Supplementary materials and data for the article "Humans can learn bimodal priors in complex sensorimotor behaviour"

1 Supplementary Material for the paper: “Humans can learn bimodal priors in complex sensorimotor behaviour” Stephan Zahno, Damian Beck, Ernst-Joachim Hossner, Konrad Kording Extended data Table 1. Multilevel regression model of error estimation on day 1 in the fast condition. Fixed effects B SE B 95% CI t(2673) p two-sided Intercept –18.32 4.01 [–26.19, –10.46] –4.57 < .001 Ball position 0.16 0.06 [0.03, 0.28] 2.45 .014 Segment (0 = left, 1 = right) 3.24 1.66 [–0.01, 6.49] 1.95 .153 Random effects Intercept variance (τ 00 ) 280.98 – – – – Slope variance (τ 11 ) 0.06 – – – – Intercept-slope covariance (ρ 01 ) –0.99 – – – – Level-1 residual (σ2) 98.98 – – – – ICC 0.24 – – – – Note. B = unstandardized regression coefficients, SE = standard error, CI = confidence intervals, t(degrees of freedom). ICC = Interclass correlation coefficient. Model statistics: Nparticipants = 24, R2marginal = .148. Model comparison in Table 2. Table 2. Model comparison for multilevel regression model of error estimation on day 1 in the fast condition. Model df AIC BIC logLik Comparison χ2 p 1 Intercept 2 21171.00 21182.80 –10583.50 2 Base model 3 20755.30 20773.01 –10374.65 1 vs 2 417.697 < .001 3 Base model + RI 4 20631.67 20655.28 –10311.84 2 vs 3 125.632 < .001 4 Base model + RS 6 20211.17 20246.57 –10099.58 3 vs 4 424.502 < .001 5 Base model + RS + Bimodal segment factor 7 20209.35 20250.65 –10097.67 4 vs 5 3.820 .051 Note. df = degrees of freedom, AIC = Akaike Information Criterion, BIC = Bayesian Information Criterion, logLik = loglikelihood, RI = random intercepts, RS = random intercept and slopes. The base model includes the predictor ball position. 2 Table 3. Multilevel regression model of error estimation on day 1 in the moderate condition. Fixed effects B SE B 95% CI t(3096) p two-sided Intercept –13.02 3.72 [–20.30, –5.73] –3.50 < .001 Ball position 0.17 0.06 [0.06, 0.28] 3.04 .002 Segment (0 = left, 1 = right) 0.27 1.46 [–2.60, 3.15] 0.19 .852 Random effects Intercept variance (τ 00 ) 251.55 – – – – Slope variance (τ 11 ) 0.04 – – – – Intercept-slope covariance (ρ 01 ) –0.96 – – – – Level-1 residual (σ2) 90.90 – – – – ICC 0.29 – – – – Note. B = unstandardized regression coefficients, SE = standard error, CI = confidence intervals, t(degrees of freedom). ICC = Interclass correlation coefficient. Model statistics: Nparticipants = 24, R2marginal = .090. Model comparison in Table 4. Table 4. Model comparison for multilevel regression model of error estimation on day 1 in the moderate condition. Model df AIC BIC logLik Comparison χ2 p 1 Intercept 2 24246.25 24258.34 –12121.13 2 Base model 3 23938.17 23956.31 –11966.09 1 vs 2 310.078 < .001 3 Base model + RI 4 23570.14 22594.32 –11781.07 2 vs 3 370.034 < .001 4 Base model + RS 6 23107.90 23144.18 –11547.95 3 vs 4 466.239 < .001 5 Base model + RS + Bimodal segment factor 7 23109.87 23152.19 –11547.93 4 vs 5 0.035 .852 Note. df = degrees of freedom, AIC = Akaike Information Criterion, BIC = Bayesian Information Criterion, logLik = loglikelihood, RI = random intercepts, RS = random intercept and slopes. The base model includes the predictor ball position. 