Using PITCHf/x to model the dependence of strikeout rate on the predictability of pitch sequences

3.1Definitions

We use the measurements described in section 2 to define a vector of descriptors for a pitcher that quantifies the relationship between consecutive pitches in location, movement, and velocity. For a given pitcher in a given season, say Clayton Kershaw in 2014, we consider separately the pitches thrown to left-handed and right-handed batters after intentional balls are removed. Let (xi,xi′), i=1,2,…,N represent all pairs of consecutive pitches that Kershaw threw to a left-handed batter within a single plate appearance in 2014 where x_i is the px coordinate of the first pitch in the pair and xi′ is the px coordinate of the second pitch in the pair. We note that a pitch can appear as the second of a pair and then as the first of the next pair, so that a four-pitch plate appearance will have three pairs. N is the total number of these pairs. Kershaw’s px correlation coefficient r_x for consecutive pitches against left-handed batters in 2014 is defined by

where x¯i and x¯i′ represent the means of the x_i and xi′ values respectively over the N pairs. The correlation coefficient r_x, therefore, provides a statistical measure of the relationship between consecutive px values over the N pairs of pitches.

We can also let (xi,xi′) represent the pz coordinates for pairs of pitches to compute Kershaw’s pz correlation coefficient r_z for consecutive pitches against left-handed batters in 2014 using (1). Similarly, we can use pfx_x, pfx_z, and start-speed to compute correlation coefficients for each of these variables which we denote respectively by r_mx, r_mz, and r_s . Thus, for a given pitcher, season, and batter handedness such as Kershaw in 2014 against left-handed batters, we can compute a vector of five correlation coefficients (r_x, r_z, r_mx, r_mz, r_s) which represents the pitcher’s degree of pitch-to-pitch consistency in location, movement, and velocity.

A correlation coefficient r has several important properties. The value of r is always between -1 and +1 with the sign of r being the same as the sign of the slope of the regression line for the set of N points (xi,xi′). The absolute value |r| measures the strength of the linear relationship between x_i and xi′. If |r|=1, then the set of points (xi,xi′) lie exactly on a line and the xi′ of the second pitch in each pair can be exactly predicted using the x_i of the preceding pitch. As |r| decreases toward zero, the ability to predict xi′ from x_i using a linear model decreases. More precisely, the square of the correlation coefficient r² is the fraction of the variance in the second pitch xi′ that is accounted for by a linear model and the x_i value for the first pitch. We might expect that a pitcher with smaller values of |r| for a given pitch attribute, everything else being equal, would be more effective due to the increased uncertainty that results from using the current pitch to predict the value of that attribute for the next pitch.

3.2Data

In section 4 we will examine the dependence of a pitcher’s strikeout rate on the correlation coefficients defined in section 3.1. As with the correlation coefficients, we compute each pitcher’s strikeout rate separately for each applicable platoon configuration for each year. Before the rate is computed, however, we remove all plate appearances that resulted in a bunt or an intentional walk and we also remove all plate appearances with a pitcher as a batter. The number of remaining plate appearances is referred to as adjusted plate appearances. A pitcher’s strikeout rate P_K is then defined as the ratio of strikeouts to adjusted plate appearances. Using considerations (Healey, 2015) that were derived from reliability studies (Carleton, 2013), we consider a pitcher’s strikeout rate to be reliable for a season and platoon configuration if the pitcher had at least 150 adjusted plate appearances for that season and platoon configuration. For this study, we also removed all pitchers that were used strictly as relievers during a season as well as all pitchers who had at least twenty percent of their pitches classified as knuckleballs. Table 1 summarizes the total number of pitcher seasons that satisfy these criteria for each of the four platoon configurations over the years from 2008 to 2014 for which PITCHf/x data was widely available.

The (r_x, r_z, r_mx, r_mz, r_s) correlation coefficients were computed for all of the cases represented in Table 1 using the Brooks Baseball adjustments to the PITCHf/x measurements. Tables 2, 3, 4, and 5 present the mean, standard deviation, and minimum and maximum values for each coefficient over the pitcher seasons for each platoon configuration. The Tables also provide the pitcher and year for each minimum and maximum value. We see that the coefficients that are based on movement and speed (r_mx, r_mz, r_s) have larger ranges and standard deviations across pitchers than the coefficients that are based on location (r_x, r_z).

3.3Year-to-year analysis

An important question is the degree to which the statistics defined in section 3.1 represent distinctive and repeatable characteristics of a pitcher. One way to answer this question is to compute year-to-year correlations which measure the consistency of a statistic for pitchers from year-to-year. Specifically, for each platoon configuration we identified the instances of pitchers who satisfied the criteria described in section 3.2 for consecutive seasons. These instances were used to form pairs of consecutive pitcher seasons for each platoon configuration where the second year of a pair was allowed to be the first year of another pair. The total number of pairs for each platoon configuration is given in Table 6. For each of the statistics defined in section 3.1 we computed the year-to-year correlation coefficient for each platoon configuration using these pairs. The results are shown in Table 7. If each pitcher has the same value of a statistic for every pair of consecutive years, then the correlation coefficient will be one. The observed value of the correlation coefficients is less than one due to variation in the statistic measurements that originate from the use of limited sample sizes and from actual changes in pitcher tendencies over time. We see that the year-to-year correlations for statistics derived from movement and speed measurements (r_mx, r_mz, r_s) are larger than the year-to-year correlations for statistics derived from location measurements (r_x, r_z).

