With my following data:
structure(list(id = c("AC12-PRD-C1", "AC12-PRD-C1", "AC12-PRD-C1",
"AC12-PRD-C1", "AC12-PRD-C1", "AC12-PRD-C1", "AC12-PRD-C1", "AL13-PRD-C1",
"AL13-PRD-C1", "AL13-PRD-C1", "AL13-PRD-C1", "AL13-PRD-C1", "AL13-PRD-C1",
"AL13-PRD-C1", "AL13-PRD-C1", "BM06-PRD-S1", "BM06-PRD-S1", "BM06-PRD-S1",
"BM06-PRD-S1", "BM06-PRD-S1", "BM06-PRD-S1", "CB19-PRD-S1", "CB19-PRD-S1",
"CB19-PRD-S1", "CB19-PRD-S1", "CB19-PRD-S1", "CB19-PRD-S1", "CB19-PRD-S1",
"CB19-PRD-S1", "CB19-PRD-S1", "CB19-PRD-S1", "CB19-PRD-S1", "CB19-PRD-S1",
"CC14-PRD-S1", "CC14-PRD-S1", "CC14-PRD-S1", "CC14-PRD-S1", "CC14-PRD-S1",
"CC14-PRD-S1", "CC14-PRD-S1", "CC14-PRD-S1", "DA03-PRD-C1", "DA03-PRD-C1",
"DA03-PRD-C1", "DA03-PRD-C1", "DA03-PRD-C1", "DA03-PRD-C1", "DA03-PRD-C1",
"DA03-PRD-C1", "DA03-PRD-C1", "DA03-PRD-C1", "DA03-PRD-C1", "DA03-PRD-C1",
"DA24-PRD-S1", "DA24-PRD-S1", "DA24-PRD-S1", "DA24-PRD-S1", "DA24-PRD-S1",
"DA24-PRD-S1", "DA24-PRD-S1", "DB22-PRD-S1", "DB22-PRD-S1", "DB22-PRD-S1",
"DB22-PRD-S1", "DB22-PRD-S1", "DB42-PRD-C1", "DB42-PRD-C1", "DB42-PRD-C1",
"DB42-PRD-C1", "DB42-PRD-C1", "DB42-PRD-C1", "DB42-PRD-C1", "DB42-PRD-C1",
"DB42-PRD-C1", "DB42-PRD-C1", "DB42-PRD-C1", "DL18-PRD-S1", "DL18-PRD-S1",
"DL18-PRD-S1", "DL18-PRD-S1", "DL18-PRD-S1", "DL18-PRD-S1", "DL18-PRD-S1",
"DL18-PRD-S1", "DL18-PRD-S1", "DL18-PRD-S1", "DL18-PRD-S1", "DL18-PRD-S1",
"DL18-PRD-S1", "DR15-PRD-C1", "DR15-PRD-C1", "DR15-PRD-C1", "DR15-PRD-C1",
"DR15-PRD-C1", "DR15-PRD-C1", "DR15-PRD-C1", "DR15-PRD-C1", "DT08-PRD-S1",
"DT08-PRD-S1", "DT08-PRD-S1", "DT08-PRD-S1", "DT08-PRD-S1", "DT08-PRD-S1",
"DT08-PRD-S1", "DT08-PRD-S1", "FB44-PRD-C1", "FB44-PRD-C1", "FB44-PRD-C1",
"FB44-PRD-C1", "FB44-PRD-C1", "FB44-PRD-C1", "FB44-PRD-C1", "FG33-PRD-C1",
"FG33-PRD-C1", "FG33-PRD-C1", "FG33-PRD-C1", "FG33-PRD-C1", "FG33-PRD-C1",
"FG33-PRD-C1", "FG33-PRD-C1", "FG33-PRD-C1", "FG33-PRD-C1", "FG33-PRD-C1",
"FG33-PRD-C1", "FG40-PRD-C1", "FG40-PRD-C1", "FG40-PRD-C1", "FG40-PRD-C1",
"FG40-PRD-C1", "GC03-PRD-S1", "GC03-PRD-S1", "GC03-PRD-S1", "GC03-PRD-S1",
"GC03-PRD-S1", "GC03-PRD-S1", "GC03-PRD-S1", "GC03-PRD-S1", "GG30-PRD-C1",
"GG30-PRD-C1", "GG30-PRD-C1", "GG30-PRD-C1", "GG30-PRD-C1", "GG30-PRD-C1",
"GG30-PRD-C1", "GG30-PRD-C1", "GH05-PRD-C1", "GH05-PRD-C1", "GH05-PRD-C1",
"GH05-PRD-C1", "GH05-PRD-C1", "GH05-PRD-C1", "GH05-PRD-C1", "GH05-PRD-C1",
"GH05-PRD-C1", "GH05-PRD-C1", "GL05-PRD-S1", "GL05-PRD-S1", "GL05-PRD-S1",
"GL05-PRD-S1", "GL05-PRD-S1", "GL05-PRD-S1", "GL05-PRD-S1", "GL05-PRD-S1",
"GL05-PRD-S1", "GL05-PRD-S1", "GL05-PRD-S1", "GL05-PRD-S1", "GP22-PRD-C1",
"GP22-PRD-C1", "GP22-PRD-C1", "GP22-PRD-C1", "GP22-PRD-C1", "GP22-PRD-C1",
"GP22-PRD-C1", "GP22-PRD-C1", "GP22-PRD-C1", "GP28-PRD-C1", "GP28-PRD-C1",
"GP28-PRD-C1", "GP28-PRD-C1", "GP28-PRD-C1", "GP28-PRD-C1", "GP28-PRD-C1",
"GP28-PRD-C1", "GP28-PRD-C1", "GP28-PRD-C1", "GP28-PRD-C1", "GT04-PRD-S1",
"GT04-PRD-S1", "GT04-PRD-S1", "GT04-PRD-S1", "GT04-PRD-S1", "GT04-PRD-S1",
"GT04-PRD-S1", "GT21-PRD-S1", "GT21-PRD-S1", "GT21-PRD-S1", "GT21-PRD-S1",
"GT21-PRD-S1", "GT21-PRD-S1", "GT21-PRD-S1", "GT21-PRD-S1"),
power = c(25L, 40L, 55L, 70L, 85L, 100L, 115L, 25L, 40L,
55L, 70L, 85L, 100L, 115L, 130L, 25L, 40L, 55L, 70L, 85L,
100L, 25L, 40L, 55L, 70L, 85L, 100L, 115L, 130L, 145L, 160L,
175L, 190L, 20L, 30L, 40L, 50L, 60L, 70L, 80L, 90L, 20L,
30L, 40L, 50L, 60L, 70L, 80L, 90L, 100L, 110L, 120L, 130L,
25L, 40L, 55L, 70L, 85L, 100L, 115L, 25L, 40L, 55L, 70L,
85L, 25L, 40L, 55L, 70L, 85L, 100L, 115L, 130L, 145L, 160L,
175L, 25L, 40L, 55L, 70L, 85L, 100L, 115L, 130L, 145L, 160L,
175L, 190L, 205L, 20L, 30L, 40L, 50L, 60L, 70L, 80L, 90L,
50L, 60L, 70L, 80L, 90L, 100L, 110L, 120L, 20L, 30L, 40L,
