# Modeling Pitcher Performance and the Distribution of Runs per Inning in Major League Baseball

Rosner, Bernard, Mosteller, Frederick, Youtz, Cleo, The American Statistician

This article develops a model for the distribution of the number of batters faced and the number of runs scored against starting pitchers in the early innings of American League baseball games based on data from 1990. The simplest models, such as the negative binomial distribution for the number of batters faced with fixed probabilities of getting a batter out, had to be adjusted. By attending to the most-used starting pitchers and restricting the analysis to the first three innings, we can defer dealing with the effects of tiring and of the introduction of relief pitchers for later research.

The rationale for considering such models is to introduce a stochastic element in the assessment of pitcher performance. Most traditional baseball statistics [such as the earned run average (ERA)] do not allow for random variation in assessing performance. Thus one can rank pitchers according to ERA, but one cannot translate such rankings into assessing the probability that pitcher A will allow fewer runs than pitcher B on any given day. Such stochastic paired comparisons offer a richer set of consequences in their descriptions than the usual point estimate that assesses pitcher performance.

1. MODEL

For an inning under consideration let

x = number of runs scored

h(x) = probability of exactly x runs being scored

b = number of men left on base, 0 [less than or equal to] b [less than or equal to] 3

N = number of batters the pitcher faced

f(N) = probability of facing exactly N batters in the inning

g(x[where]N) = conditional probability of x runs given N batters.

We ignore unfinished innings in games interrupted by such matters as darkness, weather, acts of God, or local laws except as discussed below. The probability of scoring exactly x runs in an inning is given by

h(x) = [summation of] f(N)g(x[where]N) where N=x+3 to x+6. (1)

The reason for the constraint on N in the summation is that by the end of an inning, a batter must either score, be left on base, or be put out. (A player who comes to the plate but does not meet one of these three conditions is not counted as a batter. Replacement batters or runners are regarded as indistinguishable from the original batter or runner.) The number of batters is therefore N = x + b + 3, and so for a given value of x the smallest and largest numbers of batters are x + 3 and x + 6, respectively.

For parsimony and interpretability we want to specify a parametric form for f and g, and thus reduce the number of parameters needed to fit a model. For this purpose let

p = probability of an out

(where, without loss of generality, we assume that the last batter faced is an out). Because the length of an inning is determined mainly (although not entirely) by "at bats" that are sorted into "outs" and "not outs," and because three outs end an inning, the negative binomial distribution is a natural distribution to explore because its main feature is trials, here at bats, until a predetermined number of successes occurs. Therefore, we consider a negative binomial model for f, namely

[Mathematical Expression Omitted] (2)

Upon examining actual data we found that model (2) underestimated the probability of facing exactly three batters and overestimated the probability of facing exactly four batters, with other outcomes being closely predicted. These deviations from the negative binomial may be primarily due to the occurrence of double plays, where two outs occur with a single at-bat, or where a runner is caught stealing second after successfully reaching first base. Therefore, we modified the model in (2) as follows:

[Mathematical Expression Omitted] (3)

where [Lambda] (usually positive for a given pitcher) is a rough indicator of the propensity of a pitcher and his team to either achieve double plays and/or to prevent a runner on first from obtaining a big lead. Appendix A shows that the MLE's of [f. …

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