Assigning numerical measures to various aspects of a ball player's performance is a convenient and perhaps useful, if not completely defensible, practice. A numerical measure certainly eases the problem of rank ordering players. Hitting measures such as batting average and it's cousins are the best known known of these measures. The ease and finality of such comparisons guarantees that these measures will always be with us even if what is being ranked is not entirely clear. So, if we are going to do this, we should use the best measures available. My purpose with this paper is to to systematically compare a number of well known and a few less well known measures that have been proposed to summarize a player's offensive capabilities. This is not a new activity. Both the Bennett&Flueck paper and HGB book contain such analysis.
The starting point for any player evaluation measure are the counting statistics collected over some period of time, perhaps a season or a career. Examples of counting statistics include the number of stolen bases, number of hits and number of assists on defense. Any event that be adequately described in baseball can be counted and the resulting number potentially can be used as a component part of the calculation of a player evaluation measure.
Intuitively, if a measure does a good job of rating the offense, it should also work well on defense. A secondary purpose of the paper is to show that offensive evaluation measures are equally applicable to the defense, if the same counting statistics are provided.
There are two types of measures in common use. The first form is the linear weights method which is a linear combination of basic counting statistics using empirically determined coefficients or weights. The second, and more common method is to define a complex and usually nonlinear combination of the basic quantities (various sums, products and ratios) that empirically reproduces individual or team runs.
The data used in the following discussions are the 1348 team season records derived from full season event files for both leagues for the 1954 to 2008 seasons obtained from Retrosheet, Inc. I have written a parser for these data sets and it is used to provide the team summary data used in the following calculations.
A formula where each coefficient appears linearly can have these coefficients determined by the linear regression procedure (Numerical Recipes in C, 2nd Edition, 1992, Chapter 15). As an example, consider the following simple linear weights formula:
RUNS = W0*TotalBases + W1*(Walks+HitByPitch)+ W2*(Ab-hits)+W3*Errors
This particular linear weights formulation will be referred to as BWOE which is suggested by its component parts, Bases, Walks plus hit by pitch, Outs and Errors.
The linear regression procedure minimizes the sum of squared differences between the actual runs scored by a team and the value estimated from the formula, the chi-square value, by determining the 4 coefficients, W0 - W3. For obvious reasons this procedure is also called Least Squares Fitting. The regression coefficients, Wi, effectively give the number of runs each single event contributes to the team runs. The following table shows the results of the regression based on these four terms.
param weight average contrib 0 BASES 0.3761 2143.4 806.06 1 BB+HP 0.3124 515.5 161.03 2 AB-HITS -0.0940 4006.9 -376.52 3 ERRS 0.6157 178.2 109.73 Table 1. Four Parameter Linear Weights Formula, "BWOE"
The square root of Chi Square (the sum of the squares of differences between the data value and calculated value from the regression) divided by the number of data points (1348) is the Standard Deviation and is abbreviated STD. Smaller values of the STD or Chi Square indicate a better representation of the actual runs scored. The standard deviation for this regression is 22.43 runs. Column "average" is the per season value for the parameter averaged over the entire data set. "contrib" is the weight for the parameter times the average number thus represents the number of runs due to the particular parameter. Table 4, which is a summary of the various runs estimators, also includes this result.
To achieve the best results, defined here as the smallest STD (18.8 runs) for the regression, a number of additional counting statistics are used in Table 2. One of the advantages of using the full season events files is the ease of determining non traditional events counting statistics. Three such statistics have been defined and determined. ROH is the count of runners out on a hit, either base runners or the batter trying unsuccessfully for an additional base. RAO is runner advance on an out which includes the traditional sacrifice. Official scoring only credits a sacrifice when there is intent on the part of the hitter. Advances on an out are much more common than official sacrifices suggest. Any base advance increases the potential for scoring thus all such events must be counted. Similarly RSO, Runs Scored on an Out, generalizes the sacrifice fly. ER_BF and ER_RA are the number of times the batter is safe on first and the number of times a runner was able to advance on an error. The OUT category includes outs that are not counted in other categories (CS, ROH, second out on a GDP, K). The meaning of the other counting statistics should be obvious.
