Modeling the Performance of a Baseball Player's Offensive Production

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1 Brgham Young Unversty BYU ScholarsArchve All Theses and Dssertatons Modelng the Performance of a Baseball Player's Offensve Producton Mchael Ross Smth Brgham Young Unversty - Provo Follow ths and addtonal works at: Part of the Statstcs and Probablty Commons BYU ScholarsArchve Ctaton Smth, Mchael Ross, "Modelng the Performance of a Baseball Player's Offensve Producton" (006. All Theses and Dssertatons Ths Selected Project s brought to you for free and open access by BYU ScholarsArchve. It has been accepted for ncluson n All Theses and Dssertatons by an authorzed admnstrator of BYU ScholarsArchve. For more nformaton, please contact scholarsarchve@byu.edu.

2 MODELING THE PERFORMANCE OF A BASEBALL PLAYER S OFFENSIVE PRODUCTION by Mchael R. Smth A project submtted to the faculty of Brgham Young Unversty In partal fulfllment of the requrements for the degree of Masters of Statstcs Department of Statstcs Brgham Young Unversty Aprl 006

3 BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL Of a master s project submtted by Mchael R. Smth Ths master s project has been read by each member of the followng graduate commttee and majorty vote has been found to be satsfactory. Date Scott Grmshaw, Char Date Glbert Fellngham Date Shane Reese

4 BRIGHAM YOUNG UNIVERSITY As char of the canddate s graduate commttee, I have read the master s project of Mchael R. Smth n ts fnal form and have found that ( ts format, ctatons, and bblographcal style are consstent and acceptable and fulfll unversty and department style requrements; ( ts llustratve materals ncludng fgures, tables, and charts are n place; and (3 the fnal manuscrpt s satsfactory to the graduate commttee and s ready for submsson to the unversty lbrary. Date Scott Grmshaw Char, Graduate Commttee Accepted for the Department G. Bruce Schaalje Graduate Coordnator Accepted for the College Thomas W. Sederberg Assocate Dean, College of Physcal and Mathematcal Scences

5 ABSTRACT MODELING THE PERFORMANCE OF A BASEBALL PLAYER S OFFENSIVE PRODUCTION Mchael Smth Department of Statstcs Masters of Scence Ths project addresses the problem of comparng the offensve abltes of players from dfferent eras n Major League Baseball (MLB. We wll study players from the perspectve of an overall offensve summary statstc that s hghly lnked wth scorng runs, or the Berry Value. We wll buld an addtve model to estmate the nnate ablty of the player, the effect of the relatve level of competton of each season, and the effect of age on performance usng pecewse age curves. Usng Herarchcal Bayes methodology wth Gbbs samplng, we model each of these effects for each ndvdual. The results of the Herarchcal Bayes model permt us to lnk players from dfferent eras and to rank the players across the modern era of baseball ( on the bass of ther nnate overall offensve ablty. The top of the rankngs, of whch the top three were Babe Ruth, Lou Gehrg, and Stan Musal, nclude many Hall of Famers and some of the most productve offensve players n the hstory of the game. We also determne that trends n overall offensve ablty n Major League Baseball exst based on dfferent rule and cultural changes. Based on the model, MLB s currently at a hgh level of run producton compared to the dfferent levels of run producton over the last century.

6 TABLE OF CONTENTS Chapter : Introducton... Chapter : Lterature Revew.. Baseball Statstcal Measures. Bayesan Models.... Applcaton of Bayesan Statstcs to Baseball Problem..... Herarchcal Bayes Models Chapter 3: The Methodology and Approach of Ths Project. 3. Valdatng Berry Value over Herarchcal Bayes Model usng the Berry Value Chapter 4: Project Setup and Results.. 4. Settng Up the Problem Data Collecton and Cleanng Convergence v

7 4.4 Results Season Effects 4.4. Age Functon Ratng Player Values Accordng to Berry Value Conclusons Appendx A Appendx B Appendx C Appendx D Bblography v

8 LIST OF FIGURES Fgure 3. Berry s Measure and Berry s Measure per at bat Densty Curve for all players over all seasons. The Berry measure s very rght skewed and vares between roughly -5 and 50.. Fgure 3. Trace Plot of mean of the season parameter from the Mckey Mantle problem that shows a constant and converged value of around 90.. Fgure 3.3 Chart of Mckey Mantle s Berry Values Posteror Means by Season, wth standard devaton ncluded, that shows a vsual of the quadratc age effect and the large varablty at extreme values... Fgure 4. Berry Value vs. Age Plot for Hank Aaron and Babe Ruth that shows the general age effect from the raw Berry Values... Fgure 4. Trace Plot of the overall precson of the model and the precson of the model constant to llustrate model convergence and the level of confdence we should have n the model.. Fgure 4.3 Average Season Effect Coeffcent by Year whch llustrates the low season effects for the decades of and and hgh season effects for the and the current state of the game.. Fgure 4.4 Predctve Posteror Dstrbutons for 99 and 98 the season wth the hgh and lowest season effects to show that the season effect s sgnfcant.. Fgure 4.5 Overall Age Curve for the Herarchcal Bayes Model to llustrate how the average player had a slow and short perod of maturaton and a long and steep perod of declne.... Fgure 4.6 Multple Age Curves for Some Top Players n the model showng the three dfferent general types of ndvdual age curves that nclude the straght and steady curve, the parabolc curve, and the Barry Bonds curve. Fgure 4.7 Actual vs. Predcted Berry Values for Hank Aaron shows how well the model does at predctng Berry Value for a player wth a relatvely stable career... Fgure 4.8 Actual vs. Predcted Berry Values for Babe Ruth from shows how well the model does at predctng Berry Value for a player wth many extreme values v

9 LIST OF TABLES Table. Lndsey s approach to calculatng value for each offensve event by creatng a table of probablty for each runner-out scenaro. Table. Beta Dstrbuton (74,4 and Beta Dstrbuton (3,53. Summary Statstcs used to compare the mpact of dfferent prors on the posteror dstrbutons.. Table 3. Sx Key Offensve Productve Summary Statstcs Correlatons wth Runs Scored on a team level from 900 to 998 whch wll help determne the best offensve producton summary statstc. Table 3. Summary Statstcs from Mckey Mantle problem showng the mean and standard devaton of all parameters whch shows a general age effect and a large overall varance.. Table 4. Innate Player Rankngs based on the peak value of the age curve and the varablty of that peak value whch ncludes many top echelon Hall of Famers v

10 Chapter.0 Introducton The purpose of ths project s to create a model to evaluate the offensve performance of MLB players. In the game of baseball, the offensve goal of a team s to score as many runs (R as possble. Ths goal s accomplshed by the batter facng a ptcher (an at-bat (AB and ether gettng a ht (H, a walk (BB, sacrfce (SF, error (E, out or a ht-by-ptch (HBP. Some hts are more valuable because they allow the htter and the runners already on base to advance. A ht n whch the batter makes t safely to frst base s called a sngle (B. If the ball s ht far enough, the batter can advance more bases. If the batter makes t to second base t s called a double (B, and f the batter makes t to thrd base t s called a trple (3B. If the ball s ht over the outfeld fence, ths s called a home run (HR and the batter gets to crcle the bases. When a htter makes contact wth the ball but does not make t safely to frst base, ths s called an out (a groundout or fly out. Sometmes, a batter can be out at frst but stll advance the runners already on base or allow the runner on thrd to tag up and score (called a sacrfce ht (SH or the batter can strkeout but stll make t to frst f the catcher drops the ball. These core events have been calculated dfferently from one era to the next. For example, let s consder at-bats. At-bats ncluded walks n 887, sacrfce hts from 889 to 893, and sacrfce fles at varous tmes between 930 and 953. Before the offcal rule of the four-ball walk was adopted n 889, a walk vared from 5 to 9 balls. Some statstcs were not offcally recognzed untl after 900. The strkeout (K as an out for a batter was not offcally recognzed as a baseball

11 statstc untl after 9. Caught-stealng (CS was not adopted offcally n the majors untl 90. A fly ball that scores a runner, called a sacrfce fly (SF, became ts own category apart from a sacrfce ht n 954. The ntentonal walk (IBB was dfferentated from a walk startng n 955 (Thorn, 00. From these core baseball statstcs, the followng formulae allow calculaton of summary values: B + B + 3B + HR BA = Battng Average = AB B + B + 3B + HR + BB + HBP OBP = On Base Percentage = AB + BB + HBP + SF B + (B + 3(3B + 4( HR SLG = Sluggng Percentage = AB Many researchers have proposed ways to calculate a player s offensve contrbuton to team performance (Lndsey, 963; Pankn, 978; Bennet and Flueck, 983; Berry, 000. Battng average, home runs, and runs-batted-n have tradtonally been used to measure a player s offensve value. These tradtonal statstcs all have problems. A key problem n usng runs-batted-n or RBIs to assess player value s the statstc s dependence on the strength of the team that surrounds the ndvdual player. Runs and RBIs are a culmnaton of a seres of events. If a player gets on base, he reles on hs teammates to brng hm n to score. Smlarly, f a player gets a ht he s dependent on hs teammates beng on base so that he can drve them n. Home runs are dependent on the stadum or era n whch a player s battng. For example, home runs are much more common today than they were n the 960s or the 980s. More home runs have been ht on average at some stadums than others because of dfferng ballpark dmensons, clmate and

12 alttude condtons. The key problem wth battng average s that even though t s only dependent on what an ndvdual batter accomplshes, t does not take nto account the value of gettng on base wth a walk and does not dscrmnate between the value of a sngle, double, trple, or a home run (Hoffman, 989; Berry, 000. Because the goal of the game of baseball s to score as many runs as possble and prevent the other team from scorng, the number of runs a team scores s the key to the calculaton of offensve value. The player value should be based on a player s contrbuton to the team total of runs scored. The number of runs scored at the team level s used because at the ndvdual player level, runs scored s dependent on the other team members. There have been varous measures created over the years based on analyzng runs scored at the team level (Lndsey, 963; Pankn, 978; Bennet and Flueck, 983; Berry, 000. These measures generally assgn a value to each offensve event (such as a B, BB, or HR. These assgned values act as weghts and the resultng measure s a weghted average of these values that s hghly correlated wth runs scored on a team level. In ths thess, we use the Berry Value because t s most hghly correlated to runs scored at the team level. We propose a methodology that wll compare players offensve producton for all players from 900 to 998. Ths comparson s complcated by the fact that ballplayers are from dfferent tme perods. The prevously mentoned measures are statc for the effect of each offensve event on runs scored. A more complete model would account for these effects varyng over tme. The era n whch a ballplayer plays should be a factor n determnng hs player value. 3

