Replication code and data for "Heel-strike mechanics reveal evolutionary tradeoffs in hominin bipedalism".

Foot-Strike Force Analysis Steven Worthington 2025-09-26 Table of contents 1 Correlation of MLR and IPF 2 1.1 Between-subjects ........................................ 2 1.2 Within-subjects ......................................... 3 2 Difference between bipedal postures in chimpanzees 5 2.1 MLR................................................ 5 2.2 IPF ................................................ 8 2.3 MVF ............................................... 11 3 Hip angle in chimpanzees 14 3.1 MLR................................................ 14 3.2 IPF ................................................ 17 3.3 MVF ............................................... 20 4 Knee angle in chimpanzees 23 4.1 MLR................................................ 23 4.2 IPF ................................................ 26 4.3 MVF ............................................... 29 5 Ankle angle in chimpanzees 32 5.1 MLR................................................ 32 5.2 IPF ................................................ 35 5.3 MVF ............................................... 38 6 FSA in chimpanzees 41 6.1 MLR................................................ 41 6.2 IPF ................................................ 45 6.3 MVF ............................................... 49 7 Differences between species and postures 53 7.1 MLR................................................ 53 Table of contents 2 7.2 IPF ................................................ 57 7.3 MVF ............................................... 61 8 Difference in CoT between postures in humans 65 9 Correlation of FSA and Ankle Angle 68 9.1 Between-subjects ........................................ 68 9.2 Within-subjects ......................................... 69 10 Difference in FSA between species and gaits 71 10.1 FSA................................................ 71 11 Difference between postures in humans 75 11.1 MLR................................................ 75 11.2 IPF ................................................ 78 11.3 MVF ............................................... 81 1 CORRELATION OF MLR AND IPF 3 1 Correlation of MLR and IPF −2 −1 0 1 2 −2 −1 0 1 2 MLR (standardized) IPF (standardized) Chimp Human Figure 1: Correlation between MLR and IPF. Dashed line indicates an identity relationship. Polygon ellipses encompass 50% and 95% of the data, respectively. 1.1 Between-subjects (MLR𝑖𝑗 |𝑏MLR 0𝑗 )∼Normal(𝜇MLR 𝑖𝑗 ,𝜎MLR) 𝜇MLR 𝑖𝑗 =𝑏MLR 0𝑗 𝑏MLR 0𝑗 ∼Normal(0,𝜏MLR) (IPF𝑖𝑗 |𝑏IPF 0𝑗 )∼Normal(𝜇IPF 𝑖𝑗 ,𝜎IPF) 𝜇IPF 𝑖𝑗 =𝑏IPF 0𝑗 𝑏IPF 0𝑗 ∼Normal(0,𝜏IPF) log 𝜎MLR,log 𝜎IPF ∼Student(3,0,2.5) log 𝜏MLR,log 𝜏IPF ∼Student(3,0,2.5) Between-subject correlation: 𝜌𝑏=cor(𝑏MLR 0𝑗 ,𝑏IPF 0𝑗 ) 1 CORRELATION OF MLR AND IPF 4 0.00 0.25 0.50 0.75 1.00 0.900 0.925 0.950 0.975 1.000 Between−subjects correlation of MLR and IPF Posterior Density Figure 2: Between-subjects correlation between MLR and IPF. Table 1: Between-subjects correlation coefficient for MLR versus IPF lower_95_HPDI upper_95_HPDI 0.928 0.985 1.2 Within-subjects (MLR𝑖𝑗,IPF𝑖𝑗 |𝑏0𝑗)∼Multivariate Normal ((𝜇MLR 𝑖𝑗 𝜇IPF 𝑖𝑗 ),(𝜎2 MLR 𝜌𝜖𝜎MLR𝜎IPF 𝜌𝜖𝜎MLR𝜎IPF 𝜎2 IPF )) 𝜇MLR 𝑖𝑗 =𝑏MLR 0𝑗 𝜇IPF 𝑖𝑗 =𝑏IPF 0𝑗 𝑏MLR 0𝑗 ,𝑏IPF 0𝑗 ∼Multivariate Normal ((0 0),(𝜏2 MLR 𝜌𝑏𝜏MLR𝜏IPF 𝜌𝑏𝜏MLR𝜏IPF 𝜏2 IPF )) log 𝜎MLR,log 𝜎IPF ∼Student(3,0,2.5) log 𝜏MLR,log 𝜏IPF ∼Student(3,0,2.5) Within-subject correlation: 𝜌𝜖=cor(𝜖MLR 𝑖𝑗 ,𝜖IPF 𝑖𝑗 ) 1 CORRELATION OF MLR AND IPF 5 0.00 0.25 0.50 0.75 1.00 0.70 0.75 0.80 0.85 Within−subjects correlation of MLR and IPF Posterior Density Figure 3: Within-subjects correlation