param weight average contrib 0 OUTS -0.0988 3031.1 -299.44 1 SNGL 0.4827 981.6 473.78 2 DOUB 0.6347 246.1 156.22 3 TRIP 1.0233 35.0 35.82 4 HRUN 1.4413 141.1 203.41 5 BB+HP 0.3366 515.5 173.50 6 IBB 0.1522 48.5 7.39 7 SB 0.0555 96.2 5.33 8 CS -0.1628 55.9 -9.10 9 ROH -0.3578 26.3 -9.39 10 GDP -0.4707 149.2 -70.24 11 ER_BF 0.5907 73.1 43.19 12 ER_RA 0.3037 105.1 31.92 13 RAO 0.0100 196.1 1.95 14 RSO 0.6546 81.9 53.62 15 K -0.1075 908.2 -97.62 Table 2. The 16 Parameter Linear Weights Formula, "ALL"
Table 2 provides considerable insight into the value of these events. It is a reasonably complete list of offensive events. Stolen bases only make a modest contribution to team runs. Each caught stealing event has about three times the influence as a stolen base. Errors of any kind contribute significantly to the offense. Errors allowing the batter to advance to first are somewhat more beneficial to the offense compared to errors that allow a base runner to advance. Fielding outs and strike outs, K, have approximately the same influence on the outcome of a game. BB includes only unintentional bases on balls as intentional bases on balls are included as a separate term (IBB). The much smaller weight for IBBs compared to BBs suggests that the practice is a useful one for the defense even if the net results is to slightly increase runs scored. The category OUT does not include outs included in other categories: K, CS, RSO and the second out in GDB. The weight of RSO (Run Scored on an Out) is less than one as the negative cost of the out must be accounted for. A linear weights formula can provide great insight into the importance of particular events as well as providing a means for measuring the offense.
Table 3 shows the regression results for a group of commonly available statistics. As might be expected the STD for this lists is between what is shown in Tables 1 and 2.
param weight average contrib 0 SNGL 0.4843 981.6 475.38 1 DOUB 0.7393 246.1 181.96 2 TRIP 1.1697 35.0 40.95 3 HRUN 1.4281 141.1 201.54 4 BB+HP 0.3334 515.5 171.86 5 AB-HITS -0.1145 4006.9 -458.97 6 ERRS 0.4918 178.2 87.65 Table 3. The 7 Parameter Linear Weights Formula, "Basic"
The methodology for this offensive measure survey, following closely the HGB, pp 57-61, and Bennett&Flueck presentations on the same topic, is to determine the constants SLOPE and INTERCEPT in the following linear regression formula:
RUNS = SLOPE*MEASURE + INTERCEPT
The resulting formula is used to evaluate the goodness of the fit by evaluating the Chi-square statistic over the entire team season data set. The direct linear weights measures ("ALL", "Palmer+errors", "BASIC", "BWOE") are also subjected to the linear fitting procedure even though they are, by definition, already linear regression formulas. BASIC is defined in Table 3. This results in slopes near 1 and intercepts near 0. If a constant term had been included in the linear weight formulas, the slope would be exactly 1 and the intercept would be 0. The linear fit also allows the calculation of the correlation coefficient which is useful for the comparisons. The linear fit can be considered a conversion of the measure into runs. The correlation and standard deviation results are intrinsic to the measure. The linear fit makes the relationship explicit.