3 Table 5. Multilevel regression model of error estimation on day 1 in the slow condition. Fixed effects B SE B 95% CI t(3049) p two-sided Intercept 1.53 3.25 [–4.84, 7.90] 0.47 .638 Ball position 0.08 0.05 [–0.02, 0.18] 1.62 .105 Segment (0 = left, 1 = right) –2.24 1.39 [–4.97, 0.49] –1.61 .216 Random effects Intercept variance (τ 00 ) 181.19 – – – – Slope variance (τ 11 ) 0.03 – – – – Intercept-slope covariance (ρ 01 ) –0.94 – – – – Level-1 residual (σ2) 79.84 – – – – ICC 0.29 – – – – Note. B = unstandardized regression coefficients, SE = standard error, CI = confidence intervals, t(degrees of freedom). ICC = Interclass correlation coefficient. Model statistics: Nparticipants = 24, R2marginal = .003. Model comparison in Table 6. Table 6. Model comparison for multilevel regression model of error estimation on day 1 in the slow condition. Model df AIC BIC logLik Comparison χ2 p 1 Intercept 2 23242.37 23254.44 –11619.19 2 Base model 3 23229.23 23247.33 –11611.62 1 vs 2 15.141 < .001 3 Base model + RI 4 22743.84 22767.97 –11367.92 2 vs 3 487.399 < .001 4 Base model + RS 6 22366.15 22402.34 –11177.08 3 vs 4 381.693 < .001 5 Base model + RS + Bimodal segment factor 7 22365.56 22407.78 –11175.78 4 vs 5 2.595 .108 Note. df = degrees of freedom, AIC = Akaike Information Criterion, BIC = Bayesian Information Criterion, logLik = loglikelihood, RI = random intercepts, RS = random intercept and slopes. The base model includes the predictor ball position. 4 Table 7. Multilevel regression model of error estimation on day 2+3 in the fast condition. Fixed effects B SE B 95% CI t(5571) p one-sided Intercept –27.10 3.09 [–33.16, –21.04] –8.76 < .001 Ball position 0.36 0.05 [0.27, 0.45] 7.79 < .001 Segment (0 = left, 1 = right) –2.36 1.04 [–4.39, –0.33] –2.28 .023 Random effects Intercept variance (τ 00 ) 190.39 – – – – Slope variance (τ 11 ) 0.04 – – – – Intercept-slope covariance (ρ 01 ) –0.98 – – – – Level-1 residual (σ2) 78.39 – – – – ICC 0.23 – – – – Note. B = unstandardized regression coefficients, SE = standard error, CI = confidence intervals, t(degrees of freedom). ICC = Interclass correlation coefficient. Model statistics: Nparticipants = 24, R2marginal = .273. Model comparison in Table 8. Table 8. Model comparison for multilevel regression model of error estimation on day 2+3 in the fast condition. Model df AIC BIC logLik Comparison χ2 p 1 Intercept 2 43268.33 43281.59 –21632.17 2 Base model 3 41641.60 41661.49 –20817.80 1 vs 2 1628.732 < .001 3 Base model + RI 4 41273.01 41273.53 –20619.51 2 vs 3 396.587 < .001 4 Base model + RS 6 40521.26 40521.04 –20234.63 3 vs 4 769.756 < .001 5 Base model + RS + Bimodal segment factor 7 40524.07 40524.48 –20232.04 4 vs 5 5.184 .023 Note. df = degrees of freedom, AIC = Akaike Information Criterion, BIC = Bayesian Information Criterion, logLik = loglikelihood, RI = random intercepts, RS = random intercept and slopes. The base model includes the predictor ball position. 