between MLR and IPF. Table 2: Within-subjects correlation coefficient for MLR versus IPF lower_95_HPDI upper_95_HPDI 0.737 0.844 2 DIFFERENCE BETWEEN BIPEDAL POSTURES IN CHIMPANZEES 6 2 Difference between bipedal postures in chimpanzees 2.1 MLR Difference in MLR between postures in chimps. MLR𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(posture𝑖𝑗)+𝛽2(froude𝑖𝑗)+𝛽3(bm𝑗) log 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−3 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) 0.00 0.25 0.50 0.75 1.00 5k 20k 35k 50k MLR Posterior density Midfoot Heel Figure 4: Predicted MLR by posture for chimpanzees. 2 DIFFERENCE BETWEEN BIPEDAL POSTURES IN CHIMPANZEES 7 0.00 0.25 0.50 0.75 1.00 10k 15k 20k 25k MLR Geometric Mean Posterior density Midfoot Heel Figure 5: Predicted MLR by posture for chimpanzees. Table 3: Geometric mean MLR by posture for chimpanzees. posture lower_95_HPDI upper_95_HPDI Midfoot 10,383 12,410 Heel 18,033 25,346 0.00 0.25 0.50 0.75 1.00 50% 100% 150% ∆% MLR (geometric mean) Midfoot to Heel Posterior Density Figure 6: Percent change in posture effect for chimpanzees. 2 DIFFERENCE BETWEEN BIPEDAL POSTURES IN CHIMPANZEES 8 Table 4: Percent change (%) in geometric mean MLR between postures (midfoot to heel) lower_95_HPDI upper_95_HPDI 52.3% 131.1% 0.00 0.25 0.50 0.75 1.00 ±0% ±50% ±100% ±150% ROPE threshold (% change in MLR geometric mean: Midfoot −> Heel) Posterior probability outside ROPE Figure 7: Posterior probabilities for posture effect for chimpanzees. 2 DIFFERENCE BETWEEN BIPEDAL POSTURES IN CHIMPANZEES 9 2.2 IPF Difference in IPF between postures in chimps. IPF𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(posture𝑖𝑗)+𝛽2(froude𝑖𝑗)+𝛽3(bm𝑗) log 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−3 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) 0.00 0.25 0.50 0.75 1.00 20 120 220 320 IPF Posterior density Midfoot Heel Figure 8: Predicted IPF by posture for chimpanzees. 0.00 0.25 0.50 0.75 1.00 130 150 170 190 IPF Geometric Mean Posterior density Midfoot Heel Figure 9: Predicted IPF by posture for chimpanzees. 3 HIP ANGLE IN CHIMPANZEES 16 Table 9: Geometric mean MLR by posture for chimpanzees (averaged over hip angle). posture lower_95_HPDI upper_95_HPDI Midfoot 7,636 11,031 Heel 20,800 35,588 0.00 0.25 0.50 0.75 1.00 −2% −1% 0% 1% 2% 3% Instantaneous % change in MLR (geometric mean) per ° hip angle Posterior Density Figure 17: Percent change in hip angle effect for chimpanzees. Table 10: Instantaneous % change in MLR (geometric mean) per ° hip angle. lower_95_HPDI upper_95_HPDI -1.2% 2.1% 3 HIP ANGLE IN CHIMPANZEES 17 0.00 0.25 0.50 0.75 1.00 ±0% ±1% ±2% ±3% ROPE threshold (% change in MLR geometric mean per ° hip angle) Posterior probability outside ROPE Figure 18: Posterior probabilities for hip angle effect for chimpanzees. Midfoot Heel −500 0 500 1k −500 0 500 1k 0.00 0.25 0.50 0.75 1.00 Instantaneous change in MLR (geometric mean) per ° hip angle Posterior Density Hip Angle 50° to 60° 60° to 70° >70° Figure 19: Change in hip angle groups effect for chimpanzees. 3 HIP ANGLE IN CHIMPANZEES 18 3.2 IPF Change in IPF for change in hip angle in chimps (bipedal gait). IPF𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(hip angle𝑖𝑗)+𝛽2(posture𝑖𝑗) +𝛽3(froude𝑖𝑗)+𝛽4(bm𝑗) log 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−4 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) 100 150 200 250 50° 60° 70° 80° Hip Angle IPF (geometric mean) Midfoot HPDI 0.95 0.8 0.5 Heel HPDI 0.95 0.8 0.5 Figure 20: Predicted IPF by hip angle and posture for chimpanzees. 