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SUPPORTING INFORMATION “Bowled Over Or Over-Bowled? Age-Related Changes in the Performance of Bowlers in Test Match Cricket” Jack Thorley

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SUPPORTING INFORMATION “Bowled Over Or Over-Bowled? Age-Related Changes in the Performance of Bowlers in Test Match Cricket” Jack Thorley SUPPORTING INFORMATION “Bowled over or over-bowled? Age-related changes in the performance of bowlers in Test match cricket” Jack Thorley Figure S1. Posterior correlation between economy rate and wicket-taking ability in a) fast and b) slow bowlers, as estimated from multivariate models, including Test cricketers that are still playing, or have only recently stopped playing. Points denote the player-specific deviations in intercept for each performance metric with 95% CI (grey lines). The posterior among-individual correlation is displayed by the solid line. Estimates are taken from models that controlled for other confounding population-level effects. Individual players with particularly low economy rate or particularly high wicket-taking ability have been highlighted; current players or bowlers that have only recently retired are coloured red. 1 Table S1: Best fitting linear mixed effects model for the economy rate of fast bowlers and slow bowlers. Significance (α < 0.05) of fixed effects was assessed by likelihood ratio tests when respective terms were dropped from the main model. We do not present the significance of main effects when present in interactions. Continuous variables were z-score transformed prior to model fitting. 95% confidence intervals for the fixed effects means were calculated by parametric bootstrapping. FAST BOWLERS ECONOMY Fixed Effect Estimate SE 95% CI t-value Χ2, df p- value Intercept 3.094 0.055 2.984 – 3.201 Age -0.105 0.025 -0.155 – -0.057 -4.19 Innings- 2 0.063 0.032 -0.003 – 0.126 1.96 3 0.049 0.027 0.001 – 0.103 1.79 4 0.301 0.036 0.222 – 0.370 8.26 Home or Away- Home -0.120 0.028 -0.177 – -0.071 -4.28 18.24, 1 < 0.001 Terminal Effect 0.126 0.036 0.055 – 0.194 3.50 12.11, 1 < 0.001 Last Game Age -0.080 0.030 -0.135 – 0.021 -2.68 6.69, 1 0.010 Age:Innings2 0.010 0.029 -0.049 – 0.069 0.35 12.46, 3 0.006 Age:Innings3 -0.013 0.027 -0.066 – 0.040 -0.46 Age:Innings4 -0.094 0.033 -0.158 – -0.026 -2.88 Random Effect Variance Std Dev PlayerID (n = 227) 0.104 0.323 MatchID (n = 2581) 0.196 0.443 Opposition:Decade (n = 42) 0.050 0.220 Country:Decade (n = 40) 0.006 0.077 Residual 1.113 1.055 SLOW BOWLERS ECONOMY Fixed Effect Estimate SE 95% CI t-value Χ2, df p- value Intercept 3.064 0.078 2.912 – 3.210 Innings- 2 -0.178 0.050 -0.273 – -0.083 -3.59 31.26, 3 < 0.001 3 -0.067 0.047 -0.162 – 0.021 -1.43 4 0.108 0.057 -0.011 – 0.220 1.88 Home or Away- Home -0.169 0.041 -0.245 – -0.092 -4.12 16.86, 1 < 0.001 Terminal Effect 0.121 0.053 0.010 – 0.216 2.32 5.32, 1 0.021 Last Game Age -0.155 0.039 -0.231 – -0.077 -3.99 15.44, 1 < 0.001 Random Effect Variance Std Dev PlayerID (n =128) 0.142 0.388 MatchID (n = 2044) 0.151 0.376 Opposition:Decade (n = 42) 0.082 0.285 Country:Decade (n = 41) 0.024 0.155 Residual 1.420 1.191 2 Table S2: Best fitting generalised linear mixed effects model for the wicket-taking ability of fast bowlers and slow bowlers. Models were fitted to a negative binomial error distribution with a zero- inflation parameter applied across all data points (~1). 