50L, 60L, 70L, 80L, 20L, 30L, 40L, 50L, 60L, 70L, 80L, 90L,
100L, 110L, 120L, 130L, 20L, 30L, 40L, 50L, 60L, 25L, 40L,
55L, 70L, 85L, 100L, 115L, 130L, 25L, 40L, 55L, 70L, 85L,
100L, 115L, 130L, 20L, 30L, 40L, 50L, 60L, 70L, 80L, 90L,
100L, 110L, 40L, 60L, 80L, 100L, 120L, 140L, 160L, 180L,
200L, 220L, 240L, 260L, 25L, 40L, 55L, 70L, 85L, 100L, 115L,
130L, 145L, 25L, 40L, 55L, 70L, 85L, 100L, 115L, 130L, 145L,
160L, 175L, 60L, 80L, 100L, 120L, 140L, 160L, 180L, 25L,
40L, 55L, 70L, 85L, 100L, 115L, 130L), VE = c(22.75, 23.75,
26.75, 32.5, 39.5, 48.25, 55.75, 24, 26.5, 29, 36.5, 44.75,
55.25, 63.75, 78, 19, 20.75, 25, 29, 39.25, 51, 25, 25.5,
28.5, 34.25, 35.5, 40.25, 47, 52.5, 59.5, 64.5, 83, 92.25,
18.25, 19.75, 20.75, 23.5, 26.25, 28, 32.75, 37.25, 24.25,
26.75, 29.5, 31, 32, 39, 43.25, 46.25, 51.75, 52.75, 64.5,
72.75, 15.75, 16.25, 20.25, 28, 34.75, 45.75, 64.75, 18.25,
21.25, 22.25, 24.75, 29.5, 21.25, 20.75, 26, 30.75, 34, 38.5,
46.75, 53.75, 70.5, 82.75, 98.25, 17.25, 22, 25.75, 29.75,
33.75, 38.75, 44.5, 50.75, 56.25, 65, 71.75, 81, 99, 30,
29, 29.75, 38.25, 40.75, 47.25, 52.75, 65.5, 22.75, 24.75,
28.75, 34, 48, 53.75, 61.5, 73.75, 22, 21.75, 24.25, 28,
34.5, 40, 48.5, 23.75, 23.75, 25.75, 29, 31.25, 33.5, 35.25,
38.5, 40.5, 50, 62.5, 77.25, 19.5, 21.25, 24.25, 30.75, 38.5,
17, 17.25, 20.75, 24.25, 28.5, 32.25, 38.75, 54.25, 21, 23.25,
28.25, 32.75, 37, 46.5, 56.25, 65.5, 24.5, 25.75, 26.75,
29.5, 33.25, 37.75, 40, 45.75, 50, 56.5, 23.75, 26.75, 29.75,
38.25, 43.75, 50.5, 58.5, 69.75, 78.25, 92.75, 109, 123.25,
22.75, 23.75, 28.5, 32.75, 39.75, 46.5, 56.5, 65.75, 76,
26.75, 26.5, 30.5, 31, 40.25, 44.75, 52.5, 60, 65, 70.75,
78, 40.5, 43.75, 50.25, 63, 75, 82, 95.25, 25.75, 27.5, 33,
37.75, 44, 53.5, 58.75, 69), absVO2 = c(0.739, 0.81975, 0.95125,
1.07525, 1.199, 1.34575, 1.49775, 0.66125, 0.7485, 0.855,
0.9845, 1.1495, 1.3465, 1.451, 1.5985, 0.61675, 0.717, 0.86275,
0.98575, 1.13, 1.262, 0.8835, 0.94575, 1.08125, 1.244, 1.30475,
1.4735, 1.679, 1.79075, 1.96, 2.0405, 2.34425, 2.4435, 0.5925,
0.661, 0.7435, 0.87875, 0.9435, 0.99675, 1.11425, 1.20275,
0.9255, 1.02925, 1.11675, 1.19725, 1.24775, 1.42625, 1.54225,
1.59425, 1.69675, 1.776, 1.94525, 2.0395, 0.535, 0.5845,
0.76875, 0.982, 1.09975, 1.259, 1.3265, 0.7205, 0.86825,
0.9235, 1.01075, 1.17275, 0.7435, 0.76575, 0.96075, 1.0975,
1.21975, 1.34525, 1.50625, 1.6755, 1.86325, 2.0465, 2.1395,
0.613, 0.85525, 0.98, 1.076, 1.287, 1.4615, 1.59325, 1.6965,
1.884, 1.998, 2.1425, 2.31275, 2.474, 0.81075, 0.8035, 0.8645,
1.076, 1.119, 1.255, 1.33825, 1.499, 0.8105, 0.8865, 1.03725,
1.134, 1.26675, 1.36275, 1.4455, 1.52875, 0.64675, 0.6645,
0.742, 0.81675, 0.983, 1.06875, 1.155, 1.02475, 1.0335, 1.09525,
1.21675, 1.283, 1.39025, 1.438, 1.575, 1.63075, 1.8005, 1.95575,
2.047, 0.708, 0.7245, 0.82675, 0.924, 0.99275, 0.6775, 0.69625,
0.806, 0.95825, 1.07425, 1.2515, 1.37325, 1.62675, 0.79325,
0.841, 0.97275, 1.04925, 1.16925, 1.41225, 1.53125, 1.65475,
0.7105, 0.74875, 0.85225, 0.92375, 1.05575, 1.155, 1.228,
1.33625, 1.4605, 1.5715, 1.02625, 1.23, 1.326, 1.57525, 1.7225,
1.907, 2.13825, 2.35175, 2.52325, 2.75525, 2.95425, 3.117,
0.727, 0.76075, 0.9335, 1.0545, 1.229, 1.37075, 1.44675,
1.59225, 1.7365, 0.99625, 0.991, 1.199, 1.20625, 1.4295,
1.52425, 1.587, 1.71075, 1.88425, 1.9755, 2.10025, 1.279,
1.41725, 1.547, 1.8155, 1.976, 2.1635, 2.3375, 0.80825, 0.87425,
1.01375, 1.1475, 1.286, 1.432, 1.56, 1.748), VCO2 = c(0.667,
0.715, 0.854, 1.047, 1.25125, 1.5185, 1.76025, 0.5805, 0.66425,
0.79425, 0.99925, 1.23475, 1.48375, 1.65775, 1.89075, 0.58225,
0.6595, 0.8275, 1.0455, 1.30525, 1.6115, 0.80475, 0.829,
0.958, 1.14275, 1.19925, 1.42225, 1.68475, 1.8795, 2.13525,