rank measure slope intrcpt R^2 chisq std 1 All (Table 2) 0.9982 1.31 0.96561 477592 18.8 2 Palmer+errors 0.9973 1.91 0.96094 542474 20.1 3 Basic (Table 3) 0.9996 0.26 0.95691 598499 21.1 4 EqR(Davenport) 0.9779 15.03 0.95373 642628 21.8 5 XR(Furtado) 0.9773 28.11 0.95123 677268 22.4 6 BWOE (Table 1) 0.9952 3.46 0.95121 677649 22.4 7 LWTS(unnormed) 0.9948 5.03 0.95048 687687 22.6 8 OERA+EFBF 0.8701 11.63 0.95008 693372 22.7 9 ERP(Johnson) 0.9737 20.11 0.94740 730542 23.3 10 ERPA(B&F) 0.9581 45.41 0.94676 739477 23.4 11 RC(1998) 0.9317 54.69 0.94564 754906 23.7 12 XRR(Furtado) 0.9741 33.58 0.94553 756505 23.7 13 RC(Tech-1) 0.9300 52.43 0.94547 757357 23.7 14 Base Runs(Smyth) 0.9714 22.46 0.94518 761305 23.8 15 ERP-2(Johnson) 0.9607 34.04 0.94393 778724 24.0 16 LWTS(KenSchmidt) 0.8958 -50.42 0.94316 789390 24.2 17 OERA(Cover) 0.8902 51.99 0.94261 797002 24.3 18 OPA3(Pankin) 0.3954 7.09 0.94080 822143 24.7 19 RC(basic) 0.9504 36.71 0.93798 861287 25.3 20 LWTS(fixed) 0.9658 22.71 0.93788 862728 25.3 21 BOP(Codel) 0.2278 -115.48 0.93316 928241 26.2 22 AB*TA(Boswell) 0.2272 -74.52 0.92393 1056506 28.0 23 AB*DX(Cook) 1.0171 36.40 0.92198 1083529 28.4 24 AB*BRA(OBP*SLG) 1.0481 28.84 0.92092 1098347 28.5 25 BWA(Cramer) 0.8545 13.22 0.91564 1171672 29.5 26 BASES+BB+HP 0.3238 -176.19 0.90766 1282440 30.8 27 OBP+(3/4)ISP(Rickey) 0.3984 -230.39 0.89572 1448251 32.8 28 AB*PRO 0.2444 -258.66 0.87678 1711283 35.6 29 BASES 0.3825 -119.53 0.86989 1806946 36.6 30 HITS+BB+HP 0.5074 -298.13 0.79057 2908619 46.5 31 AB*OBP 0.5858 -343.15 0.76977 3197452 48.7 32 ISP 0.5986 257.66 0.74554 3534045 51.2 33 OPS=SLG+OBP 1862.1326 -649.56 0.73224 3718676 52.5 34 TBR(O'Reilly) 2625.6358 -479.22 0.71483 3960463 54.2 35 HITS 0.6763 -249.02 0.69569 4226371 56.0 36 SLG 2517.2123 -295.92 0.68757 4339060 56.7 37 BA 5682.6046 -773.36 0.52894 6542241 69.7 38 AVERAGE 0.0000 700.37 0.00000 13888291 101.5 Table 4. A Survey of Baseball Offense Evaluation Measures.
It is necessary to distinguish between COUNTING stats, the number of times the particular event being counted occurred and RATE stats, a counting stat that has been divided by the number of chances used to get the particular counted value. The best known rate stat is BA, the batting average, defined as hits/atbats. Since a rate stat does not reflect the number of chances used it will not be a good predictor of counting stats. Since team runs scored is the counting stat being estimated in Table 4 the poor showing of the rate stats (BA, SLG, OPS and TBR) is not unexpected. The counting version of BA is total hits (HITS) which is a somewhat better predictor of team runs. Similarly compare SLG with BASES and TBR with BASES+BB+HP. In Table 4 several other rate stats have been multiplied by AB indicated by AB* in the measure column.
The simple AVERAGE of all the runs scored is given as this establishes a maximum for both the Chi-square and Standard Deviation thus provides a reference for evaluating the other measures. The column R^2 is the square of the linear correlation coefficient between the measure and observed runs. The conventional interpretation is that the correlation coefficient squared is the fraction of variance (Chi-square) in runs explained by the measure.
The first seven entries in Table 4 are linear weights formulas. ALL and BASIC are defined in the discussions of Tables 2 and 3. "Palmer+errors" is a linear regression that uses the same terms as the Palmer linear weights formula and an additional errors term. "BWOE", is a four parameter linear weights formula that summarizes all hitting with total bases instead of the four individual hit type counts (Table 1). Total bases in combination with at bats, errors and walks plus hit by pitch is a simple and effective measure for estimating team runs from basic hitting statistics. When used as an offensive runs measure for a player, the error term would be dropped. The component of these linear weights formulas leading to their excellent performance is the inclusion of a negative weight for outs made. The negative value of an out can be explicitly seen in Tables 1 - 3.
The four linear weights formulas that had their weights explicitly determined by regressions clearly provide the best representation of runs scored from the available team season data. Not surprisingly, including a larger selection of counting statistics results in a more accurate formula.
A Linear Weights formula determined from team season data should include an team errors term. This is evident as error contributions to runs scored are estimated to be around 10% of total runs allowed depending on the details of the particular formula being used. The effect of errors is larger than several other terms often used. The use of an explicit error term preserves the accuracy of the other terms in the regression.