3 HIP ANGLE IN CHIMPANZEES 19 Table 11: Geometric mean IPF by posture for chimpanzees (averaged over hip angle). posture lower_95_HPDI upper_95_HPDI Midfoot 106 157 Heel 179 259 0.00 0.25 0.50 0.75 1.00 −2% −1% 0% 1% 2% 3% Instantaneous % change in IPF (geometric mean) per ° hip angle Posterior Density Figure 21: Percent change in hip angle effect for chimpanzees. Table 12: Instantaneous % change in IPF (geometric mean) per ° hip angle. lower_95_HPDI upper_95_HPDI -1.1% 1.9% 3 HIP ANGLE IN CHIMPANZEES 20 0.00 0.25 0.50 0.75 1.00 ±0% ±1% ±2% ±3% ROPE threshold (% change in IPF geometric mean per ° hip angle) Posterior probability outside ROPE Figure 22: Posterior probabilities for hip angle effect effect for chimpanzees. Midfoot Heel −5 0 5 10 −5 0 5 10 0.00 0.25 0.50 0.75 1.00 Instantaneous change in IPF (geometric mean) per ° hip angle Posterior Density Hip Angle 50° to 60° 60° to 70° >70° Figure 23: Change in hip angle groups effect for chimpanzees. 3 HIP ANGLE IN CHIMPANZEES 21 3.3 MVF Change in MVF for change in hip angle in chimps (bipedal gait). MVF𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(hip angle𝑖𝑗)+𝛽2(posture𝑖𝑗) +𝛽3(froude𝑖𝑗)+𝛽4(bm𝑗) log 𝜎=𝛾0 𝑏0𝑗 ∼Normal(0,𝜏0) 𝛽1−4 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) 280 300 320 340 360 380 50° 60° 70° 80° Hip Angle MVF (geometric mean) Midfoot HPDI 0.95 0.8 0.5 Heel HPDI 0.95 0.8 0.5 Figure 24: Predicted MVF by hip angle and posture for chimpanzees. Table 13: Geometric mean MVF by posture for chimpanzees (averaged over hip angle). posture lower_95_HPDI upper_95_HPDI Midfoot 274 305 Heel 340 382 3 HIP ANGLE IN CHIMPANZEES 22 0.00 0.25 0.50 0.75 1.00 0% 0.2% 0.4% 0.6% 0.8% Instantaneous % change in MVF (geometric mean) per ° hip angle Posterior Density Figure 25: Percent change in hip angle effect for chimpanzees. Table 14: Instantaneous % change in MVF (geometric mean) per ° hip angle. lower_95_HPDI upper_95_HPDI 0.1% 0.6% 0.00 0.25 0.50 0.75 1.00 ±0.0% ±0.2% ±0.4% ±0.6% ±0.8% ROPE threshold (% change in MVF geometric mean per ° hip angle) Posterior probability outside ROPE Figure 26: Posterior probabilities for hip angle effect for chimpanzees. 3 HIP ANGLE IN CHIMPANZEES 23 Midfoot Heel 024024 0.00 0.25 0.50 0.75 1.00 Instantaneous change in MVF (geometric mean) per ° hip angle Posterior Density Hip Angle 50° to 60° 60° to 70° >70° Figure 27: Change in hip angle groups effect for chimpanzees. 4 KNEE ANGLE IN CHIMPANZEES 24 4 Knee angle in chimpanzees 4.1 MLR Change in MLR for change in knee angle in chimps (bipedal gait). MLR𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(knee angle𝑖𝑗)+𝛽2(posture𝑖𝑗) +𝛽3(froude𝑖𝑗)+𝛽4(bm𝑗) log 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−4 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) 10k 20k 30k 40k 15° 20° 25° 30° 35° Knee Angle MLR (geometric mean) Midfoot HPDI 0.95 0.8 0.5 Heel HPDI 0.95 0.8 0.5 Figure 28: Predicted MLR by knee angle and posture for chimpanzees. 4 KNEE ANGLE IN CHIMPANZEES 25 Table 15: Geometric mean MLR by posture for chimpanzees (averaged over knee angle). posture lower_95_HPDI upper_95_HPDI Midfoot 8,734 11,353 Heel 23,661 36,052 0.00 0.25 0.50 0.75 1.00 −2% 0% 2% Instantaneous % change in MLR (geometric mean) per ° knee angle Posterior Density Figure 29: Percent change in knee angle effect for chimpanzees. Table 16: Instantaneous % change in MLR (geometric mean) per ° knee angle. lower_95_HPDI upper_95_HPDI -2.1% 1.2% 4 KNEE ANGLE IN CHIMPANZEES 32 0.00 0.25 0.50 0.75 1.00 ±0.0% ±0.2% ±0.4% ±0.6% ROPE threshold (% change in MVF geometric mean per ° knee angle) Posterior probability outside ROPE Figure 38: Posterior probabilities for knee angle effect for chimpanzees. Midfoot Heel −2 −1 0 1 2 −2 −1 0 1 2 0.00 0.25 0.50 0.75 1.00 Instantaneous change in MVF (geometric mean) per ° knee angle Posterior Density Knee Angle 10° to 20° 20° to 30° 30° to 40° Figure 39: Change in knee angle groups effect for chimpanzees. 