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 ta1, . . . , am✉
• Comparison/Competition between two objects ar and as
• Binary response

Y♣r,sq ✏ ★ 1 if ar preferred over as if as preferred over ar

• Ordinal response

Y♣r,sq ✏ ✩ ✬ ✬ ✫ ✬ ✬ ✪ 1 if ar strongly preferred over as . . . . . . K if as strongly preferred over ar

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Bradley-Terry Model

Set of objects ta1, . . . , am✉ for Y♣r,sq ✏ ★ 1 if ar preferred over as if as preferred over ar P♣Y♣r,sq ✏ 1q ✏ exp♣γr ✁ γsq 1 exp♣γr ✁ γsq ,

m

➳

r✏1

γr ✏ 0 γr attractivity/strength of object r γs attractivity/strength of object s

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From binary to ordinal response

A match between teams ar and as is treated as a paired comparison with ordinal response Y♣r,sq, with Y♣r,sq ✏ ✩ ✬ ✬ ✬ ✬ ✬ ✫ ✬ ✬ ✬ ✬ ✬ ✪ 1 if team ar wins by at least 2 goals difference 2 if team ar wins by 1 goal difference 3 if the match ends with a draw 4 if team as wins by 1 goal difference 5 if team as wins by at least 2 goals difference. P♣Y♣r,sq ↕ kq ✏ exp♣θk γr ✁ γsq 1 exp♣θk γr ✁ γsq , k ✏ 1, . . . , 5

• θk: category-specific threshold parameters, θ1 ✏ ✁θ4, θ2 ✏ ✁θ3
• γr, γs: team-specific abilities,

18

➦

r✏1

γr ✏ 0

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Assumptions and Derivation

• Unobservable random utility Ur that represents ability of team ar:

Ur ✏ γr εr,

• γr is a fixed value (the fixed ability)
• εr is a random variable (represents noise)
• Assume that ε1, . . . , εm are iid random variables with distribution function Fε.
• Given the pair ♣ar, asq, one observes

Y♣r,sq ✏ k ô θk✁1 ➔ Us ✁ Ur ➔ θk,

• Low categories k indicate dominance of ar
• High categories k indicate dominance of as

ñ Y♣r,sq is a categorized/coarsened version of the differences in latent abilities.

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Ordinal Bradley-Terry Model

From Y♣r,sq ✏ k ô θk✁1 ➔ Us ✁ Ur ➔ θk we derive Y♣r,sq ↕ k ô Us ✁ Ur ➔ θk Y♣r,sq ↕ k ô εs ✁ εr ➔ θk γr ✁ γs and P♣Y♣r,sq ↕ k⑤♣r, sqq ✏ F♣ηrskq, ηrsk ✏ θk γr ✁ γs where F♣☎q is the distribution of the differences εs ✁ εr. ♣☎q ♣

♣ q ↕

⑤♣ qq ✏ ♣

• ✁

q

• ♣
• ✁

q

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Ordinal Bradley-Terry Model

From Y♣r,sq ✏ k ô θk✁1 ➔ Us ✁ Ur ➔ θk we derive Y♣r,sq ↕ k ô Us ✁ Ur ➔ θk Y♣r,sq ↕ k ô εs ✁ εr ➔ θk γr ✁ γs and P♣Y♣r,sq ↕ k⑤♣r, sqq ✏ F♣ηrskq, ηrsk ✏ θk γr ✁ γs where F♣☎q is the distribution of the differences εs ✁ εr. With F♣☎q as the logistic distribution function we get P♣Y♣r,sq ↕ k⑤♣r, sqq ✏ exp♣θk γr ✁ γsq 1 exp♣θk γr ✁ γsq

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Ordinal Bradley-Terry Model

From Y♣r,sq ✏ k ô θk✁1 ➔ Us ✁ Ur ➔ θk we derive Y♣r,sq ↕ k ô Us ✁ Ur ➔ θk Y♣r,sq ↕ k ô εs ✁ εr ➔ θk γr ✁ γs and P♣Y♣r,sq ↕ k⑤♣r, sqq ✏ F♣ηrskq, ηrsk ✏ θk γr ✁ γs where F♣☎q is the distribution of the differences εs ✁ εr. With F♣☎q as the logistic distribution function we get P♣Y♣r,sq ↕ k⑤♣r, sqq ✏ exp♣θk γr ✁ γsq 1 exp♣θk γr ✁ γsq

0.00 0.10 0.20 Us − Ur f(Us − Ur) P(Yrs = 2) θ1 θ2 θ3 = −θ2 θ4 = −θ1 γs − γr ← ar as →

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Ordinal Bradley-Terry Model

From Y♣r,sq ✏ k ô θk✁1 ➔ Us ✁ Ur ➔ θk we derive Y♣r,sq ↕ k ô Us ✁ Ur ➔ θk Y♣r,sq ↕ k ô εs ✁ εr ➔ θk γr ✁ γs and P♣Y♣r,sq ↕ k⑤♣r, sqq ✏ F♣ηrskq, ηrsk ✏ θk γr ✁ γs where F♣☎q is the distribution of the differences εs ✁ εr. With F♣☎q as the logistic distribution function we get P♣Y♣r,sq ↕ k⑤♣r, sqq ✏ exp♣θk γr ✁ γsq 1 exp♣θk γr ✁ γsq

