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 Ochoa PakPark Webb

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

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Observed Means

Daniel 70.4375 Tourney1 68.2917 Round1 70.9792Kung 72.0938 Tourney2 70.2083 Round2 70.5417Park 69.5398 Tourney3 70.9583 Round3 69.875Webb 70.5000 Tourney4 70.9583 Round4 70.9167Ochoa 71.3438 Tourney5 69.9167Pak 69.5000 Tourney6 69.8750

Tourney7 70.9583Tourney8 73.4583 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 df SS MS F P-valueRound (WPT) 3 37.0156 12.3385 1.3041 0.2994Tournament (Block) 7 360.6198 51.5171 5.4449RxT (Error1) 21 198.6927 9.4616Golfer (SPT) 5 161.2969 32.2594 4.8690 0.0004RxG 15 83.6406 5.5760 0.8416 0.6302TxG + RxTxG (Error2) 140 927.5625 6.6254Total 191 1768.8281

There are significant differences among golfers, none among rounds, nor a golfer by round interaction

Post-hoc Comparisons Among Golfers

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Golfer Comparisons Golfer i Mean Golfer j Mean Mean i-j SE(Mean i-j) t(.05/(2*15) t*SE CI Lower CI UpperDaniel vs Kung (1v2) 70.4375 72.0938 -1.6563 0.6435 3.2062 2.0632 -3.7195 0.4070Daniel vs Park (1v3) 70.4375 69.5938 0.8438 0.6435 3.2062 2.0632 -1.2195 2.9070Daniel v Webb (1v4) 70.4375 70.5000 -0.0625 0.6435 3.2062 2.0632 -2.1257 2.0007Daniel v Ochoa (1v5) 70.4375 71.3438 -0.9063 0.6435 3.2062 2.0632 -2.9695 1.1570

Daniel v Pak (1v6) 70.4375 69.5000 0.9375 0.6435 3.2062 2.0632 -1.1257 3.0007Kung v Park (2v3) 72.0938 69.5938 2.5000 0.6435 3.2062 2.0632 0.4368 4.5632 ***

Kung v Webb (2v4) 72.0938 70.5000 1.5938 0.6435 3.2062 2.0632 -0.4695 3.6570Kung v Ochoa (2v5) 72.0938 71.3438 0.7500 0.6435 3.2062 2.0632 -1.3132 2.8132

Kung v Pak (2v6) 72.0938 69.5000 2.5938 0.6435 3.2062 2.0632 0.5305 4.6570 ***Park v Webb (3v4) 69.5938 70.5000 -0.9063 0.6435 3.2062 2.0632 -2.9695 1.1570Park v Ochoa (3v5) 69.5938 71.3438 -1.7500 0.6435 3.2062 2.0632 -3.8132 0.3132

Park v Pak (3v6) 69.5938 69.5000 0.0938 0.6435 3.2062 2.0632 -1.9695 2.1570Webb v Ochoa (4v5) 70.5000 71.3438 -0.8438 0.6435 3.2062 2.0632 -2.9070 1.2195

Webb v Pak (4v6) 70.5000 69.5000 1.0000 0.6435 3.2062 2.0632 -1.0632 3.0632Ochoa v Pak (5v6) 71.3438 69.5000 1.8438 0.6435 3.2062 2.0632 -0.2195 3.9070

Post-Hoc Comparisons

Daniel (70.44) Kung (72.09)Ochoa (71.34)Pak (69.50) Park (69.59) Webb (70.50)

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 c

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Expected Mean Squares (Fixed WP/Random SP)

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Unrestricted Model wrt GolferxCourse Interaction

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Testing for WP (Round) Fixed Effects

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63.634.12 :StatisticTest

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Testing for SP Effects and WP/SP Interaction

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0:0: :EffectsSubplot

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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;