5 ANKLE ANGLE IN CHIMPANZEES 33 5 Ankle angle in chimpanzees 5.1 MLR Change in MLR for change in ankle angle in chimps (bipedal gait). MLR𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(ankle angle𝑖𝑗)+𝛽2(posture𝑖𝑗) +𝛽3(froude𝑖𝑗)+𝛽4(bm𝑗) log 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−4 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) 10k 20k 30k 40k 90° 100° 110° 120° Ankle Angle MLR (geometric mean) Midfoot HPDI 0.95 0.8 0.5 Heel HPDI 0.95 0.8 0.5 Figure 40: Predicted MLR by ankle angle and posture for chimpanzees. 5 ANKLE ANGLE IN CHIMPANZEES 34 Table 21: Geometric mean MLR by posture for chimpanzees (averaged over ankle angle). posture lower_95_HPDI upper_95_HPDI Midfoot 8,618 12,249 Heel 26,692 38,277 0.00 0.25 0.50 0.75 1.00 −1% 0% 1% 2% Instantaneous % change in MLR (geometric mean) per ° ankle angle Posterior Density Figure 41: Percent change in ankle angle effect for chimpanzees. Table 22: Instantaneous % change in MLR (geometric mean) per ° ankle angle. lower_95_HPDI upper_95_HPDI -0.5% 1.6% 5 ANKLE ANGLE IN CHIMPANZEES 35 0.00 0.25 0.50 0.75 1.00 ±0.0% ±0.5% ±1.0% ±1.5% ±2.0% ±2.5% ROPE threshold (% change in MLR geometric mean per ° ankle angle) Posterior probability outside ROPE Figure 42: Posterior probabilities for ankle angle effect for chimpanzees. Midfoot Heel −300 0 300 600 −300 0 300 600 0.00 0.25 0.50 0.75 1.00 Instantaneous change in MLR (geometric mean) per ° ankle angle Posterior Density Ankle Angle <100° 100° to 110° 110° to 120° >120° Figure 43: Change in ankle angle groups effect for chimpanzees. 5 ANKLE ANGLE IN CHIMPANZEES 36 5.2 IPF Change in IPF for change in ankle angle in chimps (bipedal gait). IPF𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(ankle angle𝑖𝑗)+𝛽2(posture𝑖𝑗) +𝛽3(froude𝑖𝑗)+𝛽4(bm𝑗) log 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−4 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) 100 150 200 250 90° 100° 110° 120° Ankle Angle IPF (geometric mean) Midfoot HPDI 0.95 0.8 0.5 Heel HPDI 0.95 0.8 0.5 Figure 44: Predicted IPF by ankle angle and posture for chimpanzees. 5 ANKLE ANGLE IN CHIMPANZEES 37 Table 23: Geometric mean IPF by posture for chimpanzees (averaged over ankle angle). posture lower_95_HPDI upper_95_HPDI Midfoot 115 165 Heel 216 273 0.00 0.25 0.50 0.75 1.00 0% 1% 2% Instantaneous % change in IPF (geometric mean) per ° ankle angle Posterior Density Figure 45: Percent change in ankle angle effect for chimpanzees. Table 24: Instantaneous % change in IPF (geometric mean) per ° ankle angle. lower_95_HPDI upper_95_HPDI -0.2% 1.5% 5 ANKLE ANGLE IN CHIMPANZEES 38 0.00 0.25 0.50 0.75 1.00 ±0.0% ±0.5% ±1.0% ±1.5% ±2.0% ROPE threshold (% change in IPF geometric mean per ° ankle angle) Posterior probability outside ROPE Figure 46: Posterior probabilities for ankle angle effect for chimpanzees. Midfoot Heel −2 0 2 4 −2 0 2 4 0.00 0.25 0.50 0.75 1.00 Instantaneous change in IPF (geometric mean) per ° ankle angle Posterior Density Ankle Angle <100° 100° to 110° 110° to 120° >120° Figure 47: Change in ankle angle groups effect for chimpanzees. 5 ANKLE ANGLE IN CHIMPANZEES 39 5.3 MVF Change in MVF for change in ankle angle in chimps (bipedal gait). MVF𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(ankle angle𝑖𝑗)+𝛽2(posture𝑖𝑗) +𝛽3(froude𝑖𝑗)+𝛽4(bm𝑗) log 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−4 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) 300 325 350 375 90° 100° 110° 120° Ankle Angle MVF (geometric mean) Midfoot HPDI 0.95 0.8 0.5 Heel HPDI 0.95 0.8 0.5 Figure 48: Predicted MVF by ankle angle and posture for chimpanzees. 