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Sample essay from Teaching Statistics Using Baseball by Jim Albert

CASE STUDY 4-5: How Important is a Run?

TOPICS COVERED: Nonlinear regression, transformation, residuals.

In this chapter, we have discussed how it is important for a baseball team to score runs and we evaluated the goodness of different batting measures by their relationship with runs scored. But of course the objective of a baseball team is not solely to score runs --- it wins games by scoring more runs than their opponent.

That raises the interesting question. What is the importance of a single run towards the goal of winning a baseball game? If a player is responsible for scoring, say 20 runs, then how many wins for his team has he contributed?

Bill James discovered a special relationship between the number of wins (W) and losses (L) for a baseball team and the number of runs scored (R) and number of runs allowed (RA). He called this relationship “The Pythagorean Method ”. This result says that the ratio between a team’s wins and losses is approximately equal to the square of the ratio of runs scored to runs allowed. That is, approximately,

If we take logs of both sides, we get the equivalent relationship

(We take logs to convert a nonlinear equation in the runs ratio to a linear equation.) Can we demonstrate that the Pythagorean Method gives a good description of the relationship between the win/loss pattern and the runs scored/allowed for current Major League Baseball teams?

To answer this question, we look at the relevant team statistics (wins, losses, runs scored, and runs allowed) for the 30 teams for the 2000 season. Looking at Table 4-21, we see that if a team has a winning record, then it generally scores more runs than its opponent. But there is one interesting exception – Toronto had a win/loss record of 83-79 but actually allowed 47 more runs this season than they scored. (We suppose that Toronto won a lot of close games in 2000.)

Table 4-21: Team statistics for the Major League teams in the 2000 season.

Team	W	L	R	RA	Team	W	L	R	RA
Anaheim	82	80	864	869	Milwaukee	73	89	740	826
Arizona	85	77	792	754	Minnesota	69	93	748	880
Atlanta	95	67	810	714	Montreal	67	95	738	902
Baltimore	74	88	794	913	New York_AL	87	74	871	814
Boston	85	77	792	745	New York_NL	94	68	807	738
Chicago_AL	95	67	978	839	Oakland	91	70	947	813
Chicago_NL	65	97	764	904	Philadelphia	65	97	708	830
Cincinnati	85	77	825	765	Pittsburgh	69	93	793	888
Cleveland	90	72	950	816	San Diego	76	86	752	815
Colorado	82	80	968	897	San Francisco	97	65	925	747
Detroit	79	83	823	827	Seattle	91	71	907	780
Florida	79	82	731	797	St. Louis	95	67	887	771
Houston	72	90	938	944	Tampa Bay	69	92	733	842
Kansas City	77	85	879	930	Texas	71	91	848	974
Los Angeles	86	76	798	729	Toronto	83	79	861	908

To look for the Pythagorean relationship, we compute log(W/L) and log(R/RA) for all teams and construct a scatterplot of the two quantities in Figure 4-8.

Figure 4-8: Scatterplot of log runs ratio against log of ratio of wins to losses for Major League team data from the 2000 season.

We see a linear positive association in this graph, indicating that there is indeed a linear association between log(W/L) and log(R/RA).

Next we want to fit a “best line” to this graph. It seems natural to restrict this line to pass through one point. If a team scores the same number of runs against its opponents (R = RA), then we expect the team to win half of its games (W = L). In other words, the point (log(R/RA), log(W/L)) = (0, 0) should fall on the line. With this restriction, we look at line fits of the form

log(W/L) = k log(R/RA).

We choose k by using a least-squares criterion. It turns out that the sum of squared residuals is minimized when k = 1.91. Figure 4-9 shows this best line on the scatterplot and a display of the corresponding residuals . We do not see any linear trend or any other pattern in the residual plot, so it appears that our fit is satisfactory.

Figure 4-9: Least-squares fit (top) and residual plot (bottom) for (R/RA, W/L) data.

So based on our analysis, we arrive at the relationship

which is pretty close to James’ Pythagorean relationship which uses the power of 2. How useful is this rule in predicting a team’s win numbers? To check the accuracy of this relationship in prediction, Table 4-22 gives the actual number of wins, the predicted number of wins (using the above model) and the residual (actual – predicted). Figure 4-10 displays a stemplot of the absolute residuals .

Table 4-22: Number of wins, predicted number of wins, and residuals using James’ Pythagorean relationship.

Team	W	predicted	residual	Team	W	predicted	residual
Anaheim	82	80.6	1.4	Milwaukee	73	72.5	0.5
Arizona	85	84.8	0.2	Minnesota	69	68.5	0.5
Atlanta	95	90.7	4.3	Montreal	67	65.6	1.4
Baltimore	74	70.2	3.8	New York_AL	87	85.7	1.3
Boston	85	85.7	-0.7	New York_NL	94	87.9	6.1
Chicago_AL	95	92.8	2.2	Oakland	91	92.2	-1.2
Chicago_NL	65	68.1	-3.1	Philadelphia	65	68.8	-3.8
Cincinnati	85	86.8	-1.8	Pittsburgh	69	72.3	-3.3
Cleveland	90	92.7	-2.7	San Diego	76	74.8	1.2
Colorado	82	86.9	-4.9	San Francisco	97	97.3	-0.3
Detroit	79	80.6	-1.6	Seattle	91	92.6	-1.6
Florida	79	73.9	5.1	St. Louis	95	91.8	3.2
Houston	72	80.5	-8.5	Tampa Bay	69	69.9	-0.9
Kansas City	77	76.6	0.4	Texas	71	70.3	0.7
Los Angeles	86	88	-2	Toronto	83	76.9	6.1

ABSOLUTE RESIDUALS

                 0 | 23455779
                 1 | 22344668
                 2 | 027
                 3 | 12388
                 4 | 39
                 5 | 1
                 6 | 11
                 7 | 
                 8 | 5

Figure 4-10: Stemplot of the abolute residuals from the fit using James’ Pythagorean relationship.

We see from the stemplot that 24 of the 30 residuals sizes are smaller than 4. This indicates that for 80% of the teams, we can predict the number of wins to within 4 games using this formula.