2.2835, 2.7605, 2.9685, 0.54825, 0.60325, 0.66025, 0.77325,
0.89375, 0.98525, 1.16275, 1.327, 0.781, 0.8775, 0.972, 1.05425,
1.1175, 1.353, 1.5075, 1.61425, 1.76925, 1.8675, 2.17975,
2.3575, 0.45075, 0.47775, 0.6545, 0.95525, 1.18325, 1.497,
1.725, 0.58275, 0.746, 0.7955, 0.91, 1.13175, 0.6205, 0.60475,
0.791, 0.963, 1.10725, 1.30025, 1.56775, 1.804, 2.1645, 2.434,
2.67125, 0.542, 0.76325, 0.89, 1.081, 1.2545, 1.4735, 1.68575,
1.83225, 2.1025, 2.31325, 2.52175, 2.755, 3.117, 0.72925,
0.7275, 0.77375, 1.02925, 1.1305, 1.33425, 1.48125, 1.77475,
0.70875, 0.839, 0.9905, 1.206, 1.4965, 1.649, 1.7875, 1.9635,
0.6755, 0.67525, 0.751, 0.86925, 1.06025, 1.23525, 1.4025,
0.871, 0.888, 0.97825, 1.11275, 1.23175, 1.3465, 1.4595,
1.60625, 1.74075, 2.029, 2.37675, 2.64675, 0.6175, 0.64475,
0.76425, 0.9405, 1.12525, 0.5545, 0.566, 0.69775, 0.8575,
1.014, 1.2, 1.42375, 1.878, 0.61675, 0.6825, 0.82925, 0.98575,
1.14725, 1.45, 1.6995, 1.92425, 0.6365, 0.67, 0.7485, 0.81925,
0.95025, 1.08175, 1.1795, 1.3525, 1.53, 1.67925, 0.73375,
0.90225, 1.0115, 1.3055, 1.498, 1.78, 2.0795, 2.436, 2.701,
3.06675, 3.38925, 3.68425, 0.6335, 0.681, 0.8515, 1.0125,
1.2365, 1.44175, 1.613, 1.8165, 1.99725, 0.82575, 0.83625,
1.03175, 1.06925, 1.3575, 1.549, 1.7125, 1.91125, 2.06925,
2.17725, 2.33825, 1.3015, 1.4475, 1.61525, 2.0795, 2.39475,
2.6085, 2.9635, 0.73725, 0.812, 0.9935, 1.1605, 1.361, 1.60875,
1.78375, 2.08475), PETCO2 = c(39.2925, 39.62, 41.41, 42.015,
41.8425, 41.095, 40.7275, 31.2275, 32.1025, 33.665, 34.5325,
34.4225, 33.18, 32.645, 30.9, 37.6425, 39.2175, 41.215, 41.705,
41.1425, 38.8925, 40.755, 41.4675, 42.2775, 42.645, 43.36,
44.265, 44.64, 44.3425, 44.73, 43.8825, 41.6025, 40.04, 37.7175,
38.58, 39.295, 40.565, 41.3525, 42.7575, 42.5025, 42.0975,
41.8375, 41.8625, 42.1025, 42.9525, 44.08, 43.635, 43.58,
43.58, 43.2775, 44.505, 42.3325, 40.055, 37.22, 38.3375,
40.1875, 41.5175, 41.4125, 39.1475, 33.0275, 40.1275, 42.4,
43.7575, 45.1775, 47.385, 38.7075, 39.6275, 40.7, 41.605,
42.6325, 43.875, 43.4775, 42.375, 39.3975, 37.6425, 34.855,
43.31, 43.7425, 44.8275, 45.235, 46.625, 47.3825, 46.8925,
46.1575, 46.555, 45.0225, 44.5875, 43.04, 39.5725, 33.09,
33.9525, 34.9375, 34.125, 34.8425, 35.015, 34.865, 33.06,
40.8075, 42.885, 44.5, 44.9725, 40.36, 39.3375, 37.555, 34.8275,
38.86, 38.81, 39.1625, 39.23, 38.3625, 37.81, 36.0175, 45.425,
46.195, 46.755, 47.72, 48.4425, 49.245, 50.5075, 51.205,
52.215, 50.62, 47.3, 43.105, 41.2125, 40.91, 40.5525, 39.59,
37.3925, 42.4, 43.255, 43.815, 45.2925, 45.66, 47.4375, 46.5525,
41.6525, 36.2, 36.105, 36.7325, 36.7875, 37.8275, 37.8575,
36.25, 35.1775, 35.7225, 35.6725, 36.57, 37.0425, 37.7075,
37.8, 38.7675, 38.34, 38.4275, 37.645, 39.295, 40.4075, 41.625,
42.225, 42.745, 43.565, 43.5925, 42.795, 41.485, 40.585,
38.205, 36.6025, 33.9875, 36.475, 37.5425, 37.985, 38.06,
37.615, 35.725, 34.4475, 33.0825, 38.9175, 39.0425, 40.9325,
42.0775, 41.08, 40.9375, 39.445, 38.72, 38.1025, 36.835,
35.85, 40.51, 42.14, 42.9625, 42.7375, 40.8275, 40.325, 39.195,
38.125, 38.5625, 39.1325, 39.275, 39.8375, 38.88, 39.085,
38.45), percent_absVO2 = c(49.3406776831915, 54.7320981472208,
63.5119345685195, 71.791019863128, 80.0534134535136, 89.8514438324153,
100, 41.3669064748201, 46.8251485767907, 53.4876446668752,
61.588989677823, 71.9111667187989, 84.2352205192368, 90.7725993118549,
100, 48.8708399366086, 56.8145800316957, 68.3637083993661,
78.1101426307448, 89.540412044374, 100, 36.1571516267649,
38.7047268262738, 44.2500511561285, 50.9105790873747, 53.3967669326785,
60.3028442807448, 68.7129118068345, 73.2862696951095, 80.2128094945774,
83.5072641702476, 95.9382033967669, 100, 49.2621076699231,
