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Hasse Diagrams for Linear Models

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)

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Hasse Diagrams for Linear Models

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  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 • Denominator for Factor U is “leading” random term below U • No Random terms between eligible V and U • 2 or more leading eligible terms  approximate F-test • Unrestricted Model  All Random Terms below U are eligible • 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 • Representative element for each random term is its Variance Component • Representative element for fixed terms is Q=effects2/df • Contribution of term = (N/#effects)*Rep element where #effects is the superscript for that term • E(MS) for U = sum of contributions for U and all eligible random terms below U • Unrestricted Model  All Random Terms below U are eligible • 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) • Only Consider Main Effects and Interactions containing only Fixed Factors • Identify BASE TERMS and FACTORS • Main Effects: Base Term=Base Factor • Interactions: Base Term=Interaction, Base Factor=Main Effects • V(Mean) is sum over all contributing terms T of: • Unrestricted Model  All random terms contribute to variance of mean of interest • 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) • Identify BASE TERMS and FACTORS • Determine whether subscripts agree or disagree for each base factor • COV(Means) is sum over all contributing terms T of: • Unrestricted Model  All random terms contribute to covariance of means of interest except those below a base factor with disagreeing subscripts • 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

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