13 To create comparable player value system, we must determne how much of a player s performance s due to the ballpark he s playng n, the qualty of competton he s playng aganst, the qualty of hs teammates, the equpment he uses or the coaches he plays for. Further, we must consder how much of hs performance s due to hs own ablty level. In ths paper, we wll be focusng on the mpact of season (or era and age. Generally, a player s performance tends to ncrease untl they reach a certan age and then begns to declne. Also, because the game of baseball has changed so much over the years, t s mportant to be aware of the mpact of seasonal dfferences. Other factors may also be mportant, but most, lke equpment and coaches, would be dffcult to analyze because of lack of records or confoundng elements. Berry, Reese, and Larkey (999 look nto the mpact of dfferent ballparks n addton to age and era effects. The objectve n ths thess s to construct a system from whch we can smultaneously compare players of dfferent eras and ages and gan the ablty to assess them on equal grounds. Our approach to assessng offensve producton s framed wthn a Bayesan Herarchcal Model (BHM usng Markov Chan Monte Carlo (MCMC (Glks, Rchardson, and Spegelhalter, 996 as the computatonal tool. We present these methodologes and dscuss why they are approprate and benefcal to buldng a player performance model n Chapter n whch we revew current approaches and methods used to calculate the offensve productvty of a player. Addtonally, we look nto the benefts and problems wthn each of these formulatons. We then revew the statstcal foundatons of BHM. In Chapter 3, a smple BHM s dscussed n detal, and an MCMC model s presented and shows the mportance of modelng a 4

14 player s agng curve. In Chapter 4, a BHM s estmated for the offensve productvty of players that models agng and the effect of dfferent eras. By puttng players on a comparable scale, we then rank players by ther offensve value. 5

15 Chapter.0 Lterature Revew Secton. Baseball Statstcal Measures We looked at how the offensve value or worth of a player has been modeled n the past, and found that most models of player performance were formulated usng the basc httng statstcs (At-bats, Hts, Doubles, Trples, Home Runs, Walks, etc. and ther relatonshp to runs scored. Researchers used these basc httng statstcs n a varety of ways. A paper on modelng player performance could not be complete wthout the menton of Bll James (Lews, 003. James sparked a revoluton n the way that baseball statstcs were vewed wth hs 977 Baseball Abstract. Startng wth feldng n 977 and movng to httng n 979, James man focus was the gross mscalculaton of a player s value based on then current baseball statstcs and the need to create new methods that better mapped a player s value. In hs 979 Baseball Abstract, James wrote: A htter should be measured by hs success n that whch he s tryng to do, and what he s tryng to do s create runs. The new measure that he created was what he called the Runs Created formula: TB = B+ (B + 3(3B + 4( HR Runs Created = (( H + BB TB /( AB + BB Ths functon s the bass of many of the offensve producton functons n the rest of ths chapter. 6

16 Schultz (995 used structural equaton modelng to examne the stablty of ndvdual baseball player performance. Ths type of modelng allowed Schultz to create two latent constructs (POWER and AVERAGE, and analyze the year-to-year stablty of these constructs usng a combnaton of factor analyss and regresson. Hs power construct conssted of HR, SLG, and RBI. Hs average construct conssted of BA, OBP, and R. He concluded that power s very consstent over tme but average s less consstent. Bennett and Flueck (994 used Player Game Percentage (PGP to measure the value of each player to hs team n the 99 and 993 World Seres. PGP estmated player value based on contrbuton to a team s vctory, measured by the degree to whch a player ncreases or decreases hs team s probablty of vctory. A player s PGP for a game s the sum of these probablty changes for each play that a player partcpates n. Each probablty change s dvded evenly between the htter and the ptcher. On plays lke stolen bases, the probablty change s dvded evenly between the catcher and the runner. Usng PGP, Bennett was able to select a most valuable and least valuable player for the 99 and 993 World Seres. One problem wth PGP was that t overstated the defensve contrbuton of the ptcher and catcher. The ptcher should not be held defensvely responsble for all offensve contrbutons of the other team. Covers and Kelers (977 used Markov Chans, sought to fnd a good ndex of a player s offensve effectveness. Ther data was derved from the box score data of all games for a partcular player. OERA (Offensve Expected Runs Per Game was based on the total runs scored by a lne-up of a gven player, whch can be 7

17 computed both emprcally from the data and through a probablty algorthm. Emprcally, the value s computed by startng at the begnnng of a player s career. A record s created, descrbng what happened durng each at bat (out, sngle, double, trple, etc. of that player s career. Then, a game was smulated as f each of the player s career at-bats were consecutve plate appearances n the same game for the same team. Game rules were as follows: sacrfces were not counted, error was consdered as an out, 3 no runners advanced on an out, and 4 sngles and doubles would both advance runners two or three bases. Based on these rules, the number of runs ths player would score n a nne-nnng game was computed. Computatonally, the probablty of sx dfferent at-bat occurrences (outs, walks, sngles, doubles, trples, and home runs was calculated from a player s data. The followng functons were used to calculate OERA: s = one of the 4 possble states of the game for the 8 possble runners on base states and the 3 out states H = the ht type, ether a 0 for an out, a B for a walk, or,,3, or 4 for a sngle, double, trple, or home run s ' = the new state of the game after a partcular ht type R ( H, s = the runs scored by a ht at a partcular state of the game: R( s = E( s = H p( s' s = H p H p H H R( H, s f ( H, s = s' p ( E( f ( H, s + R( H, s = H s' p( s' s E( s' + R( s 8

18 E(s was a Markov Chan calculatng the expected runs n an nnng begnnng wth state (s. Usng the theory of ergodc Markov chans, f battng events were ndependent and dentcally dstrbuted random varables, then the smulated OERA approached nne E (, the expected runs scored n an nnng startng at the frst state (s=, wth the probablty of one. Covers and Kelers used ths functon to rank htters. Buket, Harold, and Palacos (997 used Markov Chans to evaluate the performance of teams and the nfluence of a player on team performance. The root of ther analyss was to create optmal battng orders for all teams. They set up a 5X5 transton matrx for each player, whch ncluded one row and column for each of the twenty-four potental states of the game (eght possble base-runner combnatons multpled by the three out combnatons plus one row and one column for three outs. The entres n ths matrx were the probabltes that a player would change the current state of the game to any other game state n a sngle plate appearance. Usng 0 as the begnnng state of the game, n as the current state of the game, U as a X5 matrx wth rows equal to the number of runs and columns equal to the current state, and p k as the transton probablty leadng to the scorng of k runs, the followng functon was used: U n+ ( rowj = U n ( rowj p + U ( rowj 3 p n 0 + U ( rowj p n 3 + U ( rowj 4 p n + U ( rowj p n 4 Ths matrx would be terated untl the last column was reached usng a set of rules to govern the transton that occurs at each ht type. The scorng ndex (U of nne startng players was used to fnd the optmal battng order. Ths could also be used 9

19 to evaluate trades by swtchng player order and scorng ndces, and calculatng runs scored. Lndsey (963 created a dataset based on over 400 baseball games occurrng n the 959 and 960 baseball seasons. Ths data ncluded the number of outs and the base runner poston durng each at-bat n those 400 games. By tabulatng the number of runs scored at each base-out combnaton, he found out when t would be advsable (when t would decrease or ncrease the probablty of scorng a run to ntentonally walk a batter, to attempt a double play, attempt a sacrfce, or attempt to steal a base. From the tabulaton of runs-scored at each base-out combnaton, he was able to create a measure of battng effcency. Usng the notaton base runner poston (B, runner on frst (, runner on second (, runner on thrd (3, runner on frst and second (, runner on frst and thrd (3, runner on second and thrd (3, runners on frst, second, and thrd (F, number of outs n the nnng (T, number of tmes that base-out scenaro occurred (N(T,B, probablty that runs were scored n that base runner-out scenaro (P(r T,B, and the expected number of runs scored n a partcular base runner-out scenaro (E(T,B, the results are summarzed n Table.. 0

20 Table.: Lndsey s approach to calculatng value for each offensve event by creatng a table of probablty for each runner-out scenaro. B T N(T,B P(0 T,B P( T,B P( T,B P(> T,B E(T,B F F F From the values ncluded n ths table, the value of a type of ht for a partcular base runner-out scenaro was calculated by takng the number of runs scored n the stuaton plus the ncrease n expected runs (E(T,B E(T,B where E(T,B was the new runner scenaro after the ht usng the followng assumptons:. runners always scored from second and thrd on any type of ht. runners scored from frst on a trple 3. runners went from frst to thrd on half of the doubles and scored from frst on the other half.

