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

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185144

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Obtaining Test Denominators1. Denominator for Factor U is “leading” random term below U2. No Random terms between eligible V and U3. 2 or more leading eligible terms approximate F-test4. Unrestricted Model All Random Terms below U are eligible5. 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)

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)

Obtaining Expected Mean Squares1. Representative element for each random term is its

Variance Component2. Representative element for fixed terms is Q=effects2/df3. Contribution of term = (N/#effects)*Rep element

where #effects is the superscript for that term4. E(MS) for U = sum of contributions for U and all eligible

random terms below U5. Unrestricted Model All Random Terms below U are

eligible6. Restricted Model All Random terms below U are

eligible, EXCEPT those containing a Fixed term not in U

Representative Elements and E(MS) Terms

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2 2 :TourneyBowler

26 26 :BowlerOil

274 274 :Tourney

230 2630 :Bowler

2674222 2674222 :Oil

Restricted stricted Unre

:SquaresMean Expected

:TourneyBowler :BowlerOil

:Tourneys :Bowlers

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F-Tests

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Analysis of Variance (Scores Divided by 100)

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555)12)(37)(3(5 :Error

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36137 :Bowler

10135 :l)Tourney(Oi

415 :Oil

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Y

ANOVA and F-TestsUnrestricted Restricted

Source df SS MS F P-value F POil 4 337.07 84.266 7.986 0.0028Tourney(Oil) 10 97.65 9.765 10.950 0.0000Bowler 36 84.44 2.346 2.612 0.0000 2.630 0.0000Oil*Bowler 144 129.32 0.898 1.007 0.4721Bowler*Tourney 360 321.05 0.892 1.671 0.0000Error 555 296.26 0.534Total 1109 1265.80

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:)Restricted and ted(Unrestric Oil

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Rules for Variances of Means (Fixed Factors)1. Only Consider Main Effects and Interactions containing

only Fixed Factors2. Identify BASE TERMS and FACTORS

a) Main Effects: Base Term=Base Factorb) 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

T termoft Superscrip

M) gT(includin above factors base all of tssuperscrip ofProduct 2T

Rules for Covariances of Means (Fixed Factors)1. Identify BASE TERMS and FACTORS2. Determine whether subscripts agree or disagree for

each base factor3. 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

T termoft Superscrip

M) gT(includin above factors base all of tssuperscrip ofProduct 2T

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)).

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Comparing All 10 Pairs of Oil Patterns

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Oil Type Mean2 15.90454 14.352611 15.435633 15.807035 15.32086

Row-Col 2 4 1 3 52 0 1.551892 0.468873874 0.097477 0.5836486494 -1.55189 0 -1.08301802 -1.45441 -0.968243241 -0.46887 1.083018 0 -0.3714 0.1147747753 -0.09748 1.454414 0.371396396 0 0.4861711715 -0.58365 0.968243 -0.11477477 -0.48617 0

4 5 1 3 2