OERA is the Cover and Keilers Offensive Earned Run Average. Their method determines the expected future runs for all 24 combinations of outs and base runners in an inning. OERA is 9 times the future runs expectation for no outs and no one on base. A separate document Implementing the Cover-Keilers Offensive Earned Run Average provides additional details of this measure including source code implementing its calculation. Expected Future Runs have been determined from the data used in this study and are compared to the OERA predictions. Since OERA was explicitly designed to be a player evaluator it did not include the effect of defensive team errors. Errors allowed a batter to reach first have a very similar effect as singles. Adding the number of such errors to the number of singles then doing the calculations is represented by the entry OERA+ER_BF. This combination provides a larger overestimate of team runs than the basic OERA but also correlates better with them.
Pete Palmer has introduced a method for evaluating the offensive contribution of a ball player known as linear weights. Three variations of this method are in Table 4, each starting LWTS. A key feature of Palmer's linear weights system is normalization. An average player or team will have a rating of 0 runs. This is accomplished by adjusting the weight of the AB-Hits term so that a league will have 0 runs for the season. A separate weight is required for each league and each season. This calculation is the (normed) variant in Table 4. The (unnormed) variant adjusts the AB-Hits weight for each league season to obtain agreement with total runs for the season. The (fixed) variant uses a single AB-Hits weight obtained by averaging the individual AB-Hits weights from the (unnormed) calculation. The (unnormed) calculation ranks among the better ones and (fixed) is only slightly inferior.
The other measures in the survey include Paul Johnson's ERP, Estimated Runs Produced, BJBA85 pp 277-281 which is also a linear formula. The Bill James Runs Created formulas Tech-1, RC(tech1), and basic, RC(basic), are defined in BJHBA(1987) pp 279 and 281. A newer version of Runs Created from the Bill James ML Handbook for 1998 is given as RC98 and is slightly more accurate than the older versions. BOP is Base Out Percentage by Barry Codell and is defined in BRJ79 pp 35-39. BWA is Richard Cramer's Batter Win Average, BRJ77 pp 74-79. TA is Tom Boswell's Total Average and is defined in TB5. Earnshaw Cook introduced gave us "Cooks Scoring index" abbreviated DX, HGB p 45. BRA is Dick Cramer's Batter Run Average defined as the product of On Base Percentage and Slugging Average. PRO is production which is defined as the sum of On Base Percentage, OBP, and Slugging Average. ISP is isolated power and is the difference between slugging average and batting average. Total Bases, total hits and OBP are also shown as hitting evaluation measures. Curiously, batting average and slugging average are not as effective a predictor of runs as are total hits and total bases. ERPA is from Bennet&Flueck. EqR is Clay Davenport's Equivalent Runs and is documented in the Baseball Prospectus. Jim Furtado's extended runs formulas are XR and XRR. OPA3 is from Mark Pankin's article in Operations Research. TBR = (BASES+BB+HP)/(AB+BB+HP+SF). While I have tried to find and cite the original references on the measures in Table 4, the best current source for such information is the Glossary of Statistical Terms in TB7.
Since the original posting of this paper, Ken Schmidt and David Smyth have sent me their evaluators which are identified by their names.
Specifically excluded from this analysis are any formulas that include runs batted in (RBI) or runs. Since runs are to be explained or predicted by the measures, using runs or RBIs in a measure is essentially circular reasoning (violates the need for linear independence of the quantities used in the regression).
Evaluating the offensive component of a baseball player appears to be straight forward. The Tech-1 version of Runs Created is an excellent measure and deserves the increasing usage it is receiving. However, a well crafted linear weights formula can provide the most accurate measure relating team counting statistics to team runs scored. A linear weights measure also allows evaluation of the same formula using individual player statistics without increasing uncertainty by more than the effects caused by the size of the statistical sample.