21 These ht values were multpled by the percentage of tmes that the base runner-out scenaro occurred (N(T,B/Σ T,B N(T,B and summed over all base runner-out scenaros. From performng these calculatons, Lndsey created coeffcents for each ht type from whch he bult the followng measure for offensve ablty: Lndsey Value = B +.97(B +.56(3B + 3.4( HR For each ndvdual batter, hs value would be the sum of the products of each of these ht type values tmes the number of that partcular ht type dvded by the number of at-bats. Ths would gve each batter a runs per at-bat value to measure httng effectveness. There were some problems wth ths measure: frst, the formula only took nto account the value of each type of ht; second, the value of a walk, sacrfce fly, or out were not ncluded; thrd, Lndsey concluded that t would be dffcult to generalze results outsde the small range of years of data that he modeled on. Pankn (978 attempted to fnd a statstcal formulaton whch ndcated how well a player performs offensvely, was ndependent of teammate performance, and 3 accounted for dfferng degrees of opportunty. Followng Lndsey s lead, Pankn created an offensve performance average (OPA wth the dea that each base runner-out combnaton had a dfferent number of expected runs. Pankn rounded the Lndsey values for each ht type and used Lndsey s approach of summng the dfferences n expected runs scored to calculate a value for BB and SB. The followng s Pankn s OPA: B + (B +.5(3B + 3.5HR + 0.8( BB + HBP + 0.5SB OPA =. AB + BB + HBP

22 Pankn also adjusted ths formula to nclude the negatve nfluence of an out as follows: OPA 3 = OPA 0.65( AB H /( AB + BB + HBP. He then compared hs functon to runs scored by team from 965 to 975. OPA had a very hgh correlaton wth runs scored and was sgnfcantly better than battng average and sluggng percentage. One problem wth Pankn s functon was that because he only compared t to data gathered from 965 to 975, t was not known f hs functon would perform well over a long tme perod or over dfferent eras of the game. Pankn s functon was also very ad hoc and not based on an analyss of the runs scored data. Fnally, Pankn s functon dd not account for other negatve mpact varables of baseball, such as beng caught stealng or strkng out. Bennett and Flueck (983 evaluated the accuracy of 0 baseball offensve performance models and created ther own model based on 969 to 976 major league baseball team statstcs. They ncluded the tradtonal baseball evaluaton functons (battng average, on-base percentage, and sluggng percentage and the functons that have been ntroduced above n ther evaluaton. In ther regresson analyss, they ncluded all offensve baseball statstcs and used a combnaton of adjusted R and Mallow s C p to select the number of varables. They came up wth the followng Expected Runs Produced (ERP model from ther regresson: ERP =.499(B +.78(B +.65(3B +.449HR +.353BB +.36HBP +.6SB +.394SF.395GIDP.085OUT 3

23 In ther analyss, Bennett and Flueck (983 found that battng average least correlated wth runs scored (R =.644 and sluggng percentage and on-base percentage correlated sgnfcantly better than battng average but were stll two of the least correlated wth runs scored (R =.8 and R =.84 respectvely. Lndsey s and Pankn s functons performed qute well ( R =.98 and R =.937 respectvely, whle Bennett and Flueck s functon ft runs scored best (R =.950. The weakness of ths approach was that because t was purely drven by data wthn a specfc tme perod, the functon was not flexble to dfferent eras. Further, extraneous effects such as ballpark effects and dfferent competton levels were not taken nto account. Berry (000 also studed the relatve worth of varous httng statstcs. In addton to the shortcomngs lsted n the ntroducton, Berry also found shortcomngs n sluggng percentage and on-base percentage. Sluggng percentage over-valued extra base hts and dd not account for walks. On-base percentage accounted for walks but dd not account for the extra value of extra base hts. Berry also performed a regresson analyss to whch he could compare the current tradtonal baseball functon. He used the 990 to 998 season team statstcs wth runs scored beng a functon of B, B, 3B, HR, SB, CS, BB, K, and Outs. The Berry measure was: Berry Value = 0.34BB (B + 0.7(B +.4(3B +.5HR + 0.6SB 0.4CS 0.0( OUT + K Although not taken nto account n hs model, Berry realzed that these coeffcents would change over tme. He combned team data over fve year perods startng n 4

24 900 and calculated the coeffcents of each term n the above functon, except for strkeouts and stolen bases. He found that the coeffcents for walks and outs were the most stable over tme, whle trples and home runs were the least stable. Berry, Reese, and Larkey (999 compared the performances of athletes from dfferent eras n baseball, hockey, and golf. In order to make ths comparson, they made a brdge of player comparsons (older players from one era competng aganst younger players n another era so that there was a lnk between dfferent eras. Berry, et al used the 998 season as a benchmark and created a brdge by comparng performances of current players and past players whose careers overlapped. Because players played aganst each other at dfferent ages, ndvdual-level agng curves were added to estmate the change n player performance as they aged. The bass of ther model was a decade-specfc Herarchcal Bayes Model where they bult n an ndcator for seasonal effects, ballpark effects, and the agng functon. The goals for the Herarchcal Bayes Model were to dscern the effect of agng, to dscover the effect of era, to fnd out f the hot hand effect exsts, and to characterze the talent of each player. They analyzed battng average and home runs to rank baseball player performance. Unlke some past researchers, the model ncluded data on every nonptcher who batted n Major League Baseball from 900 to 998, ncludng year-ofbrth, home ballpark, at-bats, hts, and home runs. Ther models used battng average (hts per at-bat and home run average (home runs per at-bat as ther offensve producton measures. In the dataset of 703 players, x j was the number of hts for the th player n hs j th season, h j was the number of home runs ht by the th player n 5

25 hs j th season, m j was the number of at-bats, a j was the player s age n that season, y j was year of play ( , and t j was player s home ballpark for that season. Lookng at just the battng average model and usng π a j as the probablty of gettng a ht for the th player n hs j th season, where a x ~ Bnomal(, π j m j j a π j a a log( = a + δ y π j a + ξ j t + j f a ( a j and a ~ Normal( µ ( d a a µ ( d a ( ( d ~ Normal( m a,( ( d,( s ~ InverseGamma( a a where was the decade-specfc condtonally-ndependent herarchcal model, a δ y j was the season ndcator, agng functon, a t j a a a, b a ξ was the home ballpark ndcator, f a was the a m was the known mean of the pror mean dstrbuton, known standard devaton of the pror mean dstrbuton, parameter of the pror standard devaton dstrbuton, and a ( j a s was the a a was the known alpha a b was the known beta parameter of the pror standard devaton dstrbuton. Berry, Reese, and Larkey s paper provded a strong bass on whch to buld a statstcal model for comparng baseball player performance. 6

26 Secton. Bayesan Models In ths paper, we are attemptng to model players performances from dfferent years, eras, and competton levels. Baseball s a game of constant change. Therefore, the data collected from year to year s smlar to gatherng data from dfferent studes. Draper, Gaver, Goel, Greenhouse, Hedges, Morrs, Tucker, and Waternaux (99 suggested that the best way to get estmates from heterogeneous sources of data s to use a BHM. The foundaton of Bayesan statstcs s the belef that unknown parameters are random varables, not fxed values. These parameters can be approxmated by mposng a dstrbuton on them, based on our pror knowledge of the parameter. When we do not know much about the parameter, we can mpose a relatvely flat dstrbuton (such as the unform dstrbuton on the parameter over a range of acceptable values. Combnng the lkelhood from the data and the pror dstrbuton, Bayes Theorem can be used to create the posteror dstrbuton. Bayes Theorem s: P( X P( P( X = = P( X P( d P( X P( P( X where P( X s the posteror dstrbuton of the parameters,, gven the data, X, P ( X s the lkelhood of the data gven the parameters, and P( s the pror dstrbuton of the parameters. If the pror s conjugate wth respect to the lkelhood, a closed form expresson of the posteror dstrbuton s avalable. A dstrbuton class of pror dstrbutons s sad to be conjugate f the posteror dstrbuton s n the same class of dstrbutons for all choces of x. For example, f the lkelhood s normal, then the conjugate pror n the normal famly results n a normal posteror 7

27 dstrbuton. If the pror dstrbuton s beta and the lkelhood s bnomal, then the posteror wll be beta wth updated parameters. (Lee, 997 Efron and Morrs (97 presented a example of a Bayesan model and the estmates that result from that model. In ther paper, Efron and Morrs looked nto the propertes of Bayes estmators, usng the followng Bayesan model as an example. If we assume that the varance s known, we have the followng lkelhood and pror dstrbutons: ~ Normal(, ~ Normal( µ, τ From ths model, the followng s the posteror dstrbuton, X j (/ τ µ + ( n / x ~ Normal(, / τ + n / / τ + n / X where n x = x n. j= j / Accordng to Lndley and Smth (97, Bayes estmates performed better than the frequentst least squares estmates when estmatng the expected value for general fxed effect lnear models because the mean squared error for the Bayes estmate was smaller than the mean squared error for the least squares estmate. Ths concluson holds only f the pror dstrbutons are exchangeable (the pror dstrbuton does not change for all and we are lookng at estmates over multple samples (n>4. Bayes models have been used n the past to model baseball data. Sobel (993 used World Seres battng averages from as the data to make comparsons between parameters from dfferent populatons. He created parameters 8

28 to compare ranks based on maxmum lkelhood and Bayes estmates. By analyzng the frst 45 at-bats and the fnal battng averages n the 970 season for a group of htters, Efron and Morrs (975 were able to compare Bayes estmators created by Sten to tradtonal maxmum lkelhood estmaton methods. For each batter, Efron and Morrs llustrated that the Bayes estmator outperformed the maxmum lkelhood estmate. Secton.. Applcaton of Bayesan Statstcs to Baseball Problem In 00 Barry Bonds set a new major league home run record wth 73 home runs n one season. Consderng Barry Bonds home run rate based just on the 00 baseball season may bas the analyst s percepton of Barry Bonds true home run rate s: addtonal hstorcal nformaton would be benefcal to a complete analyss. Because of the ablty to add addtonal nformaton usng a pror dstrbuton, ths case lends tself to the use of Bayesan data analyss. In settng up ths problem, we have data from all 59 games that Barry Bonds played n the 00 season. For each game, we have the number of home runs he ht and the number of hs plate appearances n that game. At each plate appearance, Bonds had a specfc probablty of httng a home run. The probablty of Bonds httng a home run n a partcular plate appearance was assgned the value, and the probablty of that he would not ht a home run was (-. He ether ht a home run (y= or ddn t (y=0. Each plate appearance has a Bernoull dstrbuton, P(y = y (- (-y, where y can be ether or 0. If we look at each plate appearance as an ndependent and dentcally dstrbuted Bernoull dstrbuton, the number of home runs Bonds hts n a season s a Bnomal dstrbuton wth the two parameters beng 9

29 the total number of at-bats n the season and Bonds home run rate. In choosng the lkelhood for the data, we need to focus on whether the bnomal dstrbuton s approprate. The home run rate has a probablty between 0 and. Therefore, we would expect the dstrbuton of home run rate to not be symmetrcal at the low and hgh ends. Ths s caused by the dstrbuton of rates wth hgh or low probablty httng the bounds of 0 and. Of the dstrbutons n the exponental famly, the Bnomal dstrbuton fts these propertes the best. In usng ths dstrbuton, we are assumng that each plate appearance s an ndependent and dentcally dstrbuted event. For the purposes of ths example, we are not takng nto account the tme of season, dfferent ptchers Bonds faces, dfferent ballparks he plays n, or the dfferent teams he s playng. Those varables are addressed n the larger scope of ths thess project. To ease the computaton of the posteror dstrbuton moments, t s helpful to choose a pror such that the posteror dstrbutons stay n a closed exponental form. A pror that has ths property s called a conjugate pror. In order to formulate the conjugate pror dstrbuton, we frst need to recognze the features of the dstrbuton of y (number of home runs Barry Bonds hts n a season. N! y P( y = ( ( y!( N y! N y We can break ths down nto a functon of y, a functon of, and a combned functon of y and n the exponental famly as follows: T P( y = f ( y g( exp( φ( u( y 0

