4 79 TRANSFERRING TARGETED MAXIMUM LIKELIHOOD ESTIMATION INTO SPORT SCIENCE soccer, and from the field of targeted maximum likelihood estimation. In Section 3 we present the methods used in this paper. This section defines the causal roadmap and its application to the current problem. Section 4 presents the results of our study. We present both the results of our simulation study as well as our application of TMLE to substitutions in soccer. Finally, in Section 5 and Section 6 we discuss and conclude the work. 2. RELATED WORK The related work on TMLE and causal modelling and the standard statistical methods to study substitution are the basis for our research on the applicability of causal inference in sport science. 2.1. Statistics and performance of substitutes in soccer Research of performance, substitutes, and soccer, has previously only been done using traditional statistical methods [3], [4], [14]–[16]. For example, Bradley, Lago-Penas, and Rey [4]studied the match performances of substitute players using one-way independent measures Analysis of Variance (ANOVA). The performance of the substitutes was compared with the players completing the entire match. The meaningfulness of the differences between the substitutes and full match players was indicated by the Effect Size (ES). Effect size is, as defined by Kelley and Preacher, 2012 [17], “We define effect size as a quantitative reflection of the magnitude of some phenomenon that is used for the purpose of addressing a question of interest”. The authors show, amongst others, that substitutes cover a greater total distance (ES: 0.33–0.67). Modric et al. [14] investigated the relation between Running Performance (RP) and Game Performance Indicators (GPI). The RP included the total distance covered, distance covered in five speed categories, and the GPI were determined by the position specific InStat index (InStat, Moscow, Russia). The InStat index is calculated based on a unique set of parameters for each playing position, with a higher numerical value indicating better performance. The exact calculations are only known by the manufacturer of the platform. The associations between RP and GPI were identified by calculating Pearson’s product moment correlation coefficient. Correlations were found between RP and GPI for different positions. For instance, the total running distance and high-intensity accelerations were correlated with the InStat index for Central Defenders (r = 0.42 and r = 0.49, respectively). Hills et al. [3] profiled the match-day physical activities performed by substitutes, focusing separately on the pre- and post-pitch-entry periods. Linear mixed modelling
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