Understanding Hasse Diagrams in Linear Models for Professional Bowlers Association Qualifying Scores
This study explores the application of Hasse diagrams in analyzing linear models for the 2006 Professional Bowlers Association qualifying scores. It delves into the structure of the tournaments, oil patterns used, replicational design, and interactions between factors. The diagrams help visualize the nested relationships between bowlers, oil patterns, and tournaments, aiding in understanding the variance components and expected mean squares for the models.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
E N D
Presentation Transcript
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 1 1 M ( ) B 37 36 5 4 O ( ) T 15 10 185 144 ) ( OB 555 360 ( ) TB ( )1110 E 555
Obtaining Test Denominators 1. Denominator for Factor U is leading random term below U 2. No Random terms between eligible V and U 3. 2 or more leading eligible terms approximate F-test 4. Unrestricted Model All Random Terms below U are eligible 5. 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)
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
Representative Elements and E(MS) Terms : Model + + + + + = kj ik k i j i ijkl Y + = ; 3 , 2 , 1 = = 2 , 1 = ,..., 1 ; 5 ,..., 1 37 ; i j k l ( ) ( ) ( ) i l ijk 5 = i 2 i o = 1 Representa tive Elements Oil : : Q 4 2 2 Bowlers : Tourneys : 2 2 Oil Bowler : Bowler Tourney : Expected Mean Squares : Unre stricted + + + Restricted + + + + + + + 2 2 2 2 2 2 2 2 Oil : 222 74 6 2 222 74 6 2 Q Q o o + + + 2 2 2 2 2 2 2 Bowler : 30 6 2 30 2 + + + + 2 2 2 2 2 2 Tourney : 74 2 74 2 + + + + 2 2 2 2 2 2 Oil Bowler : 6 2 6 2 + + 2 2 2 2 Bowler Tourney : 2 2
F-Tests Oil (Unrestric ted and Restricted : ) ( ) ( ) 2 2 + + + MS MS MS MS MS MS = = = O BT O BT T OB F ( ) ( ) ( ) ( ) 1 2 O + 2 2 2 2 MS MS MS MS MS MS T OB + + O OB BT T df df df df O BT T OB Bowler : MS MS = = Unrestrict ed : Restricted : B B F F B B MS ted MS OB and BT Tourney (Unrestric Restricted : ) MS = T F T MS Bowler x BT Oil (Unrestric ted and Restricted : ) MS = OB F OB MS BT Tourney Bowler x (Unrestric ted and Restricted : ) MS = BT F BT MS E
Analysis of Variance (Scores Divided by 100) = + + + + + + Y ( ) ( ) ( ) ijkl i j i k ik kj i l ijk ( ) 5 3 37 2 = = i j 2 = = = Oil : 5 1 4 SS Y Y df i O O = = 1 1 1 1 k l ( ) 5 3 37 2 ( ) = = i j 2 = = = Tourney(Oi l) : 5 3 1 10 SS Y Y df ij i ( ) ( ) T O T O = = 1 1 1 1 k l ( ) 5 3 37 2 = = i j 2 = = = Bowler : 37 1 36 SS Y Y df k B B = = 1 1 1 1 k l ( ) 5 3 37 2 = = i j 2 = + = = Oil Bowler : ( 4 36 ) 144 SS Y Y Y Y df i k i k OB OB = = 1 1 1 1 k l ( ) 5 3 37 2 = = i j 2 = + = = Bowler Tourney : 36 10 ( ) 360 SS Y Y Y Y df ijk ij i k i BT BT = = 1 1 1 1 k l ( ) 5 3 37 2 = = i j 2 = 3 ( 5 = ) 1 = Error : )( 37 )( 2 555 SS Y Y df ijk E ijkl E = = 1 1 1 1 k l ( ) 5 3 37 2 = = i j 2 = 3 ( 5 = = Total : )( 37 ) 2 )( 1 1109 SS Y Y df Total ijkl Total = = 1 1 1 1 k l
ANOVA and F-Tests Unrestricted F 7.986 10.950 2.612 1.007 1.671 Restricted F Source Oil Tourney(Oil) Bowler Oil*Bowler Bowler*Tourney Error Total df 4 10 36 144 360 555 1109 SS MS 84.266 9.765 2.346 0.898 0.892 0.534 P-value 0.0028 0.0000 0.0000 0.4721 0.0000 P 337.07 97.65 84.44 129.32 321.05 296.26 1265.80 2.630 0.0000 Oil (Unrestric ted and Restricted : ) ( ) ( ) 2 2 + + + MS MS MS MS MS MS = = = = = 1 . 4 11 8 . O BT O BT T OB F ( ) ( ) ( ) ( ) 1 2 O + 2 2 2 2 MS MS MS MS MS MS T OB + + O OB BT T df df df df O BT T OB
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: factors base all of ts superscrip of Product 2 T 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: base all of ts superscrip of Product 2 T 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 1 5 5 5 = + + + + 2 2 2 2 2 V Y i 15 37 185 555 1110 ( ) 1 = 2 , COV Y Y ' i i 37 ( ) ( ) ( ) 1 1 1 1 = = + + + 2 2 2 2 2 2 , 2 V Y Y V Y COV Y Y ' ' i i i i i 3 37 111 222 ( ) ( ) ( ) + 2 E MS E MS E MS = T OB BT 222
Comparing All 10 Pairs of Oil Patterns ) ) 111 2 2 2 BT OB T df df df ( ) ( ) ( ) ( ) ( ) ( ) ( ( + + 2 E MS E MS E MS E MS E MS E MS = = T OB BT T OB BT V Y Y ' i i 222 MS 111 ( ) + + . 9 765 . 0 898 . 0 892 MS MS ^ ^ = = = = 0.0880 0.2967 T OB BT V Y Y SE Y Y ' ' i i i i 111 111 2 + MS MS MS T OB BT ^ = = . 0 1 00 ( ) ( ) ( ) 111 111 111 MS MS MS + + OB T BT ( ) ^ = . 1 = Bonferron' MSD s : . 3 58 (0.2967) 063 05 . t SE Y Y ' i i ( 2 10 ), 10 Row-Col 2 4 1 3 5 2 0 4 1 3 5 Oil Type 2 4 1 3 5 Mean 15.9045 14.35261 15.43563 15.80703 15.32086 1.551892 0.468873874 0.097477 0.583648649 0 -1.08301802 -1.45441 -0.96824324 -0.46887 1.083018 0 -0.09748 1.454414 0.371396396 -0.58365 0.968243 -0.11477477 -0.48617 -1.55189 -0.3714 0 0.114774775 0.486171171 0 4 5 1 3 2