54.9573893161505, 61.8166701309499, 73.0617335273332, 78.4452296819788,
82.8725836624402, 92.6418623986697, 100, 45.3787693062025,
50.4658004412846, 54.7560676636431, 58.7031135082128, 61.1792105908311,
69.9313557244423, 75.6190242706546, 78.1686687913704, 83.1944103947046,
87.0801667075264, 95.3787693062025, 100, 40.3316999623068,
44.0633245382586, 57.9532604598568, 74.0294006784772, 82.9061439879382,
94.9114210327931, 100, 61.4367938605841, 74.0353869111064,
78.7465359198465, 86.1863142187167, 100, 34.7511100724468,
35.7910726805328, 44.905351717691, 51.2970320168264, 57.0109838747371,
62.8768403832671, 70.4019630754849, 78.3126898808133, 87.0881046973592,
95.653189997663, 100, 24.7776879547292, 34.569523039612,
39.6119644300728, 43.4923201293452, 52.021018593371, 59.0743734842361,
64.3997574777688, 68.57316087308, 76.1519805982215, 80.7599029911075,
86.6006467259499, 93.4822150363783, 100, 54.0860573715811,
53.6024016010674, 57.6717811874583, 71.7811874583056, 74.6497665110073,
83.7224816544363, 89.2761841227485, 100, 53.017170891251,
57.988552739166, 67.8495502861815, 74.1782502044154, 82.8618152085037,
89.1414554374489, 94.5543744889616, 100, 55.995670995671,
57.5324675324675, 64.2424242424242, 70.7142857142857, 85.1082251082251,
92.5324675324675, 100, 50.0610649731314, 50.4885197850513,
53.505129457743, 59.4406448461163, 62.6770884220811, 67.9164631167562,
70.2491450903761, 76.9418661455789, 79.6653639472399, 87.9579872984856,
95.5422569614069, 100, 71.3170486023672, 72.979098463863,
83.2787710904054, 93.0747922437673, 100, 41.6474565852159,
42.8000614722606, 49.5466420777624, 58.9057937605656, 66.0365759950822,
76.932534193945, 84.4167819271554, 100, 47.9377549478773,
50.8233872186131, 58.7853150022662, 63.4083698443874, 70.6602205771265,
85.3452183109231, 92.5366369542227, 100, 45.2115812917595,
47.6455615653834, 54.2316258351893, 58.7814190264079, 67.1810372255807,
73.4966592427617, 78.1419026407891, 85.0302258988228, 92.9366846961502,
100, 32.9242861726019, 39.4610202117421, 42.5409047160731,
50.5373756817453, 55.2614693615656, 61.1806223933269, 68.599615014437,
75.4491498235483, 80.9512351620148, 88.3942893808149, 94.7786333012512,
100, 41.8658220558595, 43.8093866973798, 53.7575583069392,
60.7255974661676, 70.7745465015837, 78.9375179959689, 83.3141376331702,
91.693060754391, 100, 47.4348291870015, 47.1848589453637,
57.0884418521605, 57.4336388525176, 68.0633257945483, 72.5746934888704,
75.5624330436853, 81.4545887394358, 89.7155100583264, 94.0602309248899,
100, 54.7165775401069, 60.6310160427808, 66.1818181818182,
77.668449197861, 84.5347593582888, 92.5561497326203, 100,
46.2385583524027, 50.0143020594966, 57.9948512585812, 65.6464530892448,
73.5697940503432, 81.9221967963387, 89.2448512585812, 100
), percent_power = c(21.7391304347826, 34.7826086956522,
47.8260869565217, 60.8695652173913, 73.9130434782609, 86.9565217391304,
100, 19.2307692307692, 30.7692307692308, 42.3076923076923,
53.8461538461538, 65.3846153846154, 76.9230769230769, 88.4615384615385,
100, 25, 40, 55, 70, 85, 100, 13.1578947368421, 21.0526315789474,
28.9473684210526, 36.8421052631579, 44.7368421052632, 52.6315789473684,
60.5263157894737, 68.4210526315789, 76.3157894736842, 84.2105263157895,
92.1052631578947, 100, 22.2222222222222, 33.3333333333333,
44.4444444444444, 55.5555555555556, 66.6666666666667, 77.7777777777778,
88.8888888888889, 100, 15.3846153846154, 23.0769230769231,
30.7692307692308, 38.4615384615385, 46.1538461538462, 53.8461538461538,
61.5384615384615, 69.2307692307692, 76.9230769230769, 84.6153846153846,
92.3076923076923, 100, 21.7391304347826, 34.7826086956522,