0.00 0.10 0.20 Us − Ur f(Us − Ur) P(Yrs = 2) θ1 θ2 θ3 = −θ2 θ4 = −θ1 γs − γr ← ar as →

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Restrictions

Symmetric restrictions of threshold parameters:

• θk ✏ ✁θK✁k, k ✏ 1, . . . , rK④2s

e.g. K ✏ 5 ñ θ1 ✏ ✁θ4, θ2 ✏ ✁θ3

• (if K is even): θK④2 ✏ 0

That means, that for teams ar and as one obtains P♣Y♣r,sq ✏ kq ✏ P♣Y♣s,rq ✏ K 1 ✁ kq. For the special case K ✏ 5 one obtains P♣Y♣r,sq ✏ 1q ✏ P♣Y♣s,rq ✏ 5q and P♣Y♣r,sq ✏ 2q ✏ P♣Y♣s,rq ✏ 4q

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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 ♣ar, asq ar wins if Ur → Us, With home/order effect ar wins if Ur δ → Us, ñ A constant δ is added to the first team (home team). ✏

• ✁

→ ✏ → ✏

• ✁

✏ ✏ ✁ ✏ ✁ ✏ ✁ ✏ ✁

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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 ♣ar, asq ar wins if Ur → Us, With home/order effect ar wins if Ur δ → Us, ñ A constant δ is added to the first team (home team). In the general case ηrsk ✏ δ θk γr ✁ γs, 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

ar) is increased ✏

• ✁

✏ ✏ ✁ ✏ ✁ ✏ ✁ ✏ ✁

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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 ♣ar, asq ar wins if Ur → Us, With home/order effect ar wins if Ur δ → Us, ñ A constant δ is added to the first team (home team). ✏

• ✁

→ ✏ → ηrsk ✏ δ θk γr ✁ γs Season 2015/16

ˆ δ ✏ 0.265 ˆ θ1 ✏ ✁ˆ θ4 ✏ ✁1.591 ˆ θ2 ✏ ✁ˆ θ3 ✏ ✁0.576 Rank Team ˆ γr Rank(ˆ γr) 1 BAY 1.899 1 2 DOR 1.598 2 3 LEV 0.433 4 4 MGB 0.475 3 5 S04 0.133 5 6 MAI 0.088 6 7 BER

• 0.001

7 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 13 BRE

• 0.361

12 14 DAR

• 0.467

15 15 HOF

• 0.448

14 16 FRA

• 0.623

16 17 STU

• 0.699

17 18 HAN

• 1.068

18

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Explanatory Variables - Effect of Budget

A simple two-step approach:

• Fit a Bradley-Terry Model
• Investigate the dependence of abilities on explanatory variables

20 40 60 80 100 120 −1 1 2 3 budget abilities x x x x x x x x x x x x x x x x x x LM : R2

adj = 0.49

AM : R2

adj = 0.58

Figure: Budgets (in millions) versus estimated abilities for all teams from the Bundesliga season 2012/2013; lines represent linear and additive model fit

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Two-step approach is not satisfactory

• BT-Model fit does not use the explanatory variables
• Inference difficult, in the second step abilities are considered as random variables,

uncertainty of estimates ignored Nowadays various explanatory variables of different types are available

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Two-step approach is not satisfactory

• BT-Model fit does not use the explanatory variables
• Inference difficult, in the second step abilities are considered as random variables,

uncertainty of estimates ignored Nowadays various explanatory variables of different types are available On-field covariates

Bayern Munich Hamburger SV Goals 5 : Goals Shots on goal 23 : 5 Shots on goal Distance 108.54 : 111.28 Distance Completion rate 90 : 64 Completion rate Ball possession 77 : 23 Ball possession Tackling rate 52 : 48 Tackling rate Fouls 10 : 12 Fouls Offside 3 : Offside

Source: German football magazine kicker (http://www.kicker.de/) 10/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Generalize Bradley-Terry Model

• Make use of all kinds of covariates in paired comparisons
• Link explanatory variables to final match outcome
• Set up paired comparison model that can include all types of covariates

simultaneously

• Sparse, interpretable model

♣

♣ q ✏

q ✏ ♣ ✁ q

• ♣

✁ q ó Ñ ñ ó ♣

♣ q ✏

q ✏ ♣ ✁ q

• ♣

✁ q

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Generalize Bradley-Terry Model

• Make use of all kinds of covariates in paired comparisons
• Link explanatory variables to final match outcome
• Set up paired comparison model that can include all types of covariates

simultaneously

• Sparse, interpretable model

P♣Y♣r,sq ✏ 1q ✏ exp♣γr ✁ γsq 1 exp♣γr ✁ γsq ó γr Ñ γir ñ γir can depend on covariates varying over subjects and/or objects ó P♣Yi♣r,sq ✏ 1q ✏ exp♣γir ✁ γisq 1 exp♣γir ✁ γisq

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Match-Specific Explanatory Variables

In classical paired comparison experiments several persons choose between objects Let person i be characterized by covariates xi (gender, age...). Covariates are subject-specific, do not vary over objects. γir ✏ βr0 xT

i βr

In Sports

• γir is the strength of team ar when meeting under circumstances captured by xi

like

• Temperature when playing, raining or not,..
• Time of the year
• Type of tournament,...
• Each team may react differently to the circumstances, βr is team-specific

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Match-Team-Specific Explanatory Variables

More often one has explanatory variables that vary both over matches and team yielding match-team-specific (subject-object-specific) covariates zir.

