9167
Overall
70.5781
Means of Golfer/Courses and Golfer/Rounds and
Courses/Rounds are on separate EXCEL spread sheet
Analysis of Variance
Source of Variation
Round (WPT)
Tournament (Block)
RxT (Error1)
Golfer (SPT)
RxG
TxG + RxTxG (Error2)
Total
df
3
7
21
5
15
140
191
SS
37.0156
360.6198
198.6927
161.2969
83.6406
927.5625
1768.8281
MS
12.3385
51.5171
9.4616
32.2594
5.5760
6.6254
F
1.3041
5.4449
P-value
0.2994
4.8690
0.8416
0.0004
0.6302
There are significant differences among golfers, none among
rounds, nor a golfer by round interaction
Post-hoc Comparisons Among Golfers
Y
.. k

 Y .. k '  t.05 /(2 (15)),140
Golfer Comparisons Golfer i Mean Golfer j Mean
Daniel vs Kung (1v2)
70.4375
72.0938
Daniel vs Park (1v3)
70.4375
69.5938
Daniel v Webb (1v4)
70.4375
70.5000
Daniel v Ochoa (1v5)
70.4375
71.3438
Daniel v Pak (1v6)
70.4375
69.5000
Kung v Park (2v3)
72.0938
69.5938
Kung v Webb (2v4)
72.0938
70.5000
Kung v Ochoa (2v5)
72.0938
71.3438
Kung v Pak (2v6)
72.0938
69.5000
Park v Webb (3v4)
69.5938
70.5000
Park v Ochoa (3v5)
69.5938
71.3438
Park v Pak (3v6)
69.5938
69.5000
Webb v Ochoa (4v5)
70.5000
71.3438
Webb v Pak (4v6)
70.5000
69.5000
Ochoa v Pak (5v6)
71.3438
69.5000
Mean i-j
-1.6563
0.8438
-0.0625
-0.9063
0.9375
2.5000
1.5938
0.7500
2.5938
-0.9063
-1.7500
0.0938
-0.8438
1.0000
1.8438
SE(Mean i-j) t(.05/(2*15)
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
0.6435
3.2062
2 MS ERROR
4(8)
t*SE
2.0632
2.0632
2.0632
2.0632
2.0632
2.0632
2.0632
2.0632
2.0632
2.0632
2.0632
2.0632
2.0632
2.0632
2.0632
CI Lower
-3.7195
-1.2195
-2.1257
-2.9695
-1.1257
0.4368
-0.4695
-1.3132
0.5305
-2.9695
-3.8132
-1.9695
-2.9070
-1.0632
-0.2195
CI Upper
0.4070
2.9070
2.0007
1.1570
3.0007
4.5632
3.6570
2.8132
4.6570
1.1570
0.3132
2.1570
1.2195
3.0632
3.9070
***
***
Post-Hoc Comparisons
Pak (69.50)
Park (69.59)
Daniel (70.44)
Webb (70.50)
Ochoa (71.34)
Kung (72.09)
Another Possibility - Mixed Model
• In reality, there are hundreds of golfers that
are “certified” members of LPGA
• Re-analyze the data as a mixed model
(rounds are still fixed)
• ANOVA hasn’t changed, but error terms
have.
• The golfer effects are now random variables
that we assume to be normal with variance
c2
Expected Mean Squares (Fixed WP/Random SP)
Unrestricted Model wrt GolferxCourse Interaction
E  MS BLOCKS   ac b2  c ab2   2
df BLOCKS  b  1
a
2
E  MSWP   c ab
 b ac2   2  bc
2
E  MS BLK WP   c ab
  2
E  MS SP   ab   
2
c
2
E  MSWPSP   b ac2   2
E  MS ERROR    2
2

 i
i 1
dfWP  a  1
a 1
df BLK WP  (a  1)(b  1)
df SP  c  1
dfWPSP  (a  1)(c  1)
df ERROR  a (b  1)(c  1)
Testing for WP (Round) Fixed Effects
H 0 : 1     4  0 H A : Not all  i  0
Note : E MSW P   E MS ERR   E MS BLK W P   E MSW PSP   bc
Test Statistic : FW P 
2

 i
a 1
MSW P  MS ERR
12.34  6.63 18.97


 1.26
MS BLK W P  MSW PSP 9.46  5.58 15.04
Approximat e Degrees of Freedom :
2

MSW P  MS ERR 
1 
MSW P 2  MS ERR 2
a 1
(12.34  6.63) 2
359.86

 6.72
2
2
(12.64)
(6.63)
53.57

4 1
4(8  1)(6  1)

a (b  1)(c  1)
2

MS BLK W P  MSW PSP 
2 
MS BLK W P 2  MSW PSP 2

(9.46  5.58) 2
226.20

 35.68
2
2
(9.46)
(5.58)
6.34

(4  1)(8  1) (4  1)(6  1)
(a  1)(b  1) (a  1)(c  1)
F.05, 6.34,35.68  F.05, 6,35  2.37
P  .301
Testing for SP Effects and WP/SP Interaction
Subplot Effects : H 0 :  c2  0 H A :  c2  0
MS SP 32.26
Test Statistic : FSP 

 4.87
MS ERR
6.63
Degrees of Freedom : 1  5  2  140
P  .0004
WP  SP Interactio n : H 0 :  ac2  0 H A :  ac2  0
Test Statistic : FW PSP
MSW PSP 5.58


 0.84
MS ERR
6.63
Degrees of Freedom : 1  15  2  140
P  .6302
SAS Program (Fixed Effects Model)
options nodate nonumber ps=54 ls=76;
data one;
infile ‘C:\lpgasplt.dat';
input golfer $ 1-24 tourney round score;
run;
proc glm;
class golfer tourney round;
model score = round tourney round*tourney golfer golfer*round;
test h=round e=round*tourney;
means golfer / bon;
run;
proc mixed;
class golfer tourney round;
model score = round golfer golfer*round;
random tourney round*tourney;
run;
quit;