4.1Variables

4.2Model estimation

We use a separate linear regression model for each platoon configuration to approximate pitcher strikeout rate P_K using the variables defined in section 4.1. The number of observations for each configuration is given by Table 1. We considered approximations of the form

for right-handed pitchers and

for left-handed pitchers where F₁ and F₂ are a linear combination of their component variables and first-order cross terms. Tables 8–11 present the models of this form for each platoon configuration that have the most significant variables where significance is defined by a p-value below 0.01. Each table includes the coefficient, standard error, t-statistic, and p-value for each significant variable along with the R² for the fit. We observe that off-speed pitches typically have a larger impact on same-sided (RHP vs. RHB, LHP vs. LHB) matchups. Therefore, the limited treatment of off-speed pitches by the model may explain the larger R² values for opposite-sided (RHP vs. LHB, LHP vs. RHB) matchups. Due to considerations presented in section 3.1, we also examined the use of variables defined by the absolute value of the pitch-to-pitch correlation coefficients. Using the original signed values of the coefficients, however, led to models with less error which suggests that a negative correlation benefits a pitcher’s strikeout rate more than a zero correlation.

Table 12 presents the mean and maximum absolute error |PK-PˆK| over the observations for each platoon configuration. We see that the average absolute error is a few percent for each configuration. The table also provides the pitcher/year associated with the maximum absolute error. For each maximum error case, the approximation PˆK underestimates P_K and the large error is a result of the pitcher having a highly effective off-speed pitch which is not accounted for by the current model. In particular, Yu Darvish benefited from an exceptional slider, Felix Hernandez from an exceptional changeup, and Clayton Kershaw from an exceptional slider and curveball.

We see from Tables 8–11 that the variable velo is significant for each platoon configuration with a positive coefficient which indicates that a one mile per hour increase in velocity corresponds to an increase in PˆK of between 0.010 and 0.014 depending on the configuration. The fastball fraction f is also significant for each platoon configuration with a negative coefficient which indicates that a larger fraction of fastballs, everything else being equal, leads to a lower strikeout rate.

For the three configurations with the most observations (Tables 8–11) the fastball vertical movement variable max_pfx_z is significant along with one of the pitch-to-pitch correlation variables. Specifically, r_mz is significant for the two configurations involving right-handed pitchers and r_mx is significant for the LHP versus RHB configuration. As expected, increased vertical movement and lower pitch-to-pitch correlation are associated with higher strikeout rates. The horizontal movement variable min_pfx_x and the cross term max_pfx_z*min_pfx_x are also significant for the RHP versus RHB configuration.

Figures 1 through 4 illustrate the distribution of the explanatory variables and the estimated F₁ and F₂ surfaces for the RHP versus LHB platoon configuration considered in Table 9. Figure 1 plots the joint distribution of fastball velocity and vertical movement for the 814 observations for this configuration. The variables are nearly uncorrelated with a correlation coefficient of 0.05. Figure 2 plots F₁(1,velo,max_pfx_z) and shows that strikeout rate increases as fastball velocity and vertical movement increase. Figure 2 plots the joint distribution of fastball fraction and pitch-to-pitch correlation in vertical movement. These variables are also nearly uncorrelated with a correlation coefficient of 0.09. Figure 3 plots F₂ (f, r_mz) and shows that strikeout rate decreases as fastball fraction and pitch-to-pitch correlation in vertical movement increase. The shape of the F₁ and F₂ surfaces will be similar for the other platoon configurations for which these variables are significant.

Fig.1

Joint distribution for maxpfxz and velo for RHP versus LHB.

Fig.2

F₁ surface for RHP versus LHB.

Fig.3

 Joint  distribution for  f and  r_mz for  RHP  versus LHB.

Fig.4

F₂ surface for RHP versus LHB.

4.3Impact of pitch-to-pitch correlation

In this section, we examine the role of the pitch-to-pitch correlation variables on strikeout rate in more detail. Table 13 considers the three platoon configurations for which a correlation variable is significant. The value c is the coefficient for the correlation variable and platoon configuration from one of Tables 8–10. The values σ and maxdiff represent the standard deviation and maximum difference over pitchers for the correlation variable and platoon configuration from one of Tables 2–4. The constant 38 is the average number of batters faced per nine innings in major league baseball in 2014. Thus, c * σ * 38 and c * maxdiff * 38 are the changes in the number of strikeouts per nine innings associated with changes of σ and maxdiff in the correlation variable if the other variables are held constant. We see that c * maxdiff * 38 is between one and two strikeouts per nine innings depending on the platoonconfiguration.

4.4The Shields/Colon example

As an example of the importance of pitch mix and sequencing we present the case of right-handed pitchers James Shields (2011) and Bartolo Colon (2012) against left-handed batters. The pitchers had similar values for the fastball parameters velo and max_pfx_z which led to identical values for F₁ for this configuration as shown in Table 14. Colon’s pitches, however, were much more predictable with a fastball fraction f = 0.902 and a vertical movement correlation r_mz = 0.237 compared to f = 0.325 and r_mz = 0.177 for Shields. As a result, F₂ and the overall PˆK approximation is 0.074 higher for Shields. The approximation PˆK underestimates the actual strikeout rate for each pitcher by a few percent and Shields actually posted a strikeout rate P_K that is 0.086 higher than Colon for this configuration and pair of years. In summary, the two pitchers have nearly identical fastball parameters and the same value for F₁ but differences in pitch mix and sequencing led to a significant advantage in strikeout rate for Shields which is predicted by the model.