• Total amount of km run by team ar in match i,
• Percentage of passes reaching team mates (team ar in match i)
• ...

Effect of match-team-specific covariates can be modelled γir ✏ βr0 zT

irα.

• r

γir ✏ βr0 zT

irαr.

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Team-Specific Explanatory Variables

Some interesting explanatory variables characterize the team and do not vary over matches (object-specific covariates).

• Budget of team ar (fixed over a season)
• Home country of the team/player
• ...

γir ✏ γr ✏ βr0 zT

r τ.

But: Effects of team-specific explanatory variables are not identifiable ! γr ✏ βr0 zT

r τ ✏ βr0 zT r c

❧♦♦♦♦♦♠♦♦♦♦♦♥

˜ βr0

zT

r ♣τ ✁ cq

❧♦♦♦♦♦♠♦♦♦♦♦♥

zT

r ˜

τ

✏ ˜ γr Therefore γr and ˜ γr yields the same model.

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Regularization by Penalty Terms

Estimation via framework of multivariate GLMs Ñ number of parameters can be huge! Example: m ✏ 18 teams, p ✏ 8 match-team-specific covariates ñ ♣m ✁ 1qp ✏ 136 additional parameters ñ Penalized likelihood estimation Replace the usual log-likelihood by lp♣βq ✏ l♣βq ✁ λJ♣βq where

• l♣βq is the log-likelihood of a GLM,
• λ is a tuning parameter
• J♣βq is a penalty term

Simple Ridge Type Shrinkage Penalty J♣βq ✏ ➳

r

β2

r

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Lasso-type Penalty: the Simple Case

Consider the simplest case without covariates P♣Y♣r,sq ↕ kq ✏ exp♣δ θk βr0 ✁ βs0q 1 exp♣δ θk βr0 ✁ βs0q Estimate by using the fusion penalty penalty J♣γq ✏ ➳

r→s

wrs⑤βr0 ✁ βs0⑤, With weights wrs (optional).

• Shows which teams have different abilities
• Clusters of teams with equal strength are identified
• Reduces number of parameters

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Lasso-type Penalty: the Simple Case

Consider the simplest case without covariates P♣Y♣r,sq ↕ kq ✏ exp♣δ θk βr0 ✁ βs0q 1 exp♣δ θk βr0 ✁ βs0q Estimate by using the fusion penalty penalty J♣γq ✏ ➳

r→s

wrs⑤βr0 ✁ βs0⑤, With weights wrs (optional).

• Shows which teams have different abilities
• Clusters of teams with equal strength are identified
• Reduces number of parameters

Bundesliga 2016/17

2.5 2.0 1.5 1.0 0.5 0.0 −1.0 −0.5 0.0 0.5 1.0 1.5 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

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Fusion for Match-Team-Specific Covariates

P♣Yi♣r,sq ↕ kq ✏ exp♣δr θk γir ✁ γisq 1 exp♣δr θk γir ✁ γisq ✏ exp♣δr θk βr0 ✁ βs0 zT

irαr ✁ zT isαsq

1 exp♣δr θk βr0 ✁ βs0 zT

irαr ✁ zT isαsq

δr team-specific home effects of team r θk category-specific threshold parameters βr0 team-specific intercepts zir p-dimensional covariate vector that varies over teams and matches αr p-dimensional parameter vector that varies over teams.

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Penalty For Fusion and Selection

• Predictor: ηrsk ✏ δr θk βr0 ✁ βs0 zT

irαr ✁ zT isαs

J♣☎q ✏ Jδ♣☎q Jα♣☎q combining the penalties Jδ♣δ1, . . . , δmq ✏ ➳

r➔s

⑤δr ✁ δs⑤ , ñ Fusion of home effects Jα♣α1, . . . , αmq ✏

p

➳

j✏1

✄ ➳

r➔s

⑤αrj ✁ αsj⑤

m

➳

r✏1

⑤αrj⑤ ☛ . ñ Fusion and selection of covariate effects

Ñ ✽ ñ ✏ ✏ ✏ ñ ✏ ✏ ✏ ñ ♣

♣ q ↕

q ✏ ♣

• ✁
• ✁

q

• ♣
• ✁
• ✁

q ✏ ♣

• ✁

q

• ♣
• ✁

q

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Penalty For Fusion and Selection