5 Table 9. Multilevel regression model of error estimation on day 2+3 in the moderate condition. Fixed effects B SE B 95% CI t(6584) p one-sided Intercept –18.20 2.90 [–24.08, –12.70] –6.28 < .001 Ball position 0.29 0.04 [0.21, 0.37] 7.26 < .001 Segment (0 = left, 1 = right) –2.64 0.91 [–4.36, –0.94] –2.90 .006 Random effects Intercept variance (τ 00 ) 171.07 – – – – Slope variance (τ 11 ) 0.03 – – – – Intercept-slope covariance (ρ 01 ) –0.97 – – – – Level-1 residual (σ2) 71.51 – – – – ICC 0.24 – – – – Note. B = unstandardized regression coefficients, SE = standard error, CI = confidence intervals, t(degrees of freedom). ICC = Interclass correlation coefficient. Model statistics: Nparticipants = 24, R2marginal = .182. Model comparison in Table 10. Table 10. Model comparison for multilevel regression model of error estimation on day 2+3 in the moderate condition. Model df AIC BIC logLik Comparison χ2 p 1 Intercept 2 50105.85 50119.45 –25050.93 2 Base model 3 48784.41 48804.80 –24389.21 1 vs 2 1323.442 < .001 3 Base model + RI 4 48022.77 48049.96 –24007.39 2 vs 3 763.638 < .001 4 Base model + RS 6 47179.99 47220.77 –23584.00 3 vs 4 846.782 < .001 5 Base model + RS + Bimodal segment factor 7 47173.56 47221.13 –23579.78 4 vs 5 8.435 .004 Note. df = degrees of freedom, AIC = Akaike Information Criterion, BIC = Bayesian Information Criterion, logLik = loglikelihood, RI = random intercepts, RS = random intercept and slopes. The base model includes the predictor ball position. 6 Table 11. Multilevel regression model of error estimation on day 2+3 in the slow condition. Fixed effects B SE B 95% CI t(6462) p one-sided Intercept –0.95 2.34 [–5.53, 3.64] –0.40 .686 Ball position 0.10 0.03 [0.03, 0.16] 2.79 .005 Segment (0 = left, 1 = right) –0.81 0.80 [–2.38, 0.76] –1.02 .154 Random effects Intercept variance (τ 00 ) 107.93 – – – – Slope variance (τ 11 ) 0.02 – – – – Intercept-slope covariance (ρ 01 ) –0.96 – – – – Level-1 residual (σ2) 54.89 – – – – ICC 0.24 – – – – Note. B = unstandardized regression coefficients, SE = standard error, CI = confidence intervals, t(degrees of freedom). ICC = Interclass correlation coefficient. Model statistics: Nparticipants = 24, R2marginal = .033. Model comparison in Table 12. Table 12. Model comparison for multilevel regression model of error estimation on day 2+3 in the slow condition. Model df AIC BIC logLik Comparison χ2 p 1 Intercept 2 46272.21 46285.77 –23134.11 2 Base model 3 46045.41 46065.74 –23019.70 1 vs 2 228.806 < .001 3 Base model + RI 4 45356.42 45383.53 –22674.21 2 vs 3 690.987 < .001 4 Base model + RS 6 44591.78 44632.44 –22289.89 3 vs 4 768.645 < .001 5 Base model + RS + Bimodal segment factor 7 44592.74 44640.19 –22289.37 4 vs 5 1.034 .309 Note. df = degrees of freedom, AIC = Akaike Information Criterion, BIC = Bayesian Information Criterion, logLik = loglikelihood, RI = random intercepts, RS = random intercept and slopes. The base model includes the predictor ball position. 7 Table 13. Multilevel regression model of error estimation for the control experiment in the fast condition. Fixed effects B SE B 95% CI t(2738) p two-sided Intercept –31.69 2.72 [–37.02, –26.35] –11.65 < .001 Ball position 0.38 0.04 [0.31, 0.45] 10.80 < .001 Random effects Intercept variance (τ 00 ) 156.34 – – – – Slope variance (τ 11 ) 0.03 – – – – Intercept-slope covariance (ρ 01 ) –0.91 – – – – Level-1 residual (σ2) 129.81 – – – – ICC 0.21 – – – – Note. B = unstandardized regression coefficients, SE = standard error, CI = confidence intervals, t(degrees of freedom). ICC = Interclass correlation coefficient. Model statistics: Nparticipants = 24, R2marginal = .207. Model comparison in Table 14. Table 14. Model comparison for multilevel regression model