Application of the Offensive Evaluation Measures to the Defense
At present, the official statistical record for the defense is not kept to the same level of detail as for the offense. However, any quantity that can be defined and counted for offensive evaluation purposes can be defined for the defense. The very great advantage of working with full season events files is the ability to define and determine non traditional statistics. All the team related counting statistics used in the preparation of Tables 1 - 4 also have been determined for the defense. The same measure survey, using exactly the same methodology as described for the offense, was done on the defensive statistics with the results shown in Table 5.
rank measure slope intrcpt r^2 chisq std 1 All 0.9970 2.14 0.96545 498562 19.2 2 Palmer+errors 0.9971 2.09 0.96127 558920 20.4 3 Basic 0.9999 0.10 0.95556 641320 21.8 4 BWOE 0.9963 2.68 0.95187 694622 22.7 5 EqR(Davenport) 0.9836 10.96 0.95019 718764 23.1 6 XR(Furtado) 0.9892 19.91 0.94802 750149 23.6 7 OERA+EFBF 0.8539 23.95 0.94656 771239 23.9 8 LWTS(unnormed) 1.0166 -10.22 0.94633 774593 24.0 9 RC(1998) 0.9362 51.46 0.94356 814502 24.6 10 RC(Tech-1) 0.9348 49.03 0.94309 821229 24.7 11 ERP(Johnson) 0.9800 15.72 0.94243 830783 24.8 12 XRR(Furtado) 0.9828 27.63 0.94172 841030 25.0 13 Base Runs(Smyth) 0.9840 13.37 0.94120 848509 25.1 14 ERPA(B&F) 0.9682 38.44 0.94118 848864 25.1 15 ERP-2(Johnson) 0.9723 25.99 0.94035 860789 25.3 16 LWTS(KenSchmidt) 0.8991 -53.23 0.93758 900856 25.9 17 OERA(Cover) 0.8840 56.08 0.93757 900886 25.9 18 OPA3(Pankin) 0.3993 0.30 0.93586 925694 26.2 19 LWTS(fixed) 0.9856 8.83 0.93284 969191 26.8 20 RC(basic) 0.9493 37.31 0.93109 994434 27.2 21 BOP(Codel) 0.2308 -126.17 0.92669 1057887 28.0 22 AB*TA(Boswell) 0.2330 -94.55 0.92405 1096086 28.5 23 BWA(Cramer) 0.8436 21.89 0.92181 1128403 28.9 24 AB*DX(Cook) 1.0191 35.00 0.91408 1239979 30.3 25 AB*BRA(OBP*SLG) 1.0463 29.86 0.91302 1255302 30.5 26 BASES+BB+HP 0.3311 -195.94 0.90291 1401084 32.2 27 OBP+(3/4)ISP(Rickey) 0.4052 -246.19 0.88779 1619331 34.7 28 AB*PRO 0.2465 -267.05 0.86999 1876233 37.3 29 BASES 0.3905 -136.52 0.85446 2100334 39.5 30 HITS+BB+HP 0.5099 -303.15 0.81297 2699078 44.7 31 AB*OBP 0.5921 -354.32 0.79403 2972460 47.0 32 ISP 0.6558 215.38 0.74313 3706919 52.4 33 HITS 0.6932 -272.77 0.73146 3875330 53.6 34 OPS=SLG+OBP 1860.1040 -648.08 0.73116 3879705 53.6 35 TBR(O'Reilly) 2716.1334 -519.88 0.71842 4063615 54.9 36 SLG 2590.4804 -324.94 0.68186 4591124 58.4 37 BA 5704.1056 -778.93 0.58088 6048508 67.0 38 AVERAGE 0.0000 700.37 0.00000 14431303 103.5 Table 5. Standard Evaluation Measures Applied to the Defense
All the discussions following Table 4 apply to Table 5. The standard deviations achieved and the relative rankings of the different measures are essentially the same using either offensive or defensive statistics.
There are two additional counting statistics available for defensive play analysis: put outs and assists. Put outs are redundant in the regressions since each out must be included in only one linear weights term. Assists are new information thus potentially could be expressed as a runs value by the regression. Table 6 shows the results obtained by adding assists to the set of parameters defining the linear regression, ALL.
param weight average contrib 0 OUTS -0.1025 3031.1 -310.55 1 SNGL 0.4828 981.6 473.91 2 DOUB 0.6778 246.1 166.82 3 TRIP 1.1277 35.0 39.48 4 HRUN 1.3949 141.1 196.86 5 BB+HP 0.3498 515.5 180.32 6 IBB 0.1178 48.5 5.71 7 CS -0.3077 55.9 -17.20 8 ROH -0.6643 26.3 -17.44 9 GDP -0.4556 149.2 -67.99 10 SB 0.1057 96.2 10.16 11 ER_BF 0.6350 73.1 46.43 12 ER_RA 0.3019 105.1 31.74 13 RAO -0.0769 196.1 -15.08 14 RSO 0.5775 81.9 47.30 15 K -0.1075 908.3 -97.63 16 ASSISTS 0.0161 1703.0 27.46 Table 6. A Linear Weights Determination for Defensive Statistics
Compared to the entry for ALL in Table 5, the Standard Deviation is unchanged. The total runs value of assists is +27.5 and in a defensive linear weights measure they would be expected to lead to reduced runs allowed. Consequently, assists do not appear to be significant at this level of analysis and demonstrably can not be associated with a runs value in a linear weight formula.