• Predictor: ηrsk ✏ δr θk βr0 ✁ βs0 zT

irαr ✁ zT isαs

J♣☎q ✏ Jδ♣☎q Jα♣☎q combining the penalties Jδ♣δ1, . . . , δmq ✏ ➳

r➔s

⑤δr ✁ δs⑤ , ñ Fusion of home effects Jα♣α1, . . . , αmq ✏

p

➳

j✏1

✄ ➳

r➔s

⑤αrj ✁ αsj⑤

m

➳

r✏1

⑤αrj⑤ ☛ . ñ Fusion and selection of covariate effects

For λ Ñ ✽: ñ Global home effect δ1 ✏ . . . ✏ δ18 ✏ δ ñ Elimination of covariate effects α1 ✏ . . . ✏ α18 ✏ 0 ñ One ends up with the basic model P ♣Yi♣r,sq ↕ kq ✏ exp♣δr θk βr0 ✁ βs0 zT

irαr ✁ zT isαsq

1 exp♣δr θk βr0 ✁ βs0 zT

irαr ✁ zT isαsq

✏ exp♣δ θk βr0 ✁ βs0q 1 exp♣δ θk βr0 ✁ βs0q

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Selection of Tuning Parameter

• Calculate model for a grid of tuning parameters λ
• Cross-validate model with respect to a certain criterion
• We use the ranked probability score (RPS) (Gneiting and Raftery, 2007)
• RPS for ordinal response y P t1, . . . , K✉ is

RPS♣y, ˆ π♣kqq ✏

K

➳

k✏1

♣ˆ π♣kq ✁ 1♣y ↕ kqq2 where π♣kq ✏ P♣y ↕ kq. Ñ takes into account the ordinal structure of the response

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Bundesliga 2015/2016

Position Team Goals For Goals Against Points 1 Bayern München 80 17 88 2 Borussia Dortmund 82 34 78 3 Bayer 04 Leverkusen 56 40 60 4

• Bor. Mönchengladbach

67 50 55 5 FC Schalke 04 51 49 52 6

• 1. FSV Mainz 05

46 42 50 7 Hertha BSC 42 42 50 8 VfL Wolfsburg 47 49 45 9

• 1. FC Köln

38 42 43 10 Hamburger SV 40 46 41 11 FC Ingolstadt 04 33 42 40 12 FC Augsburg 42 52 38 13 Werder Bremen 50 65 38 14 SV Darmstadt 98 38 53 38 15 TSG Hoffenheim 39 54 37 16 Eintracht Frankfurt 34 52 36 17 VfB Stuttgart 50 75 33 18 Hannover 96 31 62 25

• Distance Total amount of km run
• BallPossession Percentage of ball possession
• TacklingRate Rate of won tacklings
• ShotsonGoal Total number of shots on goal
• CompletionRate Percentage of passes reaching teammates
• FoulsSuffered Number of fouls suffered
• Offside Number of offsides (in attack)

Source: German football magazine kicker (http://www.kicker.de/) 20/33

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CV error against tuning parameter λ

4 3 2 1 180 200 220 240 260 280 log(λ + 1) RPS

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Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

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Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 22/33

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Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 22/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 22/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Parameterization Scale

In order to obtain a common scale covariates are transformed to variance one (over all matches and teams).

Centering

Covariates are centered around ¯ zr, the team-specific mean The strength of team r with covariates is ˜ γr ✏ δr βr0 ♣zir ✁ ¯ zrqT αr ✏ δr βr0 ✁ ♣¯ zr ✁ ¯ zqT αr ♣zir ✁ ¯ zqT αr

• Effect αr is the same if one centers around the global mean ¯

z (over all matches and teams)

• Only the strengths/intercepts are changing

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 24/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Pure team-Specific Explanatory Variables

Parameterization γir ✏ γr ✏ βr0 zT

r τ.

Penalizing only intercepts J♣β10, . . . , βm0q ✏ ➳

r➔s

⑤βr0 ✁ βs0⑤ For λ Ñ ✽:

• β10 ✏ . . . ✏ βm0 ✏ 0, enforces that all variation in strength contained in

explanatory variables, no variation left

• reduces number of parameters, identifiable for proper design matrices

Penalizing all parameters J♣β10, . . . , βm0, τq ✏ ➳

r➔s

⑤βr0 ✁ βs0⑤

m

➳

r✏1

⑤τr⑤.

• In addition selects explanatory variables

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Pure team-Specific Explanatory Variables

Parameterization γir ✏ γr ✏ βr0 zT

r τ.

Penalizing only intercepts J♣β10, . . . , βm0q ✏ ➳

r➔s

⑤βr0 ✁ βs0⑤ For λ Ñ ✽:

• β10 ✏ . . . ✏ βm0 ✏ 0, enforces that all variation in strength contained in

explanatory variables, no variation left

• reduces number of parameters, identifiable for proper design matrices

Penalizing all parameters J♣β10, . . . , βm0, τq ✏ ➳

r➔s

⑤βr0 ✁ βs0⑤

m

➳

r✏1

⑤τr⑤.