5 ANKLE ANGLE IN CHIMPANZEES 40 Table 25: Geometric mean MVF by posture for chimpanzees (averaged over ankle angle). posture lower_95_HPDI upper_95_HPDI Midfoot 290 315 Heel 364 388 0.00 0.25 0.50 0.75 1.00 0% 0.2% 0.4% Instantaneous % change in MVF (geometric mean) per ° ankle angle Posterior Density Figure 49: Percent change in ankle angle effect for chimpanzees. Table 26: Instantaneous % change in MVF (geometric mean) per ° ankle angle. lower_95_HPDI upper_95_HPDI 0% 0.4% 5 ANKLE ANGLE IN CHIMPANZEES 41 0.00 0.25 0.50 0.75 1.00 ±0.0% ±0.1% ±0.2% ±0.3% ±0.4% ±0.5% ROPE threshold (% change in MVF geometric mean per ° ankle angle) Posterior probability outside ROPE Figure 50: Posterior probabilities for ankle angle effect for chimpanzees. Midfoot Heel 012012 0.00 0.25 0.50 0.75 1.00 Instantaneous change in MVF (geometric mean) per ° ankle angle Posterior Density Ankle Angle <100° 100° to 110° 110° to 120° >120° Figure 51: Change in ankle angle groups effect for chimpanzees. 6 FSA IN CHIMPANZEES 48 Quadruped Biped 0% 1% 2% 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Instantaneous % change in IPF (geometric mean) per ° FSA Posterior Density Figure 57: Posterior densities of the instantaneous rate of change in geometric-mean IPF with respect to FSA (IPF units per degree). Table 35: Instantaneous % change in IPF (geometric mean) per ° FSA. gait lower_95_HPDI upper_95_HPDI Biped -0.1% 1.4% Quadruped 0.1% 2.1% Table 36: Posterior probabilities that the instantaneous rate of change in geometric-mean IPF with respect to FSA (IPF units per degree) is greater than 0 Gait ℙ(>0) Biped 0.952 Quadruped 0.980 6 FSA IN CHIMPANZEES 49 0.00 0.25 0.50 0.75 1.00 ±0% ±1% ±2% ROPE threshold (% change in IPF geometric mean per ° FSA) Posterior probability outside ROPE Biped Quadruped Figure 58: Posterior probabilities of the instantaneous rate of change in geometric-mean IPF with respect to FSA (IPF units per degree). Biped Quadruped −2 0 2 4 6 −2 0 2 4 6 0.00 0.25 0.50 0.75 1.00 Instantaneous change in IPF (geometric mean) per ° FSA Posterior Density FSA −10° to 0° 0° to 10° 10° to 20° >20° Figure 59: Posterior densities of the instantaneous rate of change in geometric-mean IPF with respect to grouped FSA (IPF units per degree). 6 FSA IN CHIMPANZEES 50 6.3 MVF Change in MVF for change in FSA in chimps, by gait. MVF𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(FSA𝑖𝑗)+𝛽2(gait𝑖𝑗)+𝛽3(FSA𝑖𝑗 ×gait𝑖𝑗) +𝛽4(froude𝑖𝑗)+𝛽5(bm𝑗) log 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−5 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) 200 250 300 350 −10° 0° 10° 20° 30° FSA MVF (geometric mean) Quadruped HPDI 0.95 0.8 0.5 Biped HPDI 0.95 0.8 0.5 Figure 60: Predicted MVF by FSA and gait for chimpanzees. 6 FSA IN CHIMPANZEES 51 Table 37: Geometric mean MVF by gait (averaged over FSA). gait lower_95_HPDI upper_95_HPDI Biped 301 328 Quadruped 206 251 Table 38: Geometric mean MVF by gait and FSA groups. fsa_groups gait lower_95_HPDI upper_95_HPDI -10° to 0° Biped 269 282 -10° to 0° Quadruped 230 243 0° to 10° Biped 317 328 0° to 10° Quadruped 228 241 10° to 20° Biped 363 379 10° to 20° Quadruped 213 228 >20° Biped 398 417 >20° Quadruped 223 239 Table 39: Percent change in MVF (geometric mean) from quadruped to biped, for each FSA group. fsa_groups lower_95_HPDI upper_95_HPDI -10° to 0° 12.4% 20.7% 0° to 10° 33.2% 42.3% 10° to 20° 62.2% 74.7% >20° 69.9% 83.4% 6 FSA IN CHIMPANZEES 52 Quadruped Biped −0.8% −0.6% −0.4% −0.2% 0% 0.2% 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Instantaneous % change in MVF (geometric mean) per ° FSA Posterior Density Figure 61: Posterior densities of the instantaneous rate of change in geometric-mean MVF with respect to FSA (MVF units per degree). Table 40: Instantaneous % change in MVF (geometric mean) per ° FSA. gait lower_95_HPDI upper_95_HPDI Biped -0.4% 0.1% Quadruped -0.7% -0.2% Table 41: Posterior probabilities that the instantaneous rate of change in geometric-mean MVF with respect to FSA (MVF units per degree) is greater than 0 Gait ℙ(<0) Biped 0.914 Quadruped 1.000 6 FSA IN CHIMPANZEES 53 0.00 0.25 0.50 0.75 1.00 ±0.0% ±0.2% ±0.4% ±0.6% ±0.8% ROPE threshold (% change in MVF geometric mean per ° FSA) Posterior probability outside ROPE Biped Quadruped Figure 62: Posterior probabilities of the instantaneous rate of change in geometric-mean MVF with respect to FSA (MVF units per degree). Biped Quadruped −2 −1 0 1 −2 −1 0 1 0.00 0.25 0.50 0.75 1.00 Instantaneous change in MVF (geometric mean) per ° FSA Posterior Density FSA −10° to 0° 0° to 10° 10° to 20° >20° Figure 63: Posterior densities of the instantaneous rate of change in geometric-mean MVF with respect to grouped FSA (MVF units per degree). 7 DIFFERENCES BETWEEN SPECIES AND POSTURES 54 7 Differences between species and postures 7.1 MLR Difference in MLR between species and postures for bipedal gait. MLR𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(species𝑖𝑗)+𝛽2(posture𝑖𝑗)+𝛽3(species𝑖𝑗 ×posture𝑖𝑗) +𝛽4(froude𝑖𝑗)+𝛽5(bm𝑗) log 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−5 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) (1) Heel Midfoot 5k 25k 45k 65k 85k 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 MLR Posterior density Chimp Human Figure 64: Predicted MLR by species and posture for bipedal gait. 7 DIFFERENCES BETWEEN SPECIES AND POSTURES 55 Heel Midfoot 10k 20k 30k 40k 50k 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 MLR Geometric Mean Posterior density Chimp Human Figure 65: Predicted MLR by species and posture for bipedal gait. Table 42: Geometric mean MLR by species and posture for bipedal gait. species posture lower_95_HPDI upper_95_HPDI Chimp Midfoot 12,704 23,799 Chimp Heel 24,631 46,564 Human Heel 23,958 46,422 7 DIFFERENCES BETWEEN SPECIES AND POSTURES 56 0.00 0.25 0.50 0.75 1.00 −50% 0% 50% 100% 150% ∆% MLR (geometric mean) Posterior Density Midfoot to Heel Chimp to Human Figure 66: Percent change in species effect for heel strikes and percent change in posture effects for chimps. Table 43: Percent change (%) in geometric mean MLR between species (chimp to human) for heel strikes contrast lower_95_HPDI upper_95_HPDI Chimp to Human -53.8% 70.1% Table 44: Posterior probability that the % change in the geometric mean of MLR between chimp and human heel striking is within a ±25% ROPE ℙ(<±25%) 0.595 Table 45: Percent change (%) in geometric mean MLR between postures (midfoot to heel) for chimpanzees contrast lower_95_HPDI upper_95_HPDI Midfoot to Heel 57.9% 138.1% 7 DIFFERENCES BETWEEN SPECIES AND POSTURES 57 Table 46: Posterior probability that the % change in the geometric mean of MLR between midfoot and heel striking in chimps is greater than 0 for bipedal steps ℙ(>0) 1.000 0.00 0.25 0.50 0.75 1.00 ±0% ±50% ±100% Chimp −> Human Posterior probability outside ROPE A ±0% ±50% ±100% ±150% Midfoot −> Heel B ROPE threshold (% change in MLR geometric mean) Figure 67: Posterior probabilities for species effect for heel strikes and posture effects for chimps, for bipedal gait. 7 DIFFERENCES BETWEEN SPECIES AND POSTURES 64 0.00 0.25 0.50 0.75 1.00 −20% −10% 0% 10% 20% ∆% MVF (geometric mean) Posterior Density Midfoot to Heel Chimp to Human