47.8260869565217, 60.8695652173913, 73.9130434782609, 86.9565217391304,
100, 29.4117647058824, 47.0588235294118, 64.7058823529412,
82.3529411764706, 100, 14.2857142857143, 22.8571428571429,
31.4285714285714, 40, 48.5714285714286, 57.1428571428571,
65.7142857142857, 74.2857142857143, 82.8571428571429, 91.4285714285714,
100, 12.1951219512195, 19.5121951219512, 26.8292682926829,
34.1463414634146, 41.4634146341463, 48.780487804878, 56.0975609756098,
63.4146341463415, 70.7317073170732, 78.0487804878049, 85.3658536585366,
92.6829268292683, 100, 22.2222222222222, 33.3333333333333,
44.4444444444444, 55.5555555555556, 66.6666666666667, 77.7777777777778,
88.8888888888889, 100, 41.6666666666667, 50, 58.3333333333333,
66.6666666666667, 75, 83.3333333333333, 91.6666666666667,
100, 25, 37.5, 50, 62.5, 75, 87.5, 100, 15.3846153846154,
23.0769230769231, 30.7692307692308, 38.4615384615385, 46.1538461538462,
53.8461538461538, 61.5384615384615, 69.2307692307692, 76.9230769230769,
84.6153846153846, 92.3076923076923, 100, 33.3333333333333,
50, 66.6666666666667, 83.3333333333333, 100, 19.2307692307692,
30.7692307692308, 42.3076923076923, 53.8461538461538, 65.3846153846154,
76.9230769230769, 88.4615384615385, 100, 19.2307692307692,
30.7692307692308, 42.3076923076923, 53.8461538461538, 65.3846153846154,
76.9230769230769, 88.4615384615385, 100, 18.1818181818182,
27.2727272727273, 36.3636363636364, 45.4545454545455, 54.5454545454545,
63.6363636363636, 72.7272727272727, 81.8181818181818, 90.9090909090909,
100, 15.3846153846154, 23.0769230769231, 30.7692307692308,
38.4615384615385, 46.1538461538462, 53.8461538461538, 61.5384615384615,
69.2307692307692, 76.9230769230769, 84.6153846153846, 92.3076923076923,
100, 17.2413793103448, 27.5862068965517, 37.9310344827586,
48.2758620689655, 58.6206896551724, 68.9655172413793, 79.3103448275862,
89.6551724137931, 100, 14.2857142857143, 22.8571428571429,
31.4285714285714, 40, 48.5714285714286, 57.1428571428571,
65.7142857142857, 74.2857142857143, 82.8571428571429, 91.4285714285714,
100, 33.3333333333333, 44.4444444444444, 55.5555555555556,
66.6666666666667, 77.7777777777778, 88.8888888888889, 100,
19.2307692307692, 30.7692307692308, 42.3076923076923, 53.8461538461538,
65.3846153846154, 76.9230769230769, 88.4615384615385, 100
), relVO2 = c(8.79761904761905, 9.75892857142857, 11.3244047619048,
12.8005952380952, 14.2738095238095, 16.0208333333333, 17.8303571428571,
7.73391812865497, 8.75438596491228, 10, 11.5146198830409,
13.4444444444444, 15.7485380116959, 16.9707602339181, 18.6959064327485,
10.0941080196399, 11.7348608837971, 14.120294599018, 16.133387888707,
18.494271685761, 20.6546644844517, 12.6214285714286, 13.5107142857143,
15.4464285714286, 17.7714285714286, 18.6392857142857, 21.05,
23.9857142857143, 25.5821428571429, 28, 29.15, 33.4892857142857,
34.9071428571429, 9.63414634146342, 10.7479674796748, 12.0894308943089,
14.2886178861789, 15.3414634146341, 16.2073170731707, 18.1178861788618,
19.5569105691057, 8.04782608695652, 8.95, 9.71086956521739,
10.4108695652174, 10.85, 12.4021739130435, 13.4108695652174,
13.8630434782609, 14.754347826087, 15.4434782608696, 16.9152173913043,
17.7347826086956, 10.0187265917603, 10.9456928838951, 14.3960674157303,
18.3895131086142, 20.5945692883895, 23.5767790262172, 24.8408239700375,
10.7859281437126, 12.997754491018, 13.8248502994012, 15.1309880239521,
17.5561377245509, 9.56885456885457, 9.85521235521236, 12.3648648648649,
14.1248391248391, 15.6981981981982, 17.3133848133848, 19.3854568854569,
21.5637065637066, 23.9800514800515, 26.3384813384813, 27.5353925353925,
8.01307189542484, 11.1797385620915, 12.8104575163399, 14.0653594771242,
16.8235294117647, 19.1045751633987, 20.8267973856209, 22.1764705882353,