It is possible to combine both the offensive and defense data sets into a single one and perform the same survey calculation. The relative rankings and standard deviations are essentially the same providing another demonstration that all events have the same meaning to both the offense and defense.
Linear weights is best considered a technique for generating an evaluation measure. Different combinations of parameters and differing sets of data will lead to different values for individual parameters in these formulas. When a judicious set of parameters is used, linear weights formulas appear to provide the most accurate evaluation measures available. In addition, a linear formula can be developed using one set of data, team season results for example, and be applied meaningfully to other tasks such as evaluating individual players.
The four term formula, BWOE, is a particularly good combination of accuracy and simplicity. In spite of its simplicity it is more accurate than any of the existing measures for predicting team runs for the data available to this study. When used for player evaluation only three coefficients and three readily available statistics are needed.
Errors must be accounted for when determining the coefficients in any, but especially a linear weights, evaluation measure. When the measure is applied to an individual player the error term is simply not used.
The relative ranking of offensive runs estimation measures from the basic counting statistics is essentially the same as given by Thorn and Palmer (HGB pp 58-59). The only significant difference between HGB and this paper is their linear weights result. In this investigation their formula yields standard deviations of about 24 runs when applied to single or groups of seasons having the same number of games. Offensive player evaluation is in good shape with several strong techniques vying for general acceptance.
When the same counting statistics are available for the defense as for the offense, the hitting measures do an equally good job of estimating runs allowed. All the methods used to rate offense are applicable to defensive evaluations. Assists do not improve the accuracy of linear weights formulas when they are applied to the defense.
"An Evaluation of Major League Baseball Offensive Performance
Models", Jay M. Bennett & John A. Flueck, American
Statistician Vol 37 No, 1 (Feb 1983), pp 76-82
Baseball Prospectus (Brasseys, 1998)
BJBAyy is the Bill James Baseball Abstract for year yy. (Balantine Books)
BJHBA is the Bill James Historical Baseball Abstract. (Villard Books, 1986)
BRJyy is the SABR Baseball Research Journal for year yy.
"Evaluating Offensive Performance in Baseball", Mark D. Pankin, Operations Research 26, #4, Jul-Aug 1978, pp 610-619
HGB is the Hidden Game of Baseball by John Thorn and Pete Palmer. (Doubleday, 1984)
"An Offensive Earned-Run Average for Baseball", Thomas M. Cover & Carroll W. Keilers, Operations Research, Vol 25, No 5, Sep. 1977
TB7 is Total Baseball, Seventh Edition. John Thorn, Pete Palmer and Michael, (Total Sports Publishing, 2001)
June 13, 1998: Original Posting to the Web
August 6, 1998: 1980 season statistics added to the analysis. A completely new treatment of the Palmer Linear Weights system was used. A paper copy of this version of the document is on file with the SABR Archives.
January 3, 2000: Data from 1980 -1998 seasons was used in the analysis. A few additional evaluators were included and several references were added to the bibliography.
June 2, 2000. Data from the 1979 - 1999 seasons has been used in the analysis.
June 6, 2001. Data from the 1978 - 2000 seasons has been used in the analysis. The linear weights formula BASIC was added to the analysis.
July 4, 2002. Data from the 1969 and 1974-2001 seasons has been used in the analysis.
September 29, 2003. Data from 1963, 1967 and 1968 AL and both leagues in 1969, 1972-2002 has been used in the analysis.
August 7, 2006. All RETROSHEET available seasons from 1957 to 2005 are included in the analysis.
January 28, 2008. All RETROSHEET available seasons from 1956 to 2007 are included in the analysis.
April 4, 2008. An error is the counting of CS in OUTS was corrected. Only the OUTS and CS lines of Tables 4 and 6 are changed. No changes are made to the conclusions.
January 1, 2008. All RETROSHEET available seasons from 1954 to 2008 are included in the analysis.
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