Figure 74: Percent change in species effect for heel strikes and percent change in posture effects for chimps. Table 53: Percent change (%) in geometric mean MVF between species (chimp to human) for heel strikes contrast lower_95_HPDI upper_95_HPDI Chimp to Human -20.8% 23.8% Table 54: Posterior probability that the % change in the geometric mean of MVF between chimp and human heel striking is within a ±25% ROPE ℙ(<±25%) 0.965 Table 55: Percent change (%) in geometric mean MVF between postures (midfoot to heel) for chimpanzees contrast lower_95_HPDI upper_95_HPDI Midfoot to Heel -5.9% -0.9% 7 DIFFERENCES BETWEEN SPECIES AND POSTURES 65 Table 56: Posterior probability that the % change in the geometric mean of MVF between midfoot and heel striking in chimps is less than 0 for bipedal steps ℙ(<0) 0.997 0.00 0.25 0.50 0.75 1.00 ±0% ±25% ±50% Chimp −> Human Posterior probability outside ROPE A ±0% ±4% ±8% Midfoot −> Heel B ROPE threshold (% change in MVF geometric mean) Figure 75: Posterior probabilities for species effect for heel strikes and posture effects for chimps, for bipedal gait. 8 DIFFERENCE IN COT BETWEEN POSTURES IN HUMANS 66 8 Difference in CoT between postures in humans Difference in cost of transport (CoT) in between postures in humans. CoT𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(posture𝑖𝑗)+𝛽2(age𝑗)+𝛽3(weight𝑗) log 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−3 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) 0.00 0.25 0.50 0.75 1.00 2.5 3 3.5 4 CoT Posterior density Heel Midfoot Figure 76: Predicted CoT by posture for humans. 8 DIFFERENCE IN COT BETWEEN POSTURES IN HUMANS 67 0.00 0.25 0.50 0.75 1.00 2.4 2.8 3.2 3.6 CoT Geometric Mean Posterior density Heel Midfoot Figure 77: Predicted CoT by posture for humans. Table 57: Geometric mean CoT by posture for humans. posture lower_95_HPDI upper_95_HPDI Heel 2.48 2.68 Midfoot 3.31 3.58 0.00 0.25 0.50 0.75 1.00 25% 35% 45% ∆% CoT (geometric mean) Heel −> Midfoot Posterior Density Figure 78: Percent change in posture effect for humans. 8 DIFFERENCE IN COT BETWEEN POSTURES IN HUMANS 68 Table 58: Percent change (%) in geometric mean CoT between postures (heel to midfoot) lower_95_HPDI upper_95_HPDI 26.1% 40.8% 0.00 0.25 0.50 0.75 1.00 ±0% ±10% ±20% ±30% ±40% ROPE threshold (% change in CoT geometric mean: Heel −> Midfoot) Posterior probability outside ROPE Figure 79: Posterior probabilities for posture effect for humans. 9 CORRELATION OF FSA AND ANKLE ANGLE 69 9 Correlation of FSA and Ankle Angle −3 −2 −1 0 1 2 −2 −1 0 1 2 3 Ankle angle (standardized) FSA (standardized) Midfoot Heel Figure 80: Correlation between FSA and Ankle Angle. Dashed line indicates an identity relationship. Polygon ellipses encompass 50% and 95% of the data, respectively. 9.1 Between-subjects (FSA𝑖𝑗 |𝑏FSA 0𝑗 )∼Normal(𝜇FSA 𝑖𝑗 ,𝜎FSA) 𝜇FSA 𝑖𝑗 =𝑏FSA 0𝑗 𝑏FSA 0𝑗 ∼Normal(0,𝜏FSA) (AA𝑖𝑗 |𝑏AA 0𝑗 )∼Normal(𝜇AA 𝑖𝑗 ,𝜎AA) 𝜇AA 𝑖𝑗 =𝑏AA 0𝑗 𝑏AA 0𝑗 ∼Normal(0,𝜏AA) log 𝜎FSA,log 𝜎AA ∼Student(3,0,2.5) log 𝜏FSA,log 𝜏AA ∼Student(3,0,2.5) Between-subject correlation: 𝜌𝑏=cor(𝑏FSA 0𝑗 ,𝑏AA 0𝑗 ) 9 CORRELATION OF FSA AND ANKLE ANGLE 70 0.00 0.25 0.50 0.75 1.00 −1.0 −0.5 0.0 0.5 1.0 Between subject correlation of FSA and Ankle Angle Posterior Density Figure 81: Between-subjects correlation between FSA and Ankle Angle. Table 59: Between-subjects correlation coefficient for FSA versus Ankle Angle lower_95_HPDI upper_95_HPDI -0.999 0.902 9.2 Within-subjects (FSA𝑖𝑗,AA𝑖𝑗 |𝑏0𝑗)∼Multivariate Normal ((𝜇FSA 𝑖𝑗 𝜇AA 𝑖𝑗 ),(𝜎2 FSA 𝜌𝜖𝜎FSA𝜎AA 𝜌𝜖𝜎FSA𝜎AA 𝜎2 AA )) 𝜇FSA 𝑖𝑗 =𝑏FSA 0𝑗 𝜇AA 𝑖𝑗 =𝑏AA 0𝑗 𝑏FSA 0𝑗 ,𝑏AA 0𝑗 ∼Multivariate Normal ((0 0),(𝜏2 FSA 𝜌𝑏𝜏FSA𝜏AA 𝜌𝑏𝜏FSA𝜏AA 𝜏2 AA )) log 𝜎FSA,log 𝜎AA ∼Student(3,0,2.5) log 𝜏FSA,log 𝜏AA ∼Student(3,0,2.5) Within-subject correlation: 𝜌𝜖=cor(𝜖FSA 𝑖𝑗 ,𝜖AA 𝑖𝑗 ) 9 CORRELATION OF FSA AND ANKLE ANGLE 71 0.00 0.25 0.50 0.75 1.00 −0.8 −0.7 −0.6 −0.5 Within−subjects correlation of FSA and Ankle Angle Posterior Density Figure 82: Within-subjects correlation between FSA and Ankle Angle. Table 60: Within-subjects correlation coefficient for FSA versus Ankle Angle lower_95_HPDI upper_95_HPDI -0.750 -0.507 10 DIFFERENCE IN FSA BETWEEN SPECIES AND GAITS 72 10 Difference in FSA between species and gaits 10.1 FSA Difference in FSA between species and gaits. FSA𝑖𝑗 ∼Normal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(species𝑖𝑗)+𝛽2(gait𝑖𝑗)+𝛽3(species𝑖𝑗 ×gait𝑖𝑗)+𝛽4(froude𝑖𝑗)+𝛽5(bm𝑗) 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−5 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) Biped Quadruped −5° 0° 5° 10° 15° 20° 25° 30° −5° 0° 5° 10° 15° 20° 25° 30° 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 FSA Posterior density Chimp Human Figure 83: Predicted FSA by species and gait. 10 DIFFERENCE IN FSA BETWEEN SPECIES AND GAITS 73 Biped Quadruped 0° 5° 10° 15° 20° 0° 5° 10° 15° 20° 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 FSA Posterior density Chimp Human Figure 84: Predicted FSA by species and gait. Table 61: Arithmetic mean FSA by species and gait. species gait lower_95_HPDI upper_95_HPDI Chimp Quadruped 5.25 11.83 Chimp Biped 2.95 9.45 Human Biped 10.66 19.28 Human Biped − Chimp Biped Human Biped − Chimp Quadruped Chimp Quadruped − Chimp Biped 0° 5° 10° 15° 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 ∆ FSA (arithmetic mean) Posterior Density Figure 85: Change in FSA by species and gait. 11 DIFFERENCE BETWEEN POSTURES IN HUMANS 80 Table 67: Geometric mean IPF by posture for humans. posture lower_95_HPDI upper_95_HPDI Midfoot 111.3 122.9 Heel 319.8 351.9 0.00 0.25 0.50 0.75 1.00 160% 180% 200% 220% ∆% IPF (geometric mean) Midfoot −> Heel Posterior Density Figure 93: Percent change in posture effect for humans. Table 68: Percent change (%) in geometric mean IPF between postures (midfoot to heel) lower_95_HPDI upper_95_HPDI 167.5% 205.5% 11 DIFFERENCE BETWEEN POSTURES IN HUMANS 81 0.00 0.25 0.50 0.75 1.00 ±0% ±100% ±200% ROPE threshold (% change in IPF geometric mean: Midfoot −> Heel) Posterior probability outside ROPE Figure 94: Posterior probabilities for posture effect for humans. 11 DIFFERENCE BETWEEN POSTURES IN HUMANS 82 11.3 MVF Difference in MVF between postures in humans. MVF𝑖𝑗 ∼Lognormal(𝜇𝑖𝑗,𝜎𝑗) 𝜇𝑖𝑗 =𝛽0+𝑏0𝑗 +𝛽1(posture𝑖𝑗)+𝛽2(froude𝑖𝑗)+𝛽3(bm𝑗) log 𝜎𝑗=𝛾0+𝑔0𝑗 𝑏0𝑗 ∼Normal(0,𝜏0) 𝑔0𝑗 ∼Normal(0,𝜙0) 𝛽1−3 ∼Student(3,0,2.5) 𝛾0∼Student(3,0,2.5) log 𝜏0∼Student(3,0,2.5) log 𝜙0∼Student(3,0,2.5) 0.00 0.25 0.50 0.75 1.00 700 900 1.1k MVF Posterior density Midfoot Heel Figure 95: Predicted MVF by posture for humans. 0.00 0.25 0.50 0.75 1.00 825 830 835 840 845 MVF Geometric Mean Posterior density Midfoot Heel Figure 96: Predicted MVF by posture for humans. 11 DIFFERENCE BETWEEN POSTURES IN HUMANS 83 Table 69: Geometric mean MVF by posture for humans. posture lower_95_HPDI upper_95_HPDI Midfoot 827 839 Heel 831 843 0.00 0.25 0.50 0.75 1.00 −1% 0% 1% 2% ∆% MVF (geometric mean) Midfoot −> Heel Posterior Density Figure 97: Percent change in posture effect for humans. Table 70: Percent change (%) in geometric mean MVF between postures (midfoot to heel) lower_95_HPDI upper_95_HPDI -0.6% 1.5% 11 DIFFERENCE BETWEEN POSTURES IN HUMANS 84 0.00 0.25 0.50 0.75 1.00 ±0% ±1% ±2% ROPE threshold (% change in MVF geometric mean: Midfoot −> Heel) Posterior probability outside ROPE Figure 98: Posterior probabilities for posture effect for humans.