A Bayesian analysis of the time through the order penalty in baseball

As a baseball game progresses, batters appear to perform better the more times they face a particular pitcher. The apparent drop-off in pitcher performance from one time through the order to the next, known as the Time Through the Order Penalty (TTOP), is often attributed to within-game batter learning. Although the TTOP has largely been accepted within baseball and influences many managers' in-game decision making, we argue that existing approaches of estimating the size of the TTOP cannot disentangle continuous evolution in pitcher performance over the course of the game from discontinuities between successive times through the order. Using a Bayesian multinomial regression model, we find that, after adjusting for confounders like batter and pitcher quality, handedness, and home field advantage, there is little evidence of strong discontinuity in pitcher performance between times through the order. Our analysis suggests that the start of the third time through the order should not be viewed as a special cutoff point in deciding whether to pull a starting pitcher.