Explaining Success in Sports Competitions: Paired Comparison Methods - PowerPoint PPT Presentation

Explaining Success in Sports Competitions: Paired Comparison Methods with Explanatory Variables Gerhard Tutz und Gunther Schauberger Ludwig-Maximilians-Universität München Padova June 2017 Collaboration with Andreas Groll

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Paired Comparison Data • Sports competitions, experiments, . . . • Aim: measure unobservable latent trait for set of objects t a 1 , . . . , a m ✉ • Comparison/Competition between two objects a r and a s • Binary response ★ 1 if a r preferred over a s Y ♣ r,s q ✏ 0 if a s preferred over a r • Ordinal response ✩ 1 if a r strongly preferred over a s ✬ ✬ . . ✫ Y ♣ r,s q ✏ . . . . ✬ ✬ if a s strongly preferred over a r ✪ K 2/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Bradley-Terry Model Set of objects t a 1 , . . . , a m ✉ ★ 1 if a r preferred over a s for Y ♣ r,s q ✏ 0 if a s preferred over a r m exp ♣ γ r ✁ γ s q ➳ P ♣ Y ♣ r,s q ✏ 1 q ✏ 1 � exp ♣ γ r ✁ γ s q , γ r ✏ 0 r ✏ 1 γ r attractivity/strength of object r γ s attractivity/strength of object s 3/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References From binary to ordinal response A match between teams a r and a s is treated as a paired comparison with ordinal response Y ♣ r,s q , with ✩ 1 if team a r wins by at least 2 goals difference ✬ ✬ 2 if team a r wins by 1 goal difference ✬ ✬ ✬ ✫ Y ♣ r,s q ✏ 3 if the match ends with a draw ✬ 4 if team a s wins by 1 goal difference ✬ ✬ ✬ ✬ ✪ 5 if team a s wins by at least 2 goals difference . exp ♣ θ k � γ r ✁ γ s q P ♣ Y ♣ r,s q ↕ k q ✏ 1 � exp ♣ θ k � γ r ✁ γ s q , k ✏ 1 , . . . , 5 • θ k : category-specific threshold parameters, θ 1 ✏ ✁ θ 4 , θ 2 ✏ ✁ θ 3 18 • γ r , γ s : team-specific abilities, ➦ γ r ✏ 0 r ✏ 1 4/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Assumptions and Derivation • Unobservable random utility U r that represents ability of team a r : U r ✏ γ r � ε r , • γ r is a fixed value (the fixed ability) • ε r is a random variable (represents noise) • Assume that ε 1 , . . . , ε m are iid random variables with distribution function F ε . • Given the pair ♣ a r , a s q , one observes Y ♣ r,s q ✏ k ô θ k ✁ 1 ➔ U s ✁ U r ➔ θ k , • Low categories k indicate dominance of a r • High categories k indicate dominance of a s ñ Y ♣ r,s q is a categorized/coarsened version of the differences in latent abilities. 5/33

♣☎q ♣ � ✁ q ♣ q ↕ ⑤♣ qq ✏ ♣ � ♣ � ✁ q Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Ordinal Bradley-Terry Model From Y ♣ r,s q ✏ k ô θ k ✁ 1 ➔ U s ✁ U r ➔ θ k we derive Y ♣ r,s q ↕ k ô U s ✁ U r ➔ θ k Y ♣ r,s q ↕ k ô ε s ✁ ε r ➔ θ k � γ r ✁ γ s and P ♣ Y ♣ r,s q ↕ k ⑤♣ r, s qq ✏ F ♣ η rsk q , η rsk ✏ θ k � γ r ✁ γ s where F ♣☎q is the distribution of the differences ε s ✁ ε r . 6/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Ordinal Bradley-Terry Model From Y ♣ r,s q ✏ k ô θ k ✁ 1 ➔ U s ✁ U r ➔ θ k we derive Y ♣ r,s q ↕ k ô U s ✁ U r ➔ θ k Y ♣ r,s q ↕ k ô ε s ✁ ε r ➔ θ k � γ r ✁ γ s and P ♣ Y ♣ r,s q ↕ k ⑤♣ r, s qq ✏ F ♣ η rsk q , η rsk ✏ θ k � γ r ✁ γ s where F ♣☎q is the distribution of the differences ε s ✁ ε r . With F ♣☎q as the logistic distribution function we get exp ♣ θ k � γ r ✁ γ s q P ♣ Y ♣ r,s q ↕ k ⑤♣ r, s qq ✏ 1 � exp ♣ θ k � γ r ✁ γ s q 6/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Ordinal Bradley-Terry Model From Y ♣ r,s q ✏ k ô θ k ✁ 1 ➔ U s ✁ U r ➔ θ k we derive θ 1 θ 2 θ 3 = −θ 2 θ 4 = −θ 1 Y ♣ r,s q ↕ k ô U s ✁ U r ➔ θ k Y ♣ r,s q ↕ k ô ε s ✁ ε r ➔ θ k � γ r ✁ γ s P ( Y rs = 2 ) 0.20 and f ( U s − U r ) P ♣ Y ♣ r,s q ↕ k ⑤♣ r, s qq ✏ F ♣ η rsk q , η rsk ✏ θ k � γ r ✁ γ s 0.10 where F ♣☎q is the distribution of the differences ε s ✁ ε r . 0.00 With F ♣☎q as the logistic distribution function we get γ s − γ r ← a r a s → 0 U s − U r exp ♣ θ k � γ r ✁ γ s q P ♣ Y ♣ r,s q ↕ k ⑤♣ r, s qq ✏ 1 � exp ♣ θ k � γ r ✁ γ s q 6/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Ordinal Bradley-Terry Model From Y ♣ r,s q ✏ k ô θ k ✁ 1 ➔ U s ✁ U r ➔ θ k we derive θ 1 θ 2 θ 3 = −θ 2 θ 4 = −θ 1 Y ♣ r,s q ↕ k ô U s ✁ U r ➔ θ k Y ♣ r,s q ↕ k ô ε s ✁ ε r ➔ θ k � γ r ✁ γ s P ( Y rs = 2 ) 0.20 and f ( U s − U r ) P ♣ Y ♣ r,s q ↕ k ⑤♣ r, s qq ✏ F ♣ η rsk q , η rsk ✏ θ k � γ r ✁ γ s 0.10 where F ♣☎q is the distribution of the differences ε s ✁ ε r . 0.00 With F ♣☎q as the logistic distribution function we get γ s − γ r ← a r a s → 0 U s − U r exp ♣ θ k � γ r ✁ γ s q P ♣ Y ♣ r,s q ↕ k ⑤♣ r, s qq ✏ 1 � exp ♣ θ k � γ r ✁ γ s q 6/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Restrictions Symmetric restrictions of threshold parameters: • θ k ✏ ✁ θ K ✁ k , k ✏ 1 , . . . , r K ④ 2 s e.g. K ✏ 5 ñ θ 1 ✏ ✁ θ 4 , θ 2 ✏ ✁ θ 3 • (if K is even): θ K ④ 2 ✏ 0 That means, that for teams a r and a s one obtains P ♣ Y ♣ r,s q ✏ k q ✏ P ♣ Y ♣ s,r q ✏ K � 1 ✁ k q . For the special case K ✏ 5 one obtains P ♣ Y ♣ r,s q ✏ 1 q ✏ P ♣ Y ♣ s,r q ✏ 5 q and P ♣ Y ♣ r,s q ✏ 2 q ✏ P ♣ Y ♣ s,r q ✏ 4 q 7/33

