1. Hasse Diagrams for Linear Models 2006 Professional Bowlers Association Qualifying Scores

2. 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)

3. Basic Hasse Diagram

4. 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

5. 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)

6. 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)

7. Obtaining Expected Mean Squares 1.Representative element for each random term is its Variance Component 2.Representative element for fixed terms is Q=  effects 2 /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

8. Representative Elements and E(MS) Terms

9. F-Tests

10. Analysis of Variance (Scores Divided by 100)

11. ANOVA and F-Tests

12. 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: 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

13. 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: 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

14. 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)).

15. Comparing All 10 Pairs of Oil Patterns 4 5 1 3 2