30 N! Usng the dstrbuton of y, f(y =, g( = y, φ ( = log(-, and y!( N y! u(y = N-y. From ths form, we can now create the conjugate pror dstrbuton whch s of the followng form: exp( φ ( T u( y y N y = ( Ths functon looks very smlar to a Beta dstrbuton wth densty Γ( a + b f ( = Γ( a Γ( b a ( b Therefore, the Beta dstrbuton s the conjugate pror dstrbuton for a Bnomal lkelhood and the posteror dstrbuton wll be n the Beta famly. The queston s whch Beta dstrbuton should we use n ths problem? One approach to elctng the pror dstrbuton parameters s to look at the moments of the Beta dstrbuton. The frst moment (or expected value of the Beta dstrbuton s equal to a/(a+b. Therefore, we used an approxmaton of Barry Bonds home run rate as our expected value. The second moment (or the varance of the Beta dstrbuton s related to the sze of a and b. The larger the sze of a and b, the more confdence we have n the expected value of Bonds home run rate and the smaller the varance of the Beta dstrbuton. For example, f the approxmated expected value of Barry Bonds home run rate s one out of every 0 at-bats, or., but we are not very confdent n ths value, we could use a = and b = 9. If the approxmated

31 expected value of Barry Bonds home run rate s about 5 percent and we are very confdent n ths, we could use a = 50 and b = 850. The frst thng to do to calculate the posteror dstrbuton s to set up the jont dstrbuton of Y and, whch s the product of the pror dstrbuton of Ө and the lkelhood of Y. If the pror dstrbuton s Beta(,9, then p( y, = p( p( y Γ(0( = 8 N! y ( y!( N y! Γ( Γ(9 N y The posteror dstrbuton of Barry Bonds home run rate s the dstrbuton of gven the data y. Ths s equal to the jont dstrbuton of y and dvded by the margnal dstrbuton of y or the jont dstrbuton of y and dvded by the jont dstrbuton of y and ntegrated wth respect to. That s, P( y, P( y = = P( y = Γ(0( Γ( Γ(9 8 Γ(0( Γ( Γ(9 P( y, P( y, d N! y ( y!( N y! N! y ( y!( N y! y Γ( y + + N y + 9 ( = Γ( y + Γ( N y N y+ 8 N y N y d Applyng the results of the data (y = 73 and N-y = 403, we get a posteror dstrbuton of Beta(74,4. We can also do the same calculatons to compute the posteror dstrbuton usng a more nformatve pror dstrbuton of Beta(50,850. Then,

32 p ( y, = p( p( y Γ(000 = y ( ( N, y ( Γ(50 Γ(850 N y y+ 49 Γ( y N y ( p( y = Γ( y + 50 Γ(( N y N y+ 849 For the more nformatve pror, we get a new posteror dstrbuton of Beta(3,53. These Beta Dstrbutons have the summary statstcs n Table.. Table.: Beta Dstrbuton (74,4 and Beta Dstrbuton (3,53. Summary Statstcs used to compare the mpact of dfferent prors on the posteror dstrbutons. Mean Varance Standard Devaton Mode Medan By comparng these two dfferent posteror dstrbutons, we can come to some conclusons based on the data and the choce of pror dstrbuton. Frst, the more nformatve the pror dstrbuton, the stronger the pror dstrbuton s mpact wll be on the center of the posteror dstrbuton. Second, the more nformatve the pror dstrbuton, the smaller the varance s n the posteror dstrbuton. The nverse s also true. The weaker the pror, the closer the posteror dstrbuton s to the data (mean = Secton.. Herarchcal Bayes Models The Herarchcal Bayes Model s called herarchcal because t has multple levels of connected pror dstrbutons. The parameters of the pror dstrbuton have ther own pror dstrbutons, known as hyperprors (Lee, 997. Usng hyperprors allows for the use of nformaton from multple levels of observatonal unts and 3

33 allows for the exchangeablty of parameters n dfferent but related studes. For example, f we were to perform a study on whether asprn s an approprate choce of treatment for post-heart attack patents, we would want to fnd out f there s a dfference between the death rate n the asprn group and the death rate n the placebo group. Multple replcatons of ths study would be performed at dfferent places and tmes. We would want to combne the nformaton gathered from each of these study locatons. Ths s called a mult-staton clncal tral. Knowng that the studes are dfferent, we could not aggregate the data drectly. To get estmates wth heterogeneous studes, we would need to account for the dfferences whle modelng the smlartes. The herarchcal model has two stages, the ndvdual stage and the general stage. The frst or ndvdual stage of ths herarchcal model would be one that states that the number of people who ded when takng asprn n a partcular study ( x s Bnomally dstrbuted wth a certan death rate proporton ( π and the number of people n the study ( n as the parameters. That s, wth π havng x Bn(, π ~ n π ~ Beta( α, β. The general stage of ths herarchcal model would be that the pror dstrbuton parameters of the ndvdual study death rate have a pror dstrbuton (or a hyperpror of the orgnal dstrbuton. Thus, we could assume that theα and β parameters have dstrbutons as follows: 4

34 α ~ α α Gamma ( a, b and the β parameter wth β ~ Gamma ( aβ, bβ where a α, b α, a β,and b β are all specfed constants. In ths herarchcal model, we assume that each study comes from a populaton of studes whch has a certan dstrbuton, nstead of beng ndependent. The estmates for a partcular study borrow strength from the other studes. Each partcular study helps to estmate ts own parameters and also the general parameters. Ths s because each of the other studes gves us more nformaton about the partcular study for whch we want to fnd estmates. In the smplest case (a lnear fxed effect model, the estmates from a herarchcal model are a weghted average between the ndvdual study values and the overall mean of the combned studes. (Draper et al, 99 Herarchcal Bayes models have been used to analyze dfferent aspects of the game of baseball. Usng player data from the 988 to 99 seasons, Albert (994 used Herarchcal Bayes models to see whch stuatonal varables (home vs. away, grass vs. turf, etc. explaned a sgnfcant amount of varaton n a player s httng. The model he proposed was for h j, the number of hts by player n the j th stuaton, and o j, the number of outs by player n the j th stuaton: y y j j j hj = log( o j ~ Normal( µ, µ = µ + α j j j 5

35 where α ~ Students t( µ,, υ j ~ Inverse Gamma( k, α and µ α and µ have a unform dstrbuton. Lookng at the posteror means from ths Herarchcal model, the ahead n the count vs. two strkes n the count effect s the largest and most sgnfcant wth the medan of the battng average dfferences of 3 ponts (on a realstc scale of 0 to 45 ponts. α α 6

36 Chapter 3.0 The Methodology and Approach of Ths Project Secton 3. Valdatng Berry Value over The purpose of ths chapter s to ft a smple BHM for the Berry measure of offensve capacty of a major league baseball player. By buldng a model wth just one player, we nvestgate the propertes of the Berry measure and verfy the presence of an age effect. Before pursung the model, t s nterestng to compare the Berry measure to fve other summary statstcs of a baseball player s offensve productvty. In buldng our BHM model, we want to use the offensve productvty summary statstc that s most hghly correlated wth the amount of runs scored. Usng the ndvdual player data from , we collapse the data by year and team so that we have team totals for each year from 900 to 998. Each of the sx summary statstcs s also calculated for each team from 900 to 998. The resultng correlatons of the functons on runs scored are n Table 3.. Table 3.: Sx Key Offensve Productve Summary Statstcs Correlatons wth Runs Scored on a team level from 900 to 998 whch wll help us determne the best offensve producton summary statstc. Correlaton OPS 85.6% Lndsey 87.67% OPA 87.78% OPA3 85.5% ERP 90.38% Berry 96.4% Table 3. shows that Berry s functon s the most hghly correlated wth runs scored at the team level (r = 96.4%. 7

37 In order to dentfy a reasonable dstrbuton of the Berry measure, consder kernel densty estmators on the Berry measure values calculated at an ndvdual level for all of the players over each year. Fgure 3.: Berry s Measure and Berry s Measure per at bat Densty Curve for all players over all seasons. The Berry measure s very rght skewed and vares between roughly -5 and 50. From Fgure 3., the Berry Value s very rght skewed because t s hghly correlated wth the number of at-bats a player receved. The more at-bats a player receves, the more runs he may produce. Another perspectve s to standardze the Berry Value to a per at bat value as also seen n Fgure 3.. It should be noted that the Berry measure can result n a negatve value when an ndvdual player creates sgnfcantly more outs than he does hts. Ths result means that the player has a negatve mpact on hs team scorng runs. From ths pont forward, we wll focus on the raw Berry Value. The frst reason for usng the raw Berry Value s that the number of at-bats s a very mportant ndcator of a player s value. The more games an above average player plays, the more the team benefts. The second reason for usng raw Berry Values s that the number of games played s also clearly a functon 8

38 of age. Generally the older a player gets, the more hs njures and declnng sklls mpact playng tme. Secton 3. Herarchcal Bayes Model usng the Berry Value As stated above, the Berry Value s the baseball measure most hghly correlated wth runs scored at the team level. Also seen above, the dstrbuton of the Berry Value s rght skewed and the values tend to fall between -0 and 50 wth the mean beng around 40. We assume that wthn each season and on an ndvdual level, the dstrbuton of Berry Value s approxmately Normal. In ths example, we choose one player s Berry Values to analyze. Mckey Mantle was chosen for the model because of hs long career (8 years and hs hghly recognzed name. We model hs Berry Values as havng a dfferent mean per season. We assume that Mckey Mantle s dstrbuton of mean Berry Values for each season are Normal dstrbutons wth parameters and. These parameters also have ther own pror dstrbutons. Followng are the dstrbutons of the lkelhood and pror dstrbutons wth ndcatng the ndvdual season of Mantle s Berry Values, usng the conventon Normal(µ,τ where µ s the mean and τ s the precson (the nverse of the varance: y µ, ~ Normal(µ, µ, ~ Normal(, ~ Inverse Gamma(,00 ~ Normal(80,.0 9