• In addition selects explanatory variables

1.5 1.0 0.5 −0.5 0.0 0.5 1.0 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA HAN HOF HSV ING KOE LEV MAI MGB S04 STU WOB 1.5 1.0 0.5 0.40 0.45 0.50 0.55 0.60 log(λ + 1) estimates

Obj−spec. Covariates

MarketValue

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

R Package BTLLasso

• Provides whole framework to incorporate all types of covariates in paired

comparison models

• Specific penalty terms to
• Select variables
• Cluster objects
• Main functions:

BTLLasso(Y, X = NULL, Z1 = NULL, Z2 = NULL, lambda = NULL, control = ctrl.BTLLasso(), trace = TRUE) and cv.BTLLasso(Y, X = NULL, Z1 = NULL, Z2 = NULL, folds = 10, lambda = NULL, control = ctrl.BTLLasso(), cores = folds, trace = TRUE, trace.cv = TRUE, cv.crit = c("RPS", "Deviance"))

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Extensions of the Bradley-Terry Model in BTLLasso

P♣Yi♣r,sq ✏ 1 ⑤ xi, zr, zir, zisq ✏ exp♣ηi♣rsqq 1 exp♣ηi♣rsqq ✏ exp♣γir ✁ γisq 1 exp♣γir ✁ γisq Covariate type Effect type γir ✏ γis ✏ ηi♣rsq ✏ intercept

• bject-spec.

βr0 βs0 βr0 ✁ βs0 subject-spec. xi

• bject-spec.

xT

i βr

xT

i βs

xT

i ♣βr ✁ βsq

• bject-spec. zr

global zT

r τ

zT

s τ

♣zr ✁ zsqTτ subject-object-spec. zir global zT

irτ

zT

isτ

♣zir ✁ zisqTτ subject-object-spec. zir

• bject-spec.

zT

irαr

zT

isαs

zT

irαr - zT isαs

• rder effect
• bject-spec.

δr δr ë incl. order effect global δ δ

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Extensions of the Bradley-Terry Model in BTLLasso

P♣Yi♣r,sq ✏ 1 ⑤ xi, zr, zir, zisq ✏ exp♣ηi♣rsqq 1 exp♣ηi♣rsqq ✏ exp♣γir ✁ γisq 1 exp♣γir ✁ γisq Covariate type Effect type γir ✏ γis ✏ ηi♣rsq ✏ intercept

• bject-spec.

βr0 βs0 βr0 ✁ βs0 subject-spec. xi

• bject-spec.

xT

i βr

xT

i βs

xT

i ♣βr ✁ βsq

• bject-spec. zr

global zT

r τ

zT

s τ

♣zr ✁ zsqTτ subject-object-spec. zir global zT

irτ

zT

isτ

♣zir ✁ zisqTτ subject-object-spec. zir

• bject-spec.

zT

irαr

zT

isαs

zT

irαr - zT isαs

• rder effect
• bject-spec.

δr δr ë incl. order effect global δ δ

27/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Extensions of the Bradley-Terry Model in BTLLasso

P♣Yi♣r,sq ✏ 1 ⑤ xi, zr, zir, zisq ✏ exp♣ηi♣rsqq 1 exp♣ηi♣rsqq ✏ exp♣γir ✁ γisq 1 exp♣γir ✁ γisq Covariate type Effect type γir ✏ γis ✏ ηi♣rsq ✏ intercept

• bject-spec.

βr0 βs0 βr0 ✁ βs0 subject-spec. xi

• bject-spec.

xT

i βr

xT

i βs

xT

i ♣βr ✁ βsq

• bject-spec. zr

global zT

r τ

zT

s τ

♣zr ✁ zsqTτ subject-object-spec. zir global zT

irτ

zT

isτ

♣zir ✁ zisqTτ subject-object-spec. zir

• bject-spec.

zT

irαr

zT

isαs

zT

irαr - zT isαs

• rder effect
• bject-spec.

δr δr ë incl. order effect global δ δ

27/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Extensions of the Bradley-Terry Model in BTLLasso

P♣Yi♣r,sq ✏ 1 ⑤ xi, zr, zir, zisq ✏ exp♣ηi♣rsqq 1 exp♣ηi♣rsqq ✏ exp♣γir ✁ γisq 1 exp♣γir ✁ γisq Covariate type Effect type γir ✏ γis ✏ ηi♣rsq ✏ intercept

• bject-spec.

βr0 βs0 βr0 ✁ βs0 subject-spec. xi

• bject-spec.

xT

i βr

xT

i βs

xT

i ♣βr ✁ βsq

• bject-spec. zr

global zT

r τ

zT

s τ

♣zr ✁ zsqTτ subject-object-spec. zir global zT

irτ

zT

isτ

♣zir ✁ zisqTτ subject-object-spec. zir

• bject-spec.

zT

irαr

zT

isαs

zT

irαr - zT isαs

• rder effect
• bject-spec.

δr δr ë incl. order effect global δ δ

27/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Extensions of the Bradley-Terry Model in BTLLasso

P♣Yi♣r,sq ✏ 1 ⑤ xi, zr, zir, zisq ✏ exp♣ηi♣rsqq 1 exp♣ηi♣rsqq ✏ exp♣γir ✁ γisq 1 exp♣γir ✁ γisq Covariate type Effect type γir ✏ γis ✏ ηi♣rsq ✏ intercept

• bject-spec.