✏ � � ✁ ✏ ✏ ✁ ✏ ✁ ✏ ✁ ✏ ✁ ✏ � � ✁ → ✏ → Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Ordinal Model With Home/Order Effect Possible order effects in sports: • playing at home (football) • serving (tennis) • playing with the white pieces (chess) Simplest case: binary response given pair ♣ a r , a s q a r wins if U r → U s , With home/order effect a r wins if U r � δ → U s , ñ A constant δ is added to the first team (home team). 8/33

✏ � � ✁ ✏ ✏ ✁ ✏ ✁ ✏ ✁ ✏ ✁ Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Ordinal Model With Home/Order Effect Possible order effects in sports: • playing at home (football) • serving (tennis) • playing with the white pieces (chess) In the general case Simplest case: binary response given pair ♣ a r , a s q η rsk ✏ δ � θ k � γ r ✁ γ s , a r wins if U r → U s , where δ → 0 represents the order/home effect. • If δ ✏ 0 no order/home effect • If δ → 0 large the probability for low categories (dominance of a r ) is increased With home/order effect a r wins if U r � δ → U s , ñ A constant δ is added to the first team (home team). 8/33

✏ � � ✁ → ✏ → Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Ordinal Model With Home/Order Effect Possible order effects in sports: η rsk ✏ δ � θ k � γ r ✁ γ s • playing at home (football) Season 2015/16 Rank Team γ r ˆ Rank( ˆ γ r ) • serving (tennis) ˆ δ ✏ 0 . 265 1 BAY 1.899 1 • playing with the white pieces (chess) θ 1 ✏ ✁ ˆ ˆ 2 DOR 1.598 2 θ 4 ✏ ✁ 1 . 591 3 LEV 0.433 4 θ 2 ✏ ✁ ˆ ˆ θ 3 ✏ ✁ 0 . 576 4 MGB 0.475 3 Simplest case: binary response given pair ♣ a r , a s q 5 S04 0.133 5 6 MAI 0.088 6 7 BER -0.001 7 a r wins if U r → U s , 8 WOB -0.142 9 9 KOE -0.045 8 10 HSV -0.183 10 11 ING -0.228 11 12 AUG -0.363 13 With home/order effect 13 BRE -0.361 12 14 DAR -0.467 15 a r wins if U r � δ → U s , 15 HOF -0.448 14 ñ A constant δ is added to the first team (home team). 16 FRA -0.623 16 17 STU -0.699 17 18 HAN -1.068 18 8/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R -package BTLLasso References Explanatory Variables - Effect of Budget A simple two-step approach: • Fit a Bradley-Terry Model • Investigate the dependence of abilities on explanatory variables LM : R 2 adj = 0.49 3 x AM : R 2 adj = 0.58 2 abilities x x 1 x x x x x x x x x 0 x x x x x −1 x 20 40 60 80 100 120 budget Figure: Budgets (in millions) versus estimated abilities for all teams from the Bundesliga season 2012/2013; lines represent linear and additive model fit 9/33