39 ~ Inverse Gamma(,00 These pror dstrbutons were created based on the assumptons of the general dstrbuton of standardzed Berry Values for all players. We also assumed that means generally follow a normal dstrbuton and varances generally follow an nverse gamma dstrbuton. It s mportant to note that these are conjugate pror dstrbutons f we look at the margnal posteror dstrbutons. Now wth the pror and lkelhood dstrbutons of 8 seasons of Berry Values, we buld the unnormalzed jont posteror dstrbuton of the data and parameters. ( µ ( y µ ( = = p( y, µ,,, ( exp( Usng the unnormalzed jont dstrbuton, we calculate the condtonal dstrbutons for each of the parameters (at least to a constant of proportonalty. We used WnBUGS software to run our BHM, whch uses Gbbs Samplng to teratvely sample each condtonal dstrbuton. If the condtonal s n a closed form, WnBUGS recognzes the conjugate specfcatons and drectly samples from the condtonal. If t s not n closed form, WnBUGS uses adaptve rejecton samplng to update the condtonals. WnBUGS allows the user to specfy the lkelhood of the data and each pror dstrbuton, then t calculates the condtonals. Even though WnBUGS calculates ths for us, t s mportant to understand the condtonal dstrbutons. The condtonal dstrbuton s dentcal to the jont dstrbuton except that t only ncludes the terms dependent on that one parameter. The followng are a couple of the condtonal dstrbutons: 30

40 ( 80 p( y, µ,, exp( 00 p( y, µ,, ( 8 = 00 exp( ( µ 8 = ( µ As seen above, each condtonal dstrbuton has multple parameters. Therefore, t s mpossble to estmate the dstrbuton of each parameter wthout knowng the values of the other parameters. The Gbbs Sampler allows us to get estmates usng a Markov Chan smulaton. In the Gbbs Sampler, we must frst choose startng values and canddate dstrbutons for the condtonal dstrbutons that are not n a closed form. For WnBUGS, the user has the opton to specfy the startng values. If WnBUGS needs to use the Metropols-Hastngs algorthm wthn the Gbbs Sampler, t wll choose ts own canddate dstrbutons. The startng values for all of the parameters are based on the means or expected values of ther pror dstrbutons. For example, we chose a startng value of and each µ of 80 because that s the mean of ther pror dstrbutons. After enough teratons (n ths case we used 00,000, we are able to converge on the true jont posteror dstrbuton. The complcatng factor of the Gbbs Sampler n ths stuaton s the Herarchcal Bayes element. Frst, we must loop through each condtonal dstrbuton for the pror of the standard devaton. Then we must also loop through the condtonal dstrbuton for each season mean. As a result, we have 8 tmes more teratons than a model wth a generc overall mean. After runnng all teratons, we have posteror dstrbutons for each parameter to analyze. 3

41 Havng run all of the teratons, we determne whether or not we converged onto the correct dstrbutons for the parameters. It s common n MCMC algorthms for the parameters to take a certan number of teratons to stablze, especally when we have to approxmate dstrbutons usng Metropols-Hastngs. A trace plot s an mportant tool used to check the stablty and convergence of parameters. The trace plot tracks the accepted samples throughout the teratons. Fgure 3. detals the trace plot of the last 5000 teratons of the mean of the season parameters n the Mckey Mantle problem. Several crtera exst for examnng the parameter trace plots that help determne convergence. When the slope of the plot s relatvely flat, and no obvous trends are present, then the parameters have converged. Fgure 3.: Trace Plot of mean of the season parameter from the Mckey Mantle problem whch shows a constant and converged value of around 90. Ths trace plot shows that whle there s a large amount of varance n the value of ths parameter, t seems to be centered around a value: there are no obvous trends or slope. We conclude that the parameters have converged. However, we should keep n mnd the hgh varablty of these estmates. 3

42 After runnng the Herarchcal Bayes Gbbs Sampler that s shown n Appendx A, we can make some conclusons on the posteror dstrbutons of the man parameters (the µ s and. Table 3. ncludes some moments from these dstrbutons. Table 3.: Summary Statstcs from Mckey Mantle problem showng the mean and standard devaton of all parameters whch shows a general age effect and a large overall varance. Varable Mean Std. Devaton mu[] mu[] mu[3] mu[4] mu[5] mu[6] mu[7] mu[8] mu[9] mu[0] mu[] mu[] mu[3] mu[4] mu[5] mu[6] mu[7] mu[8] /sgma /sgma theta Fgure 3.3 plots the posteror mean estmates of µ and the standard devaton of these means from Table 3.. Season does have a sgnfcant mpact on the Berry Values for Mckey Mantle. It seems that Mckey Mantle s Berry Values sgnfcantly ncreased over the begnnng of hs career, stablzed n the mddle, and then decreased steadly untl hs retrement. The rate of ncrease n the begnnng of 33

43 hs career s steeper than the rate of decrease later n hs career. There are some years that seem to be outlers. For example, Mantle averaged between 0 and 40 games per season, but n season 3 Mantle was njured wth a broken foot and only played n 65 games. Apart from ths outler, the general trends from the Mckey Mantle model show a quadratc knd of effect. Ths valdates the use of a quadratc age curve for the overall model. We wll use a pecewse quadratc functon for the age effect n our overall model, to take nto account the dfferent slopes of maturaton and declne. Accordng to the table and chart, we see that there s plenty of varablty n the data. The overall varance s close to 00. The varances of the season means between 80 and 00 are much smaller than those outsde of that range. Fgure 3.3: Chart of Mckey Mantle s Berry Values Posteror Means by Season wth standard devaton ncluded whch shows a vsual of the quadratc age effect and the large varablty at extreme values. 34

44 Chapter 4.0 Project Setup and Results The goal of ths project s to create a Herarchcal Bayes model that wll permt the lnkng of players from dfferent eras, whch enables us to calculate each player s nnate value. We follow the paradgm of Berry, Reese, and Larkey (999 and use Herarchcal Bayes models wth season and age effects to defne a player s overall offensve value based on Berry Value, a summary statstc hghly correlated wth run scorng. We have shown that a Herarchcal Bayes model on one ndvdual player can be bult and the year-to-year effects can be nvestgated. Wth one player, the year to year effect confounds the player s changng performance over ther career and the dfferences n the game over dfferent eras. Ths secton fts a Herarchcal Bayes model wth a season effect and a functon for the physcal age of a player. We wll apply ths model to all who played Major League Baseball between 900 and 004. In Secton 4., we defne the model and explan the jont posteror and condtonal dstrbutons. We also dscuss the years n whch dramatc changes have occurred n the game of baseball. In Secton 4., we dscuss methods of data cleanng and preparaton. In Secton 4.3, we determne the valdty of the dstrbutons. In Secton 4.4, we dscuss model results, partcularly the season and age effects. We wll dscuss whether the mpact of these changes can be seen n the seasonal effects of the model. We then look more closely at the nnate player value. By calculatng the nnate player value, we can compare a player from past years lke Babe Ruth, to a current player, lke Barry Bonds, n the same context. 35

45 Secton 4.: Settng Up the Problem The Berry Value s the summary statstc most hghly correlated wth runs scored at the team level. It s also the best value used to measure a baseball player s offensve performance. The followng s a dscusson of the Berry Value ncludng some descrptve statstcs. The hgher the Berry Value, the better the player s overall. The mnmum Berry Value s 9 and the maxmum s 3, wth the average around 44. Only about one percent of Berry Values are negatve. Most current superstars and nductees to the Hall of Fame have average career Berry Values around 00. For example, some career mean Berry Values are Babe Ruth (7, Barry Bonds (33, Hank Aaron (05, Ted Wllams (9, Lou Gehrg (40, and Tony Gwynn (80. We make two assumptons n buldng our model. Frst, the Berry Value for a player n one year does not affect the Berry Value of the same player n the next year. The outcomes are ndependent. Second, the effects n the model do not have any mpact on each other. There are no nteractons. We model each player s Berry Values n two dfferent parts. The frst s a player level age functon. Based on the analyss of Mckey Mantle s career s Secton.3 and further supported by Berry Values by age for Hank Aaron and Babe Ruth (whch seem to be typcal of most players n Fgure 4., we notce that players have a perod of maturaton where ther value s ncreasng at the begnnng of ther careers. Then a player s value seems to peak and stays relatvely constant for a perod. At the end of a player s career, hs Berry Value hts a perod of declne. It 36

46 can also be noted from the above two charts that the perod of maturaton and perod of declne for each player can be dfferent n sze and magntude. Fgure 4.: Berry Value vs. Age Plot for Hank Aaron and Babe Ruth whch shows the general age effect from the raw Berry Values. Berry Value Age Babe Ruth Hank Aaron Because of the ncreasng nature of the perod of maturaton and the decreasng nature of the perod of declne, the Berry Value can be modeled by a quadratc functon n the most smplstc form. One drawback of usng a quadratc functon s that the rate and nterval of the perod of maturaton wll be symmetrc to the rate and nterval of the perod of declne. A pecewse contnuous quadratc functon s proposed to better approxmate the agng curve. Relyng on Berry, et al. (999 conclusons and some basc regresson analyss on the overall data, the peak of the age dstrbuton s found to be between 7 and 30 years. We have chosen 9 years as the knot for our pecewse age effect. The age functon of our model therefore s: β 0 + β (a j + β (a j + β 3 z j (a j -9 37

47 where z j = f age 9 and z j = 0 otherwse. It should be noted that at age 9 the age functon s contnuous and has constant slope. The second part of our model consders the effect for each season denoted by δ j. The δ j captures the effect due to changes n rules, qualty of competton, and other dfferences found over tme. A good approxmaton for the tme placement of the era effect s to use the years of major rule changes n Major League Baseball. Over the years, Major League Baseball has made rule changes that affect the way the game s played. Certan events were recorded dfferently and the compettve balance of the game tself was altered. In the early 900s, Major League Baseball made varous adjustments n how the games are recorded. For example, n 903, the foul strke rule was adopted. Before 903, when a batter ht a foul ball, t dd not count as a strke. The foul strke gave htters a slght dsadvantage, because f they ht a foul t brought them closer to three strkes and an out. In 907, the sacrfce fly rule was adopted; when a batter ht a fly ball out, but t drove n a run, t was not counted as an at-bat. The sacrfce fly gave htters a slght advantage because the out was not charged as an at-bat. In 90, many rule changes occurred. The RBI became an offcal statstc. A home run was gven to a batter n the bottom of the nnth f the player who scored the wnnng run was on base. The sptball and other freak delveres were also outlawed snce they gave the ptchers a dstnct advantage. The most monumental change n Major League Baseball occurred on Aprl 5, 947, the date Jacke Robnson broke the color barrer. The ncluson of Afrcan-Amercans n Major League Baseball dramatcally ncreased the talent level n the game. In 959, Major League Baseball 38