βr0 βs0 βr0 ✁ βs0 subject-spec. xi

• bject-spec.

xT

i βr

xT

i βs

xT

i ♣βr ✁ βsq

• bject-spec. zr

global zT

r τ

zT

s τ

♣zr ✁ zsqTτ subject-object-spec. zir global zT

irτ

zT

isτ

♣zir ✁ zisqTτ subject-object-spec. zir

• bject-spec.

zT

irαr

zT

isαs

zT

irαr - zT isαs

• rder effect
• bject-spec.

δr δr ë incl. order effect global δ δ

27/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Extensions of the Bradley-Terry Model in BTLLasso

P♣Yi♣r,sq ✏ 1 ⑤ xi, zr, zir, zisq ✏ exp♣ηi♣rsqq 1 exp♣ηi♣rsqq ✏ exp♣γir ✁ γisq 1 exp♣γir ✁ γisq Covariate type Effect type γir ✏ γis ✏ ηi♣rsq ✏ intercept

• bject-spec.

βr0 βs0 βr0 ✁ βs0 subject-spec. xi

• bject-spec.

xT

i βr

xT

i βs

xT

i ♣βr ✁ βsq

• bject-spec. zr

global zT

r τ

zT

s τ

♣zr ✁ zsqTτ subject-object-spec. zir global zT

irτ

zT

isτ

♣zir ✁ zisqTτ subject-object-spec. zir

• bject-spec.

zT

irαr

zT

isαs

zT

irαr - zT isαs

• rder effect
• bject-spec.

δr δr ë incl. order effect global δ δ

27/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Extensions of the Bradley-Terry Model in BTLLasso

P♣Yi♣r,sq ✏ 1 ⑤ xi, zr, zir, zisq ✏ exp♣ηi♣rsqq 1 exp♣ηi♣rsqq ✏ exp♣γir ✁ γisq 1 exp♣γir ✁ γisq Covariate type Effect type γir ✏ γis ✏ ηi♣rsq ✏ intercept

• bject-spec.

βr0 βs0 βr0 ✁ βs0 subject-spec. xi

• bject-spec.

xT

i βr

xT

i βs

xT

i ♣βr ✁ βsq

• bject-spec. zr

global zT

r τ

zT

s τ

♣zr ✁ zsqTτ subject-object-spec. zir global zT

irτ

zT

isτ

♣zir ✁ zisqTτ subject-object-spec. zir

• bject-spec.

zT

irαr

zT

isαs

zT

irαr - zT isαs

• rder effect
• bject-spec.

δr δr ë incl. order effect global δ δ

27/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Penalties in BTLLasso

For all model components, specific penalty terms can be applied to keep the model sparse and interpretable We can apply L1-penalties for

• clustering of intercept parameters
• clustering and selection of variable effects
• clustering and selection of order effects

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Penalties in BTLLasso

For all model components, specific penalty terms can be applied to keep the model sparse and interpretable We can apply L1-penalties for

• clustering of intercept parameters
• clustering and selection of variable effects
• clustering and selection of order effects

All penalties can be weighted according to the idea of adaptive lasso proposed by Yuan and Lin (2006)

• weights are calculated as inverse of the absolute values of the penalized term

when estimates using ML or small Ridge penalties

• high values (or large differences) in ML estimates are penalized less strong

compared to small values (or small differences)

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

4 3 2 1 −0.1 0.0 0.1 0.2 log(λ + 1) estimates

Intercepts

Object_1 Object_2 Object_3 Object_4 Object_5 Object_6 Object_7 Object_8 P ♣☎q ✏ ➳ r➔s ⑤βr0 ✁ βs0⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ 4 3 2 1 −0.15 −0.05 0.05 0.10 0.15 0.20 log(λ + 1) estimates X Object_1 Object_2 Object_3 Object_4 Object_5 Object_6 Object_7 Object_8 P ♣☎q ✏ px ➳ j✏1 ➳ r➔s ⑤βrj ✁ βsj ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ 8 6 4 2 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 log(λ + 1) estimates Obj−spec. Covariates Z1 Z2 Z3 P ♣☎q ✏ p2 ➳ j✏1 ⑤τj ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ 6 5 4 3 2 1 0.0 0.1 0.2 0.3 0.4 log(λ + 1) estimates Z Object_1 Object_2 Object_3 Object_4 Object_5 Object_6 Object_7 Object_8 P ♣☎q ✏ p1 ➳ j✏1 ➳ r➔s ⑤αrj ✁ αsj ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ 7 6 5 4 3 2 1 0.15 0.20 0.25 0.30 0.35 0.40 log(λ + 1) estimates Z Object_1 Object_2 Object_3 Object_4 Object_5 Object_6 Object_7 Object_8 P ♣☎q ✏ p1 ➳ j✏1 ➳ r➔s ⑤αrj ✁ αsj ⑤ p1 ➳ j✏1 m ➳ r✏1 ⑤αrj ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ 8 6 4 2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 log(λ + 1) estimates Order P ♣☎q ✏ ⑤δ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ 6 5 4 3 2 1 0.5 0.6 0.7 0.8 log(λ + 1) estimates Order Object_1 Object_2 Object_3 Object_4 Object_5 Object_6 Object_7 Object_8 P ♣☎q ✏ ➳ r➔s ⑤δr ✁ δs⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ⑤ ⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Overview over Penalty Terms in BTLLasso

Covariate type Effect type Penalty intercept

• bject-spec.