of error estimation for the control experiment in the fast condition. Model df AIC BIC logLik Comparison χ2 p 1 Intercept 2 22636.72 22648.57 –11316.36 2 Base model 4 21959.43 21977.20 –10976.71 1 vs 2 679.295 < .001 3 Base model + RI 5 21539.11 21562.80 –10765.55 2 vs 3 422.321 < .001 4 Base model + RS 7 21423.08 21458.62 –10705.54 3 vs 4 120.028 < .001 Note. df = degrees of freedom, AIC = Akaike Information Criterion, BIC = Bayesian Information Criterion, logLik = loglikelihood, RI = random intercepts, RS = random intercept and slopes. The base model includes the predictor ball position. 8 Table 15. Multilevel regression model of error estimation for the control experiment in the moderate condition. Fixed effects B SE B 95% CI t(3018) p two-sided Intercept –13.13 2.40 [–17.83, –8.42] –5.46 < .001 Ball position 0.15 0.03 [0.09, 0.21] 4.93 < .001 Random effects Intercept variance (τ 00 ) 124.11 – – – – Slope variance (τ 11 ) 0.02 – – – – Intercept-slope covariance (ρ 01 ) –0.88 – – – – Level-1 residual (σ2) 100.02 – – – – ICC 0.25 – – – – Note. B = unstandardized regression coefficients, SE = standard error, CI = confidence intervals, t(degrees of freedom). ICC = Interclass correlation coefficient. Model statistics: Nparticipants = 24, R2marginal = .050. Model comparison in Table 16. Table 16. Model comparison for multilevel regression model of error estimation for the control experiment in the moderate condition. Model df AIC BIC logLik Comparison χ2 p 1 Intercept 2 23660.52 23672.56 –11828.26 2 Base model 4 23465.21 23483.28 –11729.61 1 vs 2 197.304 < .001 3 Base model + RI 5 22921.17 22945.26 –11456.59 2 vs 3 546.042 < .001 4 Base model + RS 7 22798.38 22834.51 –11393.19 3 vs 4 126.790 < .001 Note. df = degrees of freedom, AIC = Akaike Information Criterion, BIC = Bayesian Information Criterion, logLik = loglikelihood, RI = random intercepts, RS = random intercept and slopes. The base model includes the predictor ball position. 9 Table 17. Multilevel regression model of error estimation for the control experiment in the slow condition. Fixed effects B SE B 95% CI t(2894) p two-sided Intercept 7.22 2.18 [2.96, 11.49] 3.32 .001 Ball position –0.05 0.02 [–0.09, 0.00] –1.96 .050 Random effects Intercept variance (τ 00 ) 101.92 – – – – Slope variance (τ 11 ) 0.01 – – – – Intercept-slope covariance (ρ 01 ) –0.88 – – – – Level-1 residual (σ2) 81.31 – – – – ICC 0.26 – – – – Note. B = unstandardized regression coefficients, SE = standard error, CI = confidence intervals, t(degrees of freedom). ICC = Interclass correlation coefficient. Model statistics: Nparticipants = 24, R2marginal = .006. Model comparison in Table 18. Table 18. Model comparison for multilevel regression model of error estimation for the control experiment in the slow condition. Model df AIC BIC logLik Comparison χ2 p 1 Intercept 2 21976.13 21988.09 –10986.07 2 Base model 3 21963.32 21981.26 –10978.66 1 vs 2 14.810 < .001 3 Base model + RI 4 21364.30 21370.22 –10669.15 2 vs 3 619.024 < .001 4 Base model + RS 6 21263.17 21299.04 –10625.58 3 vs 4 87.130 < .001 Note. df = degrees of freedom, AIC = Akaike Information Criterion, BIC = Bayesian Information Criterion, logLik = loglikelihood, RI = random intercepts, RS = random intercept and slopes. The base model includes the predictor ball position.