48 set mnmum ballpark boundary regulatons, affectng the htters because t set a mnmum dstance for a batted ball to be counted as a home run. In 969, htters ganed a huge advantage when Major League Baseball dropped the heght of the ptcher s mound by 5 nches and shrank the strke zone. Durng most of the 60s, umpres called a large strke zone (from the top of the shoulders to the bottom of the knees. In 969, ths strke zone was rescnded and the tradtonal strke zone (between the armpts and top of the knee was renstated. Ths rule change put the ptchers on a more even plane wth the htters and a ptched ball dd not have the extra advantage of gravty when t was ptched. The shrunken strke zone also gave the htters an advantage because they could be more selectve. In 97, Amercan League htters ganed an added beneft when ts commssoner adopted the desgnated htter rule. The httng ablty of a team drastcally ncreased when the ptcher was replaced by a desgnated htter n the lneup. The probablty of a htter gettng an RBI or scorng a run ncreases when the httng ablty of the teammates surroundng that htter ncreases. In 00, the htters were put back at a dsadvantage when the strke zone was altered vertcally. Over tme, the strke zone had become smaller. The umpres were asked to call the tradtonal strke zone, whch meant to call more hgh strkes and nsde strkes. Ths ncreased the area nto whch a ptcher can am and ptch a strke. The tmng of these rule changes gves us chronologcal breakng ponts by whch we can separate our era or tme perod functons. The severty of these rule changes demonstrates that an era effect s ndeed necessary to model the changes these rules brng to the game. 39

49 We assume that the dstrbutons of coeffcents for the age functon and overall season effect have ther own normal dstrbutons wth dstnct hyperparameters. These hyperparameters also have unque pror dstrbutons. The followng are the dstrbutons of the lkelhood and pror dstrbutons where y j s the Berry Value for the th player playng n the j th season; usng the conventon Normal(µ,τ where µ s the mean and τ s the precson (the nverse of the varance y j β 0, β, β, β 3, δ j, ~ Normal(β 0 + β (a j + β (a j + β 3 z j (a j -9 +δ j, β 0 β0, β 0 ~ Normal( β0, β 0 β β, β ~ Normal( β, β β β, β ~ Normal( β, β β 3 β3, β 3 ~ Normal( β3, β 3 δ j δj, δ ~ Normal( δ, ~ Inverse Gamma(, 5 β0 ~ Normal(0,.0 β 0 ~ Inverse Gamma(, 00 β ~ Normal(0,.0 β ~ Inverse Gamma(, 64 β ~ Normal(0,. δ 40

50 β ~ Inverse Gamma(, 9 β3 ~ Normal(0,.5 β 3 ~ Gamma(, 4 δ ~ Normal(0,.0 δ ~ Gamma(, 36 These pror dstrbutons were not created based on any assumptons about the general dstrbuton of Berry Values for all players. We can make assumptons about the drectonalty of the mean parameters for the pror dstrbutons, based on the need for our ndvdual age curves to peak and be concave down. We do not have enough evdence however, to make any conclusons on the magntude of ths drectonalty. All of our pror dstrbutons of the mean parameters wll therefore have a zero mean. Now that we have the pror and lkelhood dstrbutons, we create the unnormalzed jont posteror dstrbuton by multplyng the lkelhood and all pror dstrbutons (assumng ndependence. The non-normalzed jont posteror dstrbuton follows where n denotes the number of players n the model (5393, m denotes the total seasons n the model (05, and m denotes the number of seasons that each ndvdual player plays: 4

51 4 = = = = = = = = n j j j j j m j n n n n j m j m m n n n n j j a z a a y y p n ( ( ( ( ( ( ( exp( ( ( ( ( exp( exp(,,,,,,,,,,,,,,,, ( β β β β β β β β δ δ β β β β β β β β β β β β δ δ δ β β β β β β β β δ δ β β β β δ δ β β β β β β β β From the unnormalzed jont posteror dstrbuton, we can calculate the condtonal dstrbutons for each of the parameters (at least to a constant of proportonalty. The followng s an example of a condtonal dstrbuton: ( 00 exp(,,,,,,,,,,,,,,, ( β β δ δ β β β β β β β β β δ β β β β β = n j y j p Because the condtonal dstrbutons contan multple parameters, we use the Gbbs Sampler or Adaptve Rejecton Samplng wthn Gbbs Samplng to get estmates usng a Markov Chan smulaton usng WnBUGS. In the ether samplng algorthm, we have the opton of choosng startng values for the teratve process. The startng values for all of the parameters were based on the means or expected values of ther pror dstrbutons. The better the startng values, the qucker the model wll converge. Secton 4.: Data Collecton and Cleanng The data came from prepared by Sean Lahman. The frst data fle (battng.csv contans player d, year d, games, at-bats, runs, hts, doubles, trples, home runs, rbs, stolen bases, caught stealngs, walks, strkeouts,

52 ntentonal walks, sacrfce hts, sacrfce fles, and ground nto double plays for every player from 87 to 004. The Berry Value s defned as: Berry Value = 0.34BB (B + 0.7(B +.4(3B +.5HR + 0.6SB 0.4CS 0.0( OUT + K For Outs, compute Outs = AB H + SH + SF. A second fle (master.csv contanng player age nformaton was merged wth battng.csv. For each player, the Berry Value for each season, the season ndcator varable, and the player s age durng that season create the dataset used for ths analyss. The year 900 s often noted as the start of the modern game of baseball. A large majorty of the rules that created the modern game were nsttuted durng the last decade of the 9 th century, and also the frst decade of the 0 th century. To focus on the modern game, only players n the years 900 to present are modeled. As part of cleanng the data, all of the ndvduals who had mssng ages from the data fle are omtted. Also, all ndvduals who had less than three seasons wth more than 00 plate appearances are removed snce we want to avod any outlers (players wth only one fantastc season n the model. Ths elmnates all of the ptchers from the httng data. Appendx A shows the model n WnBUGS. For ths model, we ran 00,000 teratons. Secton 4.3: Convergence The most mportant crteron for testng the valdty of a BHM model that uses Gbbs Samplng or Adaptve Rejecton Samplng wthn Gbbs Samplng s the test of convergence. Because the Gbbs Sampler uses an teratve process to update the posteror dstrbutons of each parameter, t s mportant that t converges. 43

53 The frst thng to be done n checkng convergence s to look at the trace plots of the draws from the posteror dstrbuton. The trace plot charts the value of the parameter after each teraton. Fgure 4. shows the trace plots for the last 5000 teratons of two posteror dstrbuton parameters from ths model. A smple crteron for assessng convergence of draws to the jont posteror dstrbuton s to examne trace plots of the draws. We look to see f there are trends or patterns n the trace plots regardng notceable shfts n the values of the parameters. If there are no notceable trends, then t s reasonable to assume that the chan has converged to the jont posteror dstrbuton. Fgure 4.: Trace Plot of the overall precson of the model and the precson of the model constant to llustrate model convergence and the level of confdence we should have n the model. Secton 4.4: Results In ths secton, we dscuss the two man effects: the season effect and the age functon. The season effects show the relatve dffculty of scorng runs n each season. We look nto the trends of the season effect and analyze the mpact of the changes n baseball over the last 00 years. From the age effect, we estmate the 44

54 peak performance, and analyze the age curve of multple players. Snce we have lnked players from dfferent eras we can rank players accordng to ther ablty to score runs. Secton 4.4.: Season Effects Fgure 4.3: Average Season Effect Coeffcent by Year whch llustrates the low season effects for the decades of and and hgh season effects for the and the current state of the game. Fgure 4.3 llustrates the yearly season effect for Berry Values n the Herarchcal Bayes Model, and a table of the posteror means of the δ j can be found n Appendx C. The decades wth the hghest season effects were the 90s, 930s and the 000s. The Berry, et al. (999 model also had very hgh seasonal effects for home runs and battng averages durng those decades. Lowest season effects were recorded durng , the dead-ball era, and n the 980s, when offensve producton was low. Some of the drastc downward spkes seen n Fgure 4.4 represent strke- 45

55 shortened seasons (98 and 994 and those affected by war and scandal ( In Secton 4., we noted the fact that Major League Baseball has seen many rule changes over the years of ts exstence, whch necesstates the ncluson of season effects n the model. The tmng of these rule changes can be seen n the season effects. The year 90 (the year that the RBI was ntroduced s the start of a drastc ncrease n season effects. The year 947 (the year that Robnson broke the color barrer s also a part of a slght upturn n season effects; ths effect s very slght because of the very slow ntroducton of Afrcan-Amercan players nto the Major Leagues. The lowerng of the ptcher s mound n 969 caused a large ncrease n the season effect for that year. The strke zone change n 00 had a negatve effect on htters, and ths s shown by the fall n the season effects from a peak n 000 to a lower value n 00. The next ssue we address s the sgnfcance of the season effect n the model. To examne ths more closely, we chart the predctve posteror dstrbutons for the seasons wth the hghest season effect (99, , and the lowest season effect (98, Fgure 4.4 dsplays the predctve posteror dstrbutons. 46

56 Fgure 4.4: Predctve Posteror Dstrbutons for 99 and 98 the season wth the hgh and lowest season effects to show that the season effect s ndeed sgnfcant. These dstrbutons overlap, due to hgh varance. To measure the sgnfcance of ths overlap, we analyze the area shared by the dstrbutons. The two predctve posteror denstes ntersect at -60. The area n 98 season densty (the lowest season effect above -60 s.07. The area n the 99 season densty (the hghest season effect below -60 s.0. Therefore, the dstrbutons are statstcally sgnfcantly dfferent at the 5% level. We conclude that season has a very strong and sgnfcant mpact n the model. Secton 4.4.: Age Functon Fgure 4.5 plots the mean age functon usng the herarchcal agng parameters. We make three observatons: Frst, the peak of the dstrbuton s 8 years; second, the peak Berry Value s around 90, whch approaches the overall average Berry Value of 44 when the general season effect (whch s around -60 s added; and, thrd, the declne perod generally has a much steeper slope than the 47

