Likelihood Ratio, Wald, and Lagrange Multiplier (Score) Tests. Soccer Goals in European Premier Leagues

Transcription

1 Lkelhood Rato, Wald, ad Lagrage Multpler (Score) Tests Soccer Goals Europea Premer Leagues - 4

2 Statstcal Testg Prcples Goal: Test a Hpothess cocerg parameter value(s) a larger populato (or ature), based o observed sample data Data Idetfed wth respect to a (possbl hpotheszed) probablt dstrbuto that s dexed b oe or more ukow parameters Notato: Data:,..., 1 Parameter(s):,..., 1 k Jot Dest Fucto: f,...,,..., 1 1 k

3 Example Eglsh League Total Goals/Match Suppose we wsh to test whether the mea umber of goals ( a hpothetcall fte populato) of games s equal to 3. Note: all games of equal legth (o overtme regular seaso games) Data: Y=Total # of goals a radoml selected game Dstrbuto: Assume Posso wth parameter Null Hpothess: H : = 3 Alteratve Hpothess: H A : 3 Jot Probablt Dest Fucto: 1 e e f 1,...,,1,,... 1!! 1

4 Lkelhood Fucto Aother term for jot probablt dest/mass fucto. Commo Notato: L() or L(,) or L( ) Cosdered as a fucto of both the (observed) data ad the (ukow) parameter values Used estmato ad testg parameter value(s) Goal s to choose parameter value(s) that maxmze lkelhood fucto gve the observed data. Tpcall work wth the log of the lkelhood, as t s ofte easer to dfferetate to solve for maxmum lkelhood (ML) estmators for ma famles of probablt dstrbutos

5 L, e ML Estmato of Posso Mea 1 1! l l L, l l! 1 1 Takg dervatve (wrt ) ad settg to zero for maxmum: dl set d

6 Goals Frequec Total Total Goals Data Frequec of Total Goals

7 l(l) l(l) versus theta (Igorg costat term) l(l) theta

8 Lkelhood Rato Test Idetf the parameter space: W {:>} Idetf the parameter space uder H : W {: } Evaluate the maxmum log-lkelhood Evaluate the log-lkelhood uder H A terms ot volvg parameter ca be gored Take - tmes dfferece (H maxmum) Uder ull hpothess (ad large samples), statstc s approxmatel ch-square wth 1 degree of freedom (umber of costrats uder H ) l, X L l L, LR

9 L Soccer Goals Example l L, 38 l l! Uder H : 3 (Igorg l! ) : 1 l 3, 38(3) 975 l(3) Maxmum : l L, 38(.57) 975l(.57) Test Statstc: X LR l L 3, l L, ( 56.9) 5.1 >.5, We have strog evdece to coclude the true mea total umber of goals s below 3.

10 Wald Test - I B Cetral Lmt Theorem argumets, ma estmators have samplg dstrbutos that are approxmatel ormal large samples The, f we have a estmate of the varace of the estmator, we ca obta a ch-square statstc b takg the square of the dstace betwee the ML estmate ad the value uder H dvded b the estmated varace The estmated varace ca be obtaed from the secod dervatve of the log-lkelhood

11 Wald Test - II l( L) V I where: I E Wald Ch-Square Statstc: W I Posso Model: l L, l l! l L, X V 1 l( L) 1 1 E 1 1 l L, 1 1 I XW I.57 V 7.34

12 Lagrage Multpler (Score) Test Obta the frst dervatve of the log-lkelhood evaluated at the parameter uder H (Ths s the slope of the log-lkelhood, evaluated at ad s called the score) Multpl the square of the score b the varace of the ML estmate, evaluated at. Ths s the verse of the varace of the score. The ch-square test statstc s computed as follows: s, X where LM s, I l L,

13 Soccer Goals Example l L, l l! 1 1 l L, 975 s s I I 3 1,, s, 55 X LM 3.57 I Note that: XW 7.34 > X LR 5.1 > X LM 3.57

14 Log(Lkelhood) - Igorg Costat Term Log-Lkelhood versus Theta (Igorg Costat Term) LR Test LM Test Wald Test l(l) Wald/LR1 Wald/LR LM Theta

15 Geeralzato to Tests of Multple Parameters 1 R11 R 1k r 1 Parameter Vector: H : R r R r rak R g R R r k g1 gk g Maxmum Lkelhood Estmator over etre parameter space: Maxmum Lkelhood Estmator over costrat Lkelhood Rato Statstc: X LR l L, l L, 1 T 1 T W uder H : Wald statstc: X R r RI R R r Lagrage Multpler (Score) Statstc: X LM 1 T 1 s, I s, 1 where: Ij E l L, s, l L, j

16 Soccer Goals Example Premer League Games 4 for k=5 Europea Coutres: Eglad 1 = 38, Y 1 = 975 Frace = 38, Y = 86 Germa 3 = 36, Y 3 = 89 Ital 4 = 38, Y 4 = 96 Spa 5 = 38, Y 5 = 98 exp L, j1 j! 1 j1 j

17 Testg Equalt of Mea Goals Amog Coutres - I H : R r R r l L, l l j! j1 5 Uder H : l L, l l j! 1 j1 l L, l, Uder H : L l L, L L l, l, E j

18 Testg Equalt of Mea Goals Amog Coutres - II s, 3 I,

19 Lkelhood Rato Test l L, l l j! j1 5 5 l l j! 1 1 j l j! 1 j l j! 3.64 l j! 1 j1 1 j1 l, l l! 5 L j 1 j1 LR l j! l j! 1 j1 1 j1 X ( 3.64) Evdece that the true populato meas dffer ( partcular: Frace lower, Germa hgher tha the others) 4,.5

20 1 T 1 T W Wald Test Wald statstc: X R r RI R R r R r (.57) (.17) T 186(.91) RI R (.53) (.58) T RI R XW

21 Lagrage Multpler (Score) Test 1 T 1 Lagrage Multpler (Score) Statstc: LM,, X s I s s, I, s, I, X LM 36.83

22 Testg Goodess of Ft to Posso Dstrbuto All estmato ad testg has assumed that umber of goals follow Posso dstrbutos To test whether that assumpto s reasoable, we compare the observed dstrbutos of goals wth what we would expect uder the Posso model We ca check whether the observed mea ad varace are smlar (uder Posso model the are equal) We ca also obta a ch-square statstc b summg over rage of goals: (observed#-expected#) /expected# whch uder hpothess of model fts s approxmatel ch-square wth (# rage)-1 degrees of freedom

23 Dstrbutos of Goals Observed Expected (Trucated at 7) Ch-Square Statstc Goals Eglad Frace Germa Ital Spa Eglad Frace Germa Ital Spa Eglad Frace Germa Ital Spa #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A 1 1 #N/A #N/A #N/A #N/A #N/A Total Games Ch-square Total Goals CrtVal Average P-Value X obs exp 7 approx obs 7 exp All leagues, except Frace, appear to be well descrbed b the Posso dstrbuto. Especall Eglad, Germa, ad Spa