95% confidence intervals for the fixed effects means were calculated via the Wald method. FAST BOWLERS WICKETS Fixed Effect Estimate SE 95% CI z-value p- value Intercept 0.531 0.029 0.475 – 0.587 Age -0.024 0.017 -0.057 – 0.008 -1.46 0.145 Innings- 2 -0.014 0.020 -0.053 – 0.024 -0.73 0.463 3 -0.041 0.021 -0.082 – 0.000 -1.95 0.051 4 -0.150 0.027 -0.203 – -0.097 -5.56 < 0.001 Home or Away- Home 0.069 0.016 0.038 – 0.100 4.41 < 0.001 Overs 0.324 0.009 0.307 – 0.341 36.95 < 0.001 Terminal Effect -0.110 0.026 -0.160 – 0.060 -4.32 < 0.001 Last Game Age 0.040 0.017 0.007 – 0.073 2.36 0.018 Age:Innings2 0.039 0.020 0.001 – 0.078 2.02 0.043 Age:Innings3 0.045 0.021 0.005 – 0.086 2.18 0.029 Age:Innings4 0.082 0.025 0.032 – 0.132 3.23 0.001 Zero-inflation Intercept -4.97 0.843 -5.90 < 0.001 Random Effect Variance Std Dev PlayerID (n =227) 0.018 0.134 Opposition:Decade (n = 42) 0.006 0.078 Country:Decade (n = 40) 0.006 0.075 SLOW BOWLERS WICKETS Fixed Effect Estimate SE 95% CI z-value p- value Intercept 0.284 0.047 0.192 – 0.376 Innings- 2 0.044 0.033 -0.020 – 0.108 1.34 0.180 3 0.135 0.033 0.071 – 0.199 4.14 < 0.001 4 0.075 0.041 -0.004 – 0.154 1.84 0.066 Home or Away- Home 0.083 0.026 0.033 – 0.133 3.23 0.001 Overs 0.414 0.014 0.387 – 0.441 30.20 < 0.001 Terminal Effect -0.163 0.036 -0.234 – -0.092 -4.51 < 0.001 Last Game Age 0.036 0.025 -0.014 – 0.086 1.42 0.157 Zero-inflation Intercept -15.60 1243.6 -0.01 0.99 Random Effect Variance Std Dev PlayerID (n =128) 0.054 0.233 Opposition:Decade (n = 41) 0.022 0.148 Country:Decade (n = 40) 0.006 0.080 3 Table S3: Best fitting generalised linear mixed effects model for the proportion of the overs bowled in the innings. Models were fitted to a binomial error distribution with the number of overs bowler by the bowler as the numerator, and the total number of overs in the innings set as the denominator. 95% confidence intervals for the fixed effects means were calculated by parametric bootstrapping. FAST BOWLERS OVERS Fixed Effect Estimate SE 95% CI z-value p- value Intercept -1.578 0.017 -1.615 – -1.545 Age 0.011 0.005 -0.003 – 0.004 1.54 0.124 Age2 -0.032 0.003 -0.004 – -0.022 -6.48 < 0.001 Innings- 2 -0.020 0.008 -0.037 – -0.007 -2.35 0.019 3 -0.087 0.007 -0.099 – -0.073 -12.61 < 0.001 4 -0.050 0.011 -0.070 – -0.031 -4.64 < 0.001 Home or Away- Home -0.027 0.008 -0.037 – -0.007 -2.85 0.004 Terminal Effect -0.049 0.011 -0.049 – -0.005 2.41 0.016 Last Game Age 0.078 0.014 0.052 – 0.106 5.54 < 0.001 Age: Last Game Age 0.034 0.007 0.019 – 0.048 4.62 < 0.001 Age2: Last Game Age 0.003 0.003 -0.002 – 0.008 1.24 0.215 Random Effect Variance Std Dev MatchID (n = 2568) 0.020 0.144 PlayerID (n =219) 0.023 0.152 Opposition:Decade (n = 42) 0.000 0.018 Country:Decade (n = 40) 0.004 0.066 SLOW BOWLERS OVERS Fixed Effect Estimate SE 95% CI z-value p- value Intercept -1.725 0.046 -1.812 – -1.634 Age -0.035 0.018 -0.072 – 0.003 -1.92 0.055 Age2 -0.038 0.012 -0.006 – -0.015 -3.16 0.002 Innings- 2 0.092 0.016 0.059 – 0.126 5.75 < 0.001 3 0.192 0.012 0.167 – 0.216 15.43 < 0.001 4 0.250 0.020 0.205 – 0.293 12.37 < 0.001 Home or Away- Home -0.015 0.013 -0.040 – 0.012 -1.10 0.273 Terminal Effect -0.016 0.021 -0.056 – -0.022 -0.78 0.438 Last Game Age 0.149 0.037 0.080 – 0.226 4.06 < 0.001 Age:Innings2 0.023 0.011 0.002 – 0.049 2.09 0.037 Age:Innings3 0.043 0.010 0.023 – 0.065 4.27 < 0.001 Age:Innings4 0.021 0.015 0.010 – 0.051 1.47 0.141 Age2:Innings2 -0.013 0.009 -0.030 – 0.004 -1.49 0.136 Age2:Innings3 -0.007 0.008 -0.022 – 0.010 -0.86 0.392 Age2:Innings4 -0.016 0.012 -0.041 – 0.008 -1.36 0.173 Age: Last Game Age 0.085 0.017 0.051 – 0.121 4.93 < 0.001 Age2: Last Game Age -0.007 0.006 -0.020 – 0.007 -1.12 0.264 Random Effect Variance Std Dev MatchID (n = 2044) 0.053 0.230 PlayerID (n =128) 0.145 0.381 Opposition:Decade (n =42) 0.001 0.035 Country:Decade (n =41) 0.018 0.135 4 Table S4: Results from Bayesian multivariate response models investigating the posterior correlation between economy rate and wicket-taking ability in fast bowlers. Economy rate was fitted to a Gaussian distribution, wicket-taking ability to a negative binomial distribution. Estimates for the group-level and the population-level effects provide the standard deviation and the mean, respectively. Estimates for the wicket-taking ability model are provided on the link (log) scale.
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