57 maturaton perod. Baseball players tend to declne faster on average than they mature. Fgure 4.5: Overall Age Curve for the Herarchcal Bayes Model to llustrate how the average player had a slow and short perod of maturaton and a long and steep perod of declne. Age Curves for Baseball Players Berry Value (Partal Age Fgure 4.6 ncludes age curves for some of the best players n the model wth the overall curve overlad for a comparson. Barry Bonds, Babe Ruth, and Ted Wllams were chosen because they were n the top 0 as far as the peak of ther age curves. Ths can be seen by the dstance of ther curves away from the overall average age curve. These three players were also chosen because ther age curves represent the dfferent types of age curves that come out of the model. 48

58 Fgure 4.6: Multple Age Curves for Some Top Players n the model showng the three dfferent general types of ndvdual age curves whch nclude the straght and steady curve, the parabolc curve, and the Barry Bonds curve. 50 Age Curves for Baseball Players 00 Berry Value (Partal Overall Barry Bonds Babe Ruth Ted Wllams Age The frst type of age curve that s observed n ths model s the straght and consstent age curve represented by Ted Wllams. Wllams had a consstent and hgh performng career. He dd have a perod of maturaton and declne, but t s much less pronounced and farly straght. The second type of age curve s the more parabolc age curve represented by Babe Ruth. Ruth had a hgher peak, but much steeper rates of nclne and declne than the mean curve. Age clearly had a strong mpact on Ruth s performance. Lookng at the overall curve, we conclude that a majorty of the players n the model have a stronger parabolc age curve and would be n ths second group. The thrd group s unque. Most players peak at 7 or 8, near the overall model peak. Bonds age curve, however, s always ncreasng. There s no peak n 49

59 hs age functon, and he s the only player out of the top 00 that does not have a negatve pecewse coeffcent. Barry Bonds best four seasons have come n the last 4 years ncluded n the model ( Ths could be due to hs tranng, current equpment, alleged sterod use, or an unknown quantty n the current state of the game that allows a person to contnue playng at a hgh level even at older ages. The age curve n general suggests that overall offensve performance peaks at a young age. Very few ndvduals have a peak age of more than 3 (only 8 out of the top 50 offensve players. Ths agrees wth Berry, et al. who concluded that the peak age for home run httng was 9 and the peak age for battng was 7. It seems that n terms of offensve player performance, any beneft ganed by experence s offset by the deteroratng effects of age. Fgures 4.7 and 4.8 are the scatter plots of the actual and predcted Berry Values for two players (Hank Aaron and Babe Ruth. The predcted Berry Values are computed by calculatng each player s age curve and addng n the season effects for the seasons n whch he played. The connected dots are the predctve values and the unconnected dots are the actual values. 50

60 Fgure 4.7: Actual vs. Predcted Berry Values for Hank Aaron from whch we can see how well the model does at predctng Berry Value for a player wth a relatvely stable career Berry Value Age Actual Value Predcted Value Fgure 4.8: Actual vs. Predcted Berry Values for Babe Ruth from whch we can see how well the model does at predctng Berry Value for a player wth many extreme values. Berry Value Age Actual Value Predcted Value From the above fgures, we can see that the model accurately predcts Berry Value at the ndvdual level. The BHM wll not, however, catch outlers (drastc changes n Berry Value due to effects outsde the model lke we see n Babe Ruth s data. 5

61 The predcted curves ht rght n the mddle of the actual values and seem to ft the actual data qute accurately. The predcted curves do seem to predct the later years of a player s career better than the earler years. Ths makes sense because there s an addtonal parameter n the model to help predct the second half of a player s career. Consderng the borrowng of nformaton between players the model performs better when a player s consstent n hs performance. Secton 4.4.3: Ratng Player Values Accordng to Berry Value Because we have bult a model to help us predct Berry Value by lnkng players from dfferent eras, we have a predctve curve for each ndvdual. In rankng the best ndvduals based on ther Berry Value, we need a fxed number or value. In analyzng a curve that has a dfferent slope and peak for each ndvdual, we can rank the players n multple ways. The opton we use s the overall peak value of the agng curve. Because we allow each ndvdual to have hs own peak, we rank the player based on the dstrbuton of hs peak value. To calculate the peak value for each player, we use the frst dervatve of each player s age functon. As stated before, the followng s our age functon: β 0 + β (a j + β (a j + β 3 z j (a j -9 wth the z j varable takng a value of after the age of 9 years and a 0 before. We therefore, have two frst dervatve functons. The frst dervatve functon, for ages under 9, s: β + β (a j. The second dervatve functon, for ages 9 and above, s: β 58β 3 + (β + β 3 (a j. The age at whch the peak value occurs can be found by settng these functons to zero. The peak age for the frst dervatve functon s: β β. The peak age for the second dervatve functon s: 58β β 3 3 β + β. If the 5

62 peak age for the frst dervatve functon s larger than 9 years, then the peak age for the second dervatve s used to calculate peak value. After the age at whch the curve peaks s found, the peak value can then be deduced by usng the peak age n the formula. To calculate the dstrbuton of the peak value, we use the saved age coeffcents for the last 5000 teratons usng the overall peak age as the age n the functon. Therefore, each ndvdual wll have 5000 estmates for peak value. A mean and a standard devaton are calculated for ths for each ndvdual s posteror dstrbuton. To ensure the selecton of the hghest peak value wth the lowest varablty, both the mean and the standard devaton are mportant measures. The coeffcent of varaton (mean/standard devaton s used as the summary statstc of the peak value posteror dstrbuton for each ndvdual. Table 4. shows the top 5 players ranked by the value of ther coeffcent of varaton for peak value. Accordng to the model, the top fve players of all tme n respect to nnate player value are: Babe Ruth, Lou Gehrg, Stan Musal, Hank Aaron, and Ted Wllams. Ths lst s topped by players who played many seasons at a hgh level of offensve productvty, and are recognzed as the best offensve players of all tme. The model consstently dentfes current superstars, and those players nducted nto the Baseball Hall of Fame. The lst ncludes the all-tme home run leader (Hank Aaron, the all tme hts leader (Pete Rose, the all-tme doubles leader (Trs Speaker, and the all tme stolen base leader (Rckey Henderson. The top ranked players n our lst nclude the top players n all tme career sluggng percentage wth Babe Ruth, Lou Gehrg, and Ted Wllams. These rankngs are strongly correlated wth hgh Berry 53

63 Values throughout a players career. Lou Gehrg has the hghest career Berry Value average. Babe Ruth and Ted Wllams were also n the top ten. Table 4.: Innate Player Rankngs based on the peak value of the age curve and the varablty of that peak value whch ncludes many top echelon Hall of Famers. Rank Player Peak Value CV Babe Ruth 30.0 Lou Gehrg Stan Musal Hank Aaron Ted Wllams Jmme Foxx Ty Cobb Wlle Mays Trs Speaker Rogers Hornsby 5.9 Ken Grffey Jr. 5.5 Frank Robnson Paul Waner Jeff Bagwell Wade Boggs Rckey Henderson Pete Rose Rafael Palmero Mckey Mantle Blly Wllams 4.59 Honus Wagner 4.53 Edde Mathews Al Smmons Frank Thomas Harmon Kllebrew 4.9 Very few current players appear on ths lst. Because they have yet to fnsh ther careers, these contemporary players have fewer seasons n the model and thus more varablty. As varablty s part of the rankng equaton, the hgh varablty of today s players kept them off the lst. As current All-Stars lke Alex Rodrguez play more seasons at a strong offensve level, they wll move up n the rankngs. Our 54

64 focus n ths rankng s hgh peak value and hgh confdence n that peak value. An alternatve rankng s shown n Appendx D. There are a couple of surprse names n the top 5. Whle Jeff Bagwell s a perennal All-Star, hs name s not among the frst that commonly arse when thnkng of the top 5 players of all tme. Hs peak Berry Value s hgh because he s a player that gets on base. He s consstently one of the top 0 players n walks, ht by ptches, and on-base percentage. He also scores many runs and produces large numbers of home runs. Bagwell was also helped by havng many good seasons n an era of low season effects. There are a couple of names mssng whch accordng to contemporary opnon, should be n the top 5. Barry Bonds s one of the best current players n the game and s frequently compared to Babe Ruth and Hank Aaron. Bonds has had 4 seasons wth a Berry Value over 00 and the season wth the hghest Berry Value ever (3.. Barry Bonds s not n the top 5 due to the varablty assocated wth estmaton. The dlemma wth Bonds s that the fnal four seasons were the best of hs career and hs age curve has no declnng porton. Because of ths anomaly, he has hgher varablty than other players n the model that have played as long as he has. Another player that seems to be mssng s Regge Jackson, a Hall of Fame player who fnshed hs career n the top ten all tme n home runs and n the top ten n sluggng twelve tmes. There are a couple of reasons that Jackson s not as hgh n the model. Frst, he played n an era wth the hghest season effects (the 970s. Second, he s top all-tme n the category of strkeouts. 55

65 Secton 4.5: Concluson Usng Bayesan Herarchcal methodology wth a Gbbs Sampler, we have developed a model permttng the lnkng of players from dfferent eras. Usng ths model, we parse out the ntrnsc offensve value of each Major League Baseball player by accountng for dfferent sources of varaton. The effect of the relatve dffculty of each season and the effects of age on performance are modeled. We use ndvdual agng curves to account for dfferent agng effects for each ndvdual. The top 50 professonals are all Hall of Fame players, current All-Stars, or had multple All-Star game appearances. Lookng at the seasonal effects, we can conclude that the game of baseball s currently n a state of hgh run producton. The current state of the game s as httng-domnated as any perod n the hstory of the game ncludng the 90s and 930s. Ths can be due to the lack of ptchng depth caused by expanson, and the hgh amount of condtonng htters go through today. Lookng at the ranked nnate player values, only 3 of the top 5 are current players, wth the rest representng past eras equally. It s nterestng to see a representaton of all eras and the fact that even the current state (takng nto account the hgh varablty of current players does not seem too far out of lne from other tmes n baseball hstory. Some extensons of ths work nclude accountng for other areas of potental mpact, lke coaches and ballparks. Allowng another pecewse pont to show the level or stable perod of a player s career could also strengthen the age curves. The model created s successful n meetng our objectve, whch was to construct a 56