➦

r➔s

⑤βr0 ✁ βs0⑤ subject-spec. xi

• bject-spec.

px

➦

j✏1

➦

r➔s

⑤βrj ✁ βsj⑤ subject-object-spec. zir global

p2

➦

j✏1

⑤τj⑤ ë incl. object-spec. zr global

p2

➦

j✏1

⑤τj⑤ subject-object-spec. zir

• bject-spec.

p1

➦

j✏1

➦

r➔s

⑤αrj ✁ αsj⑤ ν1

p1

➦

j✏1 m

➦

r✏1

⑤αrj⑤

• rder effect

global ⑤δ⑤

• rder effect
• bject-spec.

➦

r➔s

⑤δr ✁ δs⑤ ν2

m

➦

r✏1

⑤δr⑤

♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ♣☎q ✏ ➳ ✏ ➳ ➔ ⑤ ✁ ⑤ ➳ ✏ ➳ ✏ ⑤ ⑤ ♣☎q ✏ ⑤ ⑤ ♣☎q ✏ ➳ ➔ ⑤ ✁ ⑤ 7 6 5 4 3 2 1 0.0 0.2 0.4 0.6 0.8 log(λ + 1) estimates Order Object_1 Object_2 Object_3 Object_4 Object_5 Object_6 Object_7 Object_8 P ♣☎q ✏ ➳ r➔s ⑤δr ✁ δs⑤ m ➳ r✏1 ⑤δr⑤ 29/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Further Functions of BTLLasso

• Methods to
• plot()
• coef()
• logLik()
• print()
• predict()

for objects created by cv.BTLLasso()

• Function

boot.BTLLasso(model, B = 500, lambda = NULL, cores = 1, trace = TRUE, trace.cv = TRUE, with.cv = TRUE) for bootstrap intervals of parameter estimates including plot() and print() methods

30/33

Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Further Functions of BTLLasso

• Methods to
• plot()
• coef()
• logLik()
• print()
• predict()

for objects created by cv.BTLLasso()

• Function

boot.BTLLasso(model, B = 500, lambda = NULL, cores = 1, trace = TRUE, trace.cv = TRUE, with.cv = TRUE) for bootstrap intervals of parameter estimates including plot() and print() methods

1.5 1.0 0.5 0.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA HAN HOF HSV ING KOE LEV MAI MGB S04 STU WOB

1.5 1.0 0.5 0.0 −0.5 0.0 0.5 1.0 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA HAN HOF HSV ING KOE LEV MAI MGB S04 STU WOB

1.5 1.0 0.5 0.0 0.40 0.45 0.50 0.55 0.60 log(λ + 1) estimates

Obj−spec. Covariates

MarketValue

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Further Functions of BTLLasso

• Methods to
• plot()
• coef()
• logLik()
• print()
• predict()

for objects created by cv.BTLLasso()

• Function

boot.BTLLasso(model, B = 500, lambda = NULL, cores = 1, trace = TRUE, trace.cv = TRUE, with.cv = TRUE) for bootstrap intervals of parameter estimates including plot() and print() methods

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Further Functions of BTLLasso

• Methods to
• plot()
• coef()
• logLik()
• print()
• predict()

for objects created by cv.BTLLasso()

• Function

boot.BTLLasso(model, B = 500, lambda = NULL, cores = 1, trace = TRUE, trace.cv = TRUE, with.cv = TRUE) for bootstrap intervals of parameter estimates including plot() and print() methods

• −2

−1 1 2 3 AUG BAY BER BRE DAR DOR FRA HAN HOF HSV ING KOE LEV MAI MGB S04 STU WOB

Home

• −1.5

−0.5 0.5 1.5 AUG BAY BER BRE DAR DOR FRA HAN HOF HSV ING KOE LEV MAI MGB S04 STU WOB

Intercept

• 0.0

0.2 0.4 0.6

Global Parameters

MarketValue

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Further Functions of BTLLasso

• Methods to
• plot()
• coef()
• logLik()
• print()
• predict()

for objects created by cv.BTLLasso()

• Function

boot.BTLLasso(model, B = 500, lambda = NULL, cores = 1, trace = TRUE, trace.cv = TRUE, with.cv = TRUE) for bootstrap intervals of parameter estimates including plot() and print() methods

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

Implementational Details

• Fitting procedure implemented in C++ to speed-up computation time

Ñ integrated into R via Rcpp and RcppArmadillo

• Fitting procedure is (penalized) Fisher scoring
• L1 penalties are approximated by quadratic terms to make them differentiable

Ñ following Oelker and Tutz (2017)

• Both cv.BTLLasso() and boot.BTLLasso() can be parallelized on several cores

to speed-up computation time

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Simple Paired Comparison Models Inclusion of Covariates Estimation Applications The R-package BTLLasso References

References

Bradley, R. A. and M. E. Terry (1952). Rank analysis of incomplete block designs, I: The method of pair comparisons. Biometrika 39, 324–345. Gneiting, T. and A. Raftery (2007). Strictly proper scoring rules, prediction, and

• estimation. Journal of the American Statistical Association 102(477), 359–376.

