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.

翻译：随着棒球比赛的进行，击球员在多次面对同一投手时表现似乎会提升。投手表现随顺序轮次下降的现象，即顺序轮次罚球（TTOP），通常归因于击球员在比赛中的学习效应。尽管TTOP在棒球界已广泛接受并影响许多教练的临场决策，但我们认为现有估计TTOP规模的方法无法区分投手在整个比赛过程中的持续状态演变与连续顺序轮次间的非连续性差异。通过贝叶斯多项逻辑回归模型，我们发现，在调整了击球员与投手质量、左右手习惯及主场优势等混杂因素后，投手表现随顺序轮次出现显著非连续性的证据不足。我们的分析表明，不应将第三轮次开始视为决定是否替换先发投手的特殊临界点。