66 system from whch we could smultaneously compare players of dfferent eras and ages and assess them on equal grounds. 57

67 APPENDIX A MICKEY MANTLE PROBLEM PROGRAM IN WnBUGS: model { for ( p n : 8 { berry[p] ~ dnorm(mu[p],tau mu[p] ~ dnorm(theta,tau0 } theta ~ dnorm(80,0.0 tau ~ dgamma(,00 tau0 ~ dgamma(,00 } lst(theta = 80, tau=.0, tau0 =.0, mu = c(80,80,80,80,80,80,80,80,80,80,80,80,80,80,80,80,80,80 lst(berry = c(46.75, 97.6, 8.4, 05.5, 4.36, 5.5, 53.09, 9.78, 0.07,.46, 44.5, 07.7, 43.84, 09.86, 6.37, 63.5, 73.5, 69 FINAL PROJECT PROGRAM IN WnBUGS: model { for(p n : 9570 { Berry[p] ~ dnorm(mu[p], tau mu[p] <- beta0[nd[p]] + beta[nd[p]] * X[p] + beta[nd[p]] * X[p] + beta3[nd[p]] * X3[p] + gamma[season[p]] } # Prors for betas: for (k n : 539 { beta0[k] ~ dnorm(theta0,tau0 beta[k] ~ dnorm(theta,tau beta[k] ~ dnorm(theta,tau beta3[k] ~ dnorm(theta3,tau3 } # Prors for gamma: for (j n : 05 { gamma[j] ~ dnorm(thetagamma,taugamma } # Hyper-prors: theta0 ~ dnorm(0,.0 theta ~ dnorm(0,.0 theta ~ dnorm(0,. theta3 ~ dnorm(0,.5 thetagamma ~ dnorm(0,.0 tau ~ dgamma(,50 tau0 ~ dgamma(,00 58

68 tau ~ dgamma(,8 tau ~ dgamma(,8 tau3 ~ dgamma(,8 taugamma ~ dgamma(,7 } lst(gamma = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, theta0 = 0, theta = 0, theta = 0, theta3 = 0, thetagamma = 0, tau=.04, tau0 =.0, tau =.03, tau =., tau3 =.5, taugamma =.03 SYNTAX FOR PLOTS: ## READ IN DATA## n < m <- 6 m <- 05 o <- 500 alldatag <- rep(0,n* alldatag.m <- matrx(alldatag,n, alldatag.m <- read.table("g:\\perre\\hbreg\\baseball\\taugammand-0-05.txt",header=false nts <- rep(0,m*3 nts.m <- matrx(nts,m,3 nts.m <- read.table("g:\\perre\\hbreg\\baseball\\taugamma-0-05.txt",header=false gammam <- rep(0,m for ( n :05 { gammam[] <- sum(alldatag.m[nts.m[,]:nts.m[,3],]/500 } m <- tau <- rep(0,o tau <- alldatag.m[nts.m[06,]:nts.m[06,3],] tau0 <- rep(0,o tau0 <- alldatag.m[nts.m[07,]:nts.m[07,3],] tau <- rep(0,o tau <- alldatag.m[nts.m[08,]:nts.m[08,3],] tau <- rep(0,o tau <- alldatag.m[nts.m[09,]:nts.m[09,3],] tau3 <- rep(0,o tau3 <- alldatag.m[nts.m[0,]:nts.m[0,3],] taugamma <- rep(0,o taugamma <- alldatag.m[nts.m[,]:nts.m[,3],] theta0 <- rep(0,o theta0 <- alldatag.m[nts.m[,]:nts.m[,3],] theta <- rep(0,o theta <- alldatag.m[nts.m[3,]:nts.m[3,3],] theta <- rep(0,o theta <- alldatag.m[nts.m[4,]:nts.m[4,3],] theta3 <- rep(0,o theta3 <- alldatag.m[nts.m[5,]:nts.m[5,3],] thetagamma <- rep(0,o thetagamma <- alldatag.m[nts.m[6,]:nts.m[6,3],] ## CALCULATE MEANS OF PRIOR AGE COEFFICIENT PARAMETERS TO CREATE OVERALL AGE CURVE ## theta0m <- sum(theta0/500 thetam <- sum(theta/500 thetam <- sum(theta/500 theta3m <- sum(theta3/500 ## PRIOR PARAMETER PLOTS FOR CONVERGENCE ## 59

69 plot(x[:500],tau,"l",xlab="teratons",ylab="tau-overall precson" plot(x[:500],tau0,"l",xlab="teratons",ylab="tau0-precson of constant" plot(x[:500],tau,"l",xlab="teratons",ylab="tau-precson of lnear age coeffcent" plot(x[:500],tau,"l",xlab="teratons",ylab="tau-precson of quadratc age coeffcent" plot(x[:500],tau3,"l",xlab="teratons",ylab="tau3-precson of pecewse age coeffcent" plot(x[:500],taugamma,"l",xlab="teratons",ylab="taugamma-precson of season effect" plot(x[:500],theta0,"l",xlab="teratons",ylab="theta0-mean of model constant" plot(x[:500],theta,"l",xlab="teratons",ylab="theta-mean of lnear age coeffcent" plot(x[:500],theta,"l",xlab="teratons",ylab="theta-mean of quadratc age coeffcent" plot(x[:500],theta3,"l",xlab="teratons",ylab="theta3-mean of pecewse age coeffcent" plot(x[:500],thetagamma,"l",xlab="teratons",ylab="thetagamma-mean of season effect" ## SEASON EFFECT PLOT ## plot(x[900:004],gammam,"l",xlab="year",ylab="average season effect" ## DENSITY PLOTS ## plot(densty(tau,bw=.0,xlab="tau-precson of lnear age coeffcent",ylab="densty",man="" plot(densty(theta,bw=.05,xlab="theta-mean of lnear age coeffcent",ylab="densty",man="" ## GROUPING THE SEASON EFFECTS FOR THE LAST 5000 ITERATIONS ## gamma <- rep(0,o*m gamma.m <- matrx(gamma,o,m for ( n :05 { gamma.m[,] <- alldatag.m[nts.m[,]:nts.m[,3],] } ## CREATING PREDICTIVE POSTERIOR DENSITIES FOR SEASONS WITH HIGHEST AND LOWEST SEASON EFFECT ## ppost <- rnorm(length(gamma.m[,8],gamma.m[,8],sqrt(/taugamma ppost <- rnorm(length(gamma.m[,30],gamma.m[,30],sqrt(/taugamma plot(densty(ppost,bw=5,xlab="posteror densty plots of the season effects for 99 and 98",ylab="densty",man="" lnes(densty(ppost,bw=5 > mean(ppost>-60 [] > mean(ppost>-60 [] #### PLAYER RANKING SYNTAX ### ## LOADING IN THE AGE COEFFICIENTS FOR THE LAST 5000 ITERATIONS FOR THE TOP 50 INDIVIDUALS ### n < m <- 600 o < alldatab <- rep(0,n* alldatab.m <- matrx(alldatab,n, alldatab.m <- read.table("g:\\perre\\hbreg\\baseball\\beta--05.txt",header=false nts <- rep(0,m*3 nts.m <- matrx(nts,m,3 nts.m <- read.table("g:\\perre\\hbreg\\baseball\\betand--05.txt",header=false ## CREATING AN MEAN AGE COEFFICIENT FOR EACH INDIVIDUAL ## betam <- rep(0,50*4 betam.m <- matrx(betam,50,4 for (a n :4 { for (b n :50 { betam.m[b,a] <- sum(alldatab.m[nts.m[(a-*50+b,]:nts.m[(a-*50+b,3],]/5000 } } beta <- rep(0,o*50*4 60

70 beta.m <- matrx(beta,o*50,4 for (a n :4 { for (b n :50 { beta.m[((b-*5000+:(b*5000,a] <- alldatab.m[nts.m[(a-*50+b,]:nts.m[(a-*50+b,3],] } } ## COMPUTING THE PEAK AGE DEPENDING ON THE CORRECT DERIVATIVE ## peakage <- rep(0,50 peakage <- rep(0,50 peakage <- rep(0,50 z <- rep(0,50 peakvalue <- rep(0,o*50 rpeakvalue <- rep(0,o*50 peakage <- read.table("g:\\perre\\hbreg\\baseball\\peakage.txt",header=false peakage <- read.table("g:\\perre\\hbreg\\baseball\\peakage.txt",header=false peakvalue.m <- matrx(peakvalue,o,50 rpeakvalue.m <- matrx(rpeakvalue,o,50 for ( n :50 { peakage[] <- peakage[,] f(peakage[,] > 9 { z[] <- peakage[] <- peakage[,] } ## COMPUTING THE PEAK VALUE FOR EACH ITERATION FOR EACH INDIVIDUAL ## peakvalue.m[,] <- beta.m[((-*5000+:(*5000,] + beta.m[((-*5000+:(*5000,]*peakage[] + beta.m[((- *5000+:(*5000,3]*peakage[]*peakage[] + beta.m[((-*5000+:(*5000,4]*z[]*(peakage[]-9*(peakage[]- 9 } ## RANKING THE PEAK VALUES FOR EACH ITERATION ## for (q n :5000 { rpeakvalue.m[q,] <- rank(peakvalue.m[q,] } ## DETERMINING THE MEAN, STANDARD DEVIATION, AND AVERAGE RANKING FOR EACH INDIVIDUAL S PEAK VALUE ## peakvaluem <- rep(0,50 peakvaluesd <- rep(0,50 rpeakvaluem <- rep(0,50 for (j n :50 { peakvaluem[j] <- sum(peakvalue.m[,j]/5000 rpeakvaluem[j] <- sum(rpeakvalue.m[,j]/5000 peakvaluesd[j] <- sd(peakvalue.m[,j],na.rm=true } 6

71 APPENDIX B TRACE PLOTS: β 0 and β : β, β 3 and δ 6

72 / β and / β / β 3, and / δ 63

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