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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 MSWPSP 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 dfWPSP (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 PSP 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 PSP 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 PSP 2 MS BLK W P 2 MSW PSP 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 PSP MSW PSP 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;