Oelker, M. and G. Tutz (2017). A uniform framework for the combination of penalties in generalized structured models. Advances in Data Analysis and Classification, published online (DOI 10.1007/s11634-015-0205-y) 11, 97– 120. Schauberger, G. (2017). BTLLasso: Modelling Heterogeneity in Paired Comparison

• Data. R package version 0.1-5.

Schauberger, G., A. Groll, and G. Tutz (2016). Modeling football results in the German Bundesliga using match-specific covariates. Technical Report 197, Department of Statistics, Ludwig-Maximilians-Universität München, Germany. Schauberger, G. and G. Tutz (2017). Subject-specific modelling of paired comparison data - a lasso-type penalty approach. Statistical Modelling, in press. Tutz, G. and G. Schauberger (2015). Extended ordered paired comparison models with application to football data from german bundesliga. Advances in Statistical Analysis 99, 209–227.

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Importance of Home Effect

For K=3 (ar wins, draw, as wins) and equal strength, γr ✏ γs, δ reflects the proportion of odds for winning of (home) team r and winning of team s, δ ✏ 1 2 log ✂P♣Yrs ✏ 1q④♣1 ✁ P♣Yrs ✏ 1qq P♣Yrs ✏ 3q④♣1 ✁ P♣Yrs ✏ 3q ✡ . The general odds of winning are P♣Yrs ✏ 1q 1 ✁ P♣Yrs ✏ 1q ✏ eδeθeγr✁γs, P♣Yrs ✏ 3q 1 ✁ P♣Yrs ✏ 3q ✏ e✁δeθeγs✁γr ✏

33/33

Importance of Home Effect

For K=3 (ar wins, draw, as wins) and equal strength, γr ✏ γs, δ reflects the proportion of odds for winning of (home) team r and winning of team s, δ ✏ 1 2 log ✂P♣Yrs ✏ 1q④♣1 ✁ P♣Yrs ✏ 1qq P♣Yrs ✏ 3q④♣1 ✁ P♣Yrs ✏ 3q ✡ . The general odds of winning are P♣Yrs ✏ 1q 1 ✁ P♣Yrs ✏ 1q ✏ eδeθeγr✁γs, P♣Yrs ✏ 3q 1 ✁ P♣Yrs ✏ 3q ✏ e✁δeθeγs✁γr Season 2012/2013 5-point scale: ˆ δ ✏ 0.293 For two teams with equal abilities, one obtains the probabilities

• 0.41 for a victory of the home team
• 0.31 for a draw
• 0.28 for a victory of the away team

42.5% of the matches were won by the home team, 25.5% of the matches ended with a draw and 32% of the matches were won by the away team.

33/33

Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

34/33

Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 34/33

Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 34/33

Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 34/33

Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 34/33

Parameterization Scale

In order to obtain a common scale covariates are transformed to variance one (over all matches and teams).

Centering

Covariates are centered around ¯ zr, the team-specific mean The strength of team r with covariates is ˜ γr ✏ δr βr0 ♣zir ✁ ¯ zrqT αr ✏ δr βr0 ✁ ♣¯ zr ✁ ¯ zqT αr ♣zir ✁ ¯ zqT αr

• Effect αr is the same if one centers around the global mean ¯

z (over all matches and teams)

• Only the strengths/intercepts are changing

35/33

Coefficient Paths

4 3 2 1 −1 1 2 log(λ + 1) estimates

Home

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 1 2 3 log(λ + 1) estimates

Distance

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 log(λ + 1) estimates

BallPossession

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1.0 0.0 0.5 1.0 1.5 2.0 log(λ + 1) estimates

TacklingRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 log(λ + 1) estimates

ShotsonGoal

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −1 1 2 3 4 5 6 log(λ + 1) estimates

CompletionRate

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −3 −2 −1 1 log(λ + 1) estimates

FoulsSuffered

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 4 3 2 1 −2 −1 1 log(λ + 1) estimates

Offside

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB

4 3 2 1 −4 −2 2 4 6 log(λ + 1) estimates

Intercepts

AUG BAY BER BRE DAR DOR FRA FRE HOF HSV ING KOE LEV MAI MGB RBL S04 WOB 36/33

Alternative Approach to Pure team-Specific Explanatory Variables

Turner and Firth (2012) use the parameterization γr ✏ βr0 zT

r τ,

where βr0 are iid random effects, βr0 ✒ N♣0, σ2q Estimation by quasi likelihood, which can be seen as a penalized estimation with penalty J♣β10, . . . , βm0q ✏

p

➳

r✏1

β2

r0

• The main difference is that a ridge type is used instead of a lasso-type penalty

(no fusion)

• Effects of explanatory variables are not penalized, therefore no selection of effects
• Random effects model assumes that explanatory variables and random effects are

uncorrelated (endogeneity problem)

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