4 94 CHAPTER 4 allowing the analysis and interpretation to be statistical. Section 4 shows our results and the interpretation thereof. 4. RESULTS Applying the simulation data on the defined causal model, both TMLE and TMLEH have less deviation of the true ATE of the influence of a substitute in the previous period on the total distance of the entire soccer team than GLM (Table 1 and Table 2). When the miss-specification of the causal model is applied (e.g., leaving out the substitution in the current period) , the increase of deviation of the true ATE is almost non-existent for TMLE and TMLEH, where GLM shows an increased deviation of the true ATE. Figure 5 illustrates the effect of the miss-specification, leaving out the substitute of the current period, on the resulting ATE of a substitute in a previous period. Applying the observed dataset, the influence of a substitution on the total distance of the soccer team differs per algorithm ATE: 0.0105 - 0.0149 (Table 3). Miss-specification of the causal model, leaving out the substitute of the current period, using the real dataset leads to less deviance in TMLE and TMLEH from the respective calculated ATE of the substitute in the previous period on the total distance of the soccer team than GLM (Table 3). Table 1. Simulation of the correct causal model. True ATE: 0.0646 Measure GLM TMLE TMLEH ATE 0.1442 0.0647 0.0647 Confidence Interval 95% 0.1399-0.1485 0.0628-0.0665 0.0605-0.0688 Bias 0.0797 0.0001 0.0001 Bias % 123.50 0.22 0.17 GLM = Generalized Linear Model; TMLE = Targeted Maximum Likelihood Estimation; TMLEH = Targeted Maximum Likelihood Estimation using Handpicked algorithms; ATE = Average Treatment Effect (i.e., Effect Size) of a substitute in a previous period on the total distance of a soccer team. Table 2. Simulation of miss-specified causal model. True ATE: 0.0646 Measure GLM TMLE TMLEH ATE 0.1491 0.0647 0.0646 Confidence Interval 95% 0.1399-0.1485 0.0628-0.0665 0.0613-0.0679 Bias 0.0846 0.0001 0.0000 Bias % 131.00 0.22 0.00 GLM = Generalized Linear Model; TMLE = Targeted Maximum Likelihood Estimation; TMLEH = Targeted Maximum Likelihood Estimation using Handpicked algorithms; ATE = Average Treatment Effect (i.e., Effect Size) of a substitute in a previous period on the total distance of a soccer team.
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