24.6274509803922, 26.1176470588235, 28.0065359477124, 30.2320261437909,
32.3398692810458, 10.1853015075377, 10.0942211055276, 10.8605527638191,
13.5175879396985, 14.0577889447236, 15.7663316582915, 16.8121859296482,
18.8316582914573, 10.6225425950197, 11.6186107470511, 13.5943643512451,
14.8623853211009, 16.6022280471822, 17.8604193971166, 18.9449541284404,
20.0360419397117, 9.98070987654321, 10.2546296296296, 11.4506172839506,
12.6041666666667, 15.1697530864198, 16.4930555555556, 17.8240740740741,
7.9437984496124, 8.01162790697674, 8.49031007751938, 9.43217054263566,
9.94573643410853, 10.7771317829457, 11.1472868217054, 12.2093023255814,
12.6414728682171, 13.9573643410853, 15.1608527131783, 15.8682170542636,
10.0854700854701, 10.3205128205128, 11.7770655270655, 13.1623931623932,
14.1417378917379, 10.2651515151515, 10.5492424242424, 12.2121212121212,
14.5189393939394, 16.2765151515152, 18.9621212121212, 20.8068181818182,
24.6477272727273, 11.109943977591, 11.7787114845938, 13.6239495798319,
14.6953781512605, 16.3760504201681, 19.7794117647059, 21.4460784313725,
23.1757703081232, 8.58091787439614, 9.04287439613527, 10.2928743961353,
11.1564009661836, 12.7506038647343, 13.9492753623188, 14.8309178743961,
16.1382850241546, 17.6388888888889, 18.9794685990338, 14.7027220630372,
17.621776504298, 18.9971346704871, 22.5680515759312, 24.6776504297994,
27.3209169054441, 30.6339541547278, 33.6926934097421, 36.1497134670487,
39.4734957020057, 42.3244985673352, 44.6561604584527, 9.34447300771208,
9.77827763496144, 11.9987146529563, 13.5539845758355, 15.7969151670951,
17.6188946015424, 18.5957583547558, 20.4659383033419, 22.3200514138817,
9.3107476635514, 9.26168224299065, 11.2056074766355, 11.2733644859813,
13.3598130841121, 14.2453271028037, 14.8317757009346, 15.9883177570093,
17.6098130841122, 18.4626168224299, 19.6285046728972, 21.0016420361248,
23.2717569786535, 25.4022988505747, 29.8111658456486, 32.4466338259442,
35.5254515599343, 38.3825944170772, 10.4560155239327, 11.3098318240621,
13.114489003881, 14.8447606727038, 16.6364812419146, 18.5252263906856,
20.1811125485123, 22.6131953428202), percent_relVO2 = c(49.3406776831915,
54.7320981472208, 63.5119345685195, 71.791019863128, 80.0534134535136,
89.8514438324153, 100, 41.3669064748201, 46.8251485767907,
53.4876446668752, 61.588989677823, 71.9111667187989, 84.2352205192368,
90.7725993118548, 100, 48.8708399366086, 56.8145800316957,
68.3637083993661, 78.1101426307448, 89.540412044374, 100,
36.1571516267649, 38.7047268262738, 44.2500511561285, 50.9105790873747,
53.3967669326785, 60.3028442807448, 68.7129118068345, 73.2862696951095,
80.2128094945775, 83.5072641702476, 95.9382033967669, 100,
49.2621076699231, 54.9573893161505, 61.8166701309499, 73.0617335273332,
78.4452296819788, 82.8725836624402, 92.6418623986697, 100,
45.3787693062025, 50.4658004412846, 54.7560676636431, 58.7031135082128,
61.1792105908311, 69.9313557244423, 75.6190242706546, 78.1686687913705,
83.1944103947046, 87.0801667075264, 95.3787693062025, 100,
40.3316999623068, 44.0633245382586, 57.9532604598568, 74.0294006784772,
82.9061439879382, 94.9114210327931, 100, 61.4367938605841,
74.0353869111064, 78.7465359198465, 86.1863142187167, 100,
34.7511100724468, 35.7910726805328, 44.905351717691, 51.2970320168264,
57.0109838747371, 62.8768403832671, 70.4019630754849, 78.3126898808133,
87.0881046973592, 95.653189997663, 100, 24.7776879547292,
34.569523039612, 39.6119644300727, 43.4923201293452, 52.021018593371,
59.074373484236, 64.3997574777688, 68.57316087308, 76.1519805982215,
80.7599029911075, 86.6006467259499, 93.4822150363783, 100,
