4 82 CHAPTER 4 [23]. The causal graph for the influence of a substitute in soccer is shown in Figure 1. Figure 1. The causal model representation of the system being studied. Y = the total distance of a team in five-minute periods; A = a substitute or not in the previous five-minute period; W1 = the consecutive five-minute periods in the second half of the match; W2 = the number of substitutes present; W3 = number of substitutes in the current period; U = possible unknown confounders influencing A, W3 and Y. The dashed lines indicate that this confounding effect is uncertain. The causal graph shows the causal relations between variables in the system. For example, an arrow from A to B describes a causal effect of A on B, or in other words, A causes B. This figure also gives rise to some notation that will be used throughout the paper. The nodes on the top of the graph are the W variables, which indicate the measured confounders (i.e., factors) in the model, A indicates the intervention or treatment that has been performed, Y the outcome of the model, and U any unmeasured confounders that influence our results. With this notation we aim to stay close to the notation used in other literature (e.g.,[2], [27]). Case study We concretize the aforementioned variables as follows, W = ( W1, W2, W3) are the three measured confounders in our model, in which ; W1 is the consecutive five-minute periods in the second half, W2 the number of substitutes present, and W3 whether there was a
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