54.086057371581, 53.6024016010674, 57.6717811874583, 71.7811874583055,
74.6497665110073, 83.7224816544363, 89.2761841227485, 100,
53.017170891251, 57.988552739166, 67.8495502861815, 74.1782502044154,
82.8618152085037, 89.1414554374489, 94.5543744889616, 100,
55.995670995671, 57.5324675324675, 64.2424242424242, 70.7142857142857,
85.1082251082251, 92.5324675324675, 100, 50.0610649731314,
50.4885197850513, 53.505129457743, 59.4406448461163, 62.6770884220811,
67.9164631167562, 70.2491450903761, 76.9418661455789, 79.6653639472399,
87.9579872984856, 95.5422569614069, 100, 71.3170486023672,
72.979098463863, 83.2787710904054, 93.0747922437673, 100,
41.6474565852159, 42.8000614722606, 49.5466420777624, 58.9057937605656,
66.0365759950822, 76.932534193945, 84.4167819271554, 100,
47.9377549478773, 50.8233872186131, 58.7853150022662, 63.4083698443874,
70.6602205771264, 85.3452183109231, 92.5366369542227, 100,
45.2115812917595, 47.6455615653834, 54.2316258351893, 58.7814190264079,
67.1810372255807, 73.4966592427617, 78.1419026407891, 85.0302258988228,
92.9366846961502, 100, 32.9242861726019, 39.4610202117421,
42.5409047160732, 50.5373756817453, 55.2614693615656, 61.1806223933269,
68.599615014437, 75.4491498235483, 80.9512351620148, 88.3942893808149,
94.7786333012512, 100, 41.8658220558595, 43.8093866973798,
53.7575583069392, 60.7255974661676, 70.7745465015836, 78.9375179959689,
83.3141376331702, 91.693060754391, 100, 47.4348291870016,
47.1848589453637, 57.0884418521605, 57.4336388525176, 68.0633257945483,
72.5746934888704, 75.5624330436853, 81.4545887394358, 89.7155100583264,
94.0602309248899, 100, 54.7165775401069, 60.6310160427808,
66.1818181818182, 77.668449197861, 84.5347593582888, 92.5561497326203,
100, 46.2385583524027, 50.0143020594966, 57.9948512585812,
65.6464530892449, 73.5697940503433, 81.9221967963387, 89.2448512585812,
100), group = c("CAD", "CAD", "CAD", "CAD", "CAD", "CAD",
"CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD",
"Healthy", "Healthy", "Healthy", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "CAD", "CAD", "CAD", "CAD", "CAD",
"CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy", "Healthy", "CAD",
"CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD",
"CAD", "Healthy", "Healthy", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "CAD", "CAD", "CAD", "CAD", "CAD",
"CAD", "CAD", "CAD", "Healthy", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy", "CAD", "CAD",
"CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD",
"CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD",
"CAD", "CAD", "CAD", "CAD", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy", "Healthy", "CAD",
"CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD",
"CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy", "Healthy", "CAD",
"CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD",
"CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD", "CAD",
"CAD", "Healthy", "Healthy", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy", "Healthy", "Healthy",
"Healthy", "Healthy", "Healthy", "Healthy")), class = c("grouped_df",
"tbl_df", "tbl", "data.frame"), row.names = c(NA, -202L), groups = structure(list(
id = c("AC12-PRD-C1", "AL13-PRD-C1", "BM06-PRD-S1", "CB19-PRD-S1",
"CC14-PRD-S1", "DA03-PRD-C1", "DA24-PRD-S1", "DB22-PRD-S1",
"DB42-PRD-C1", "DL18-PRD-S1", "DR15-PRD-C1", "DT08-PRD-S1",
"FB44-PRD-C1", "FG33-PRD-C1", "FG40-PRD-C1", "GC03-PRD-S1",
"GG30-PRD-C1", "GH05-PRD-C1", "GL05-PRD-S1", "GP22-PRD-C1",
"GP28-PRD-C1", "GT04-PRD-S1", "GT21-PRD-S1"), .rows = structure(list(
1:7, 8:15, 16:21, 22:33, 34:41, 42:53, 54:60, 61:65,
66:76, 77:89, 90:97, 98:105, 106:112, 113:124, 125:129,
130:137, 138:145, 146:155, 156:167, 168:176, 177:187,
188:194, 195:202), ptype = integer(0), class = c("vctrs_list_of",
"vctrs_vctr", "list"))), class = c("tbl_df", "tbl", "data.frame"
), row.names = c(NA, -23L), .drop = TRUE))
I created a plot according the 2 groups CAD and Healthy with the variable VE2:
dftest %>%
filter(percent_power < 100) %>%
filter(group == "CAD") %>%
ggscatter(x = "percent_absVO2", y = "VE", color = "group")
stat_cor(aes(color = group), label.x = 20, label.y = c(100,125))
stat_regline_equation(label.x = 20, label.y = c(90,115),
formula = y ~ poly(x, 2),
aes(color = group, label = paste(..eq.label.., ..adj.rr.label.., sep = "~~~~")),)
geom_smooth(aes(colour=group), method = "lm", formula = y ~ poly(x, 2))
xlab("Percentage of VO2 (%)")
ylab(expression(paste("VE"," (L/min)")))
Question:
I would like to get a plot like below and determine the breaking point defined as the first inflection point in VE.
In other words, from which percentage of VO2 value the slope after the breaking point starts to be significantly different compared to the slope before the breaking point ?
CodePudding user response：
Is that what you want?
library(ggplot2)
library(dplyr)
library(segmented)
df <- dftest %>%
filter(percent_power < 100) %>%
filter(group == "CAD")
df$polyvol2 <- poly(df$percent_absVO2, 2)
out.lm <- lm(VE ~ percent_absVO2 , data=df)
o <- segmented(out.lm)
breakpt <- o$psi[2]
df$group <- 42
df[df$percent_absVO2 <= o$psi[2],]$group <- 1
df[df$percent_absVO2 >= o$psi[2],]$group <- 2
df$group <- as.factor(df$group)
ggplot(df, aes(x = percent_absVO2, y = VE))
geom_point(aes(colour = group))
geom_smooth(data = df[df$group == 1,], method = "lm", formula = y ~ x)
geom_smooth(data = df[df$group == 2,], method = "lm", formula = y ~ x)
CodePudding user response：
Minimize AIC, BIC, deviance, or similar of a linear model using optimize where the two segments are defined as two functions f1 and f2.
f1 <- \(x, bp) ifelse(x < bp, bp - x, 0)
f2 <- \(x, bp) ifelse(x < bp, 0, x - bp)
main <- \(bp) BIC(lm(VE ~ f1(percent_absVO2, bp) f2(percent_absVO2, bp), dat))
(bp <- optimize(main, interval=range(dat$percent_absVO2))$min) ## break point
# [1] 71.31706
After that do a piecewise regression
a <- lm(VE ~ percent_absVO2, data=dat, subset=percent_absVO2 <= bp)
b <- lm(VE ~ percent_absVO2, data=dat, subset=percent_absVO2 > bp)
to get the intercepts and slopes.
a$coefficients
# (Intercept) percent_absVO2
# 5.1246310 0.4216362
b$coefficients
# (Intercept) percent_absVO2
# -35.078900 1.004829
To plot the segment lines, predict the two model fits along their respective predictor sequences (i.e. x-values, here percent_absVO2).
png('plot.png')
par(mar=c(5, 5, 4, 2) .1)
plot(VE ~ percent_absVO2, data=dat, main='Break point',
xlab=bquote(VO[2](Ml*'·'*min^-1)), ylab=bquote(V[E](L*'·'*min^-1)))
ola <- 0; olb <- 0 ## set values to something other than zero to make the lines overlap
lines(1:(bp ola),
predict(a, newdata=list(percent_absVO2=1:(bp ola))), lwd=2)
lines((bp - olb):max(dat$percent_absVO2),
predict(b, newdata=list(percent_absVO2=(bp - olb):max(dat$percent_absVO2))), lwd=2)
abline(v=bp, lty=2, col=4)
mtext(round(bp, 3), 3, 0, at=bp, cex=.8, col=4)
dev.off()
The confidence interval of the break point can be bootstrapped.
set.seed(42)
bs <- replicate(1e3, {
X <- dat[sample(nrow(dat), replace=TRUE), ]
main <- \(bp) BIC(lm(VE ~ f1(percent_absVO2, bp) f2(percent_absVO2, bp), X))
optimize(main, interval=range(X$percent_absVO2))$min
})
quantile(bs, c(.025, .975)) ## 95% CI
# 2.5% 97.5%
# 51.15653 93.07481
See also this Q&A on CrossValidated which does similar with a mixed model.
Data:
dat <- subset(dftest, percent_power < 100 & group == 'CAD')