Fitting a quadratic regression in R

Question

I am trying to fit a quadratic regression model in R. Here is an example of my dataframe:

dat <- structure(list(heading = c(1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L), startingpos = c(8L, 
4L, 0L, 0L, 8L, 0L, 0L, 0L, 0L, 4L, 4L, 8L, 4L, 8L, 8L, 4L, 0L, 
0L, 0L, 4L, 8L, 4L, 4L, 4L, 8L, 8L, 0L, 8L, 8L, 0L, 4L, 4L, 8L, 
4L, 0L, 8L, 8L, 0L, 0L, 0L, 8L, 4L, 4L, 8L, 0L, 4L, 8L, 4L, 8L, 
4L, 4L, 0L, 4L, 0L, 0L, 8L, 4L, 0L, 0L, 0L, 8L, 4L, 8L, 0L, 8L, 
8L, 4L, 8L, 4L, 0L, 0L, 4L, 0L, 4L, 0L, 4L, 4L, 8L, 4L, 0L, 8L, 
0L, 0L, 4L, 0L, 0L, 8L, 0L, 8L, 8L, 8L, 8L, 8L, 0L, 4L, 4L, 4L, 
0L, 0L, 4L, 4L, 8L, 0L, 8L, 4L, 0L, 8L, 8L, 4L, 4L, 0L, 8L, 4L, 
8L, 4L, 0L, 8L, 4L, 4L, 0L, 0L, 0L, 4L, 8L, 8L, 8L, 8L, 4L, 0L, 
0L, 8L, 0L, 0L, 0L, 4L, 4L, 0L, 0L, 8L, 4L, 4L, 8L, 8L, 4L, 8L, 
0L, 8L, 8L, 8L, 4L, 4L, 0L, 4L, 0L, 4L, 8L, 8L, 8L, 8L, 8L, 0L, 
4L, 4L, 0L, 4L, 8L, 0L, 4L, 0L, 0L, 8L, 8L, 8L, 4L, 4L, 4L, 4L, 
8L, 0L, 4L, 8L, 8L, 4L, 0L, 8L, 4L, 0L, 0L, 0L, 4L, 8L, 8L, 0L, 
8L, 8L, 0L, 8L, 8L, 8L, 4L, 0L, 4L, 8L, 0L, 4L, 4L, 4L, 0L, 4L, 
0L, 8L, 0L, 4L, 4L, 0L, 8L, 0L, 0L, 4L, 4L, 4L, 0L, 8L, 8L, 0L, 
8L, 8L, 8L, 0L, 8L, 4L, 4L, 8L, 0L, 0L, 0L, 0L, 8L, 4L, 8L, 8L, 
0L, 8L, 8L, 4L, 4L, 8L, 8L, 0L, 4L, 4L, 4L, 0L, 4L, 4L, 0L, 0L, 
4L, 0L, 4L, 8L, 8L, 0L, 8L, 0L, 4L, 4L, 4L, 0L, 8L, 4L, 4L, 8L, 
0L, 8L, 4L, 8L, 0L, 8L, 8L, 4L, 4L, 8L, 4L, 4L, 8L, 0L, 8L, 0L, 
0L, 4L, 4L, 8L, 8L, 8L, 0L, 0L, 4L, 0L, 8L, 8L, 4L, 0L, 4L, 0L, 
0L, 8L, 4L, 8L, 0L, 8L, 4L, 8L, 4L, 4L, 8L, 4L, 0L, 4L, 4L, 8L, 
0L, 8L, 8L, 8L, 4L, 0L, 8L, 0L, 8L, 0L, 8L, 4L, 0L, 4L, 0L, 4L, 
4L, 4L, 4L, 0L, 0L, 8L, 4L, 0L, 0L, 4L, 8L, 4L, 0L, 8L, 4L, 0L, 
8L, 0L, 8L, 8L, 4L, 8L, 8L, 0L, 8L, 0L, 4L, 0L, 8L, 0L, 4L, 4L, 
0L, 4L, 8L, 4L, 8L, 4L, 4L, 0L, 4L, 8L, 0L, 4L, 8L, 8L, 0L, 0L, 
4L, 0L, 4L, 0L, 0L, 8L, 8L, 8L, 8L, 8L, 4L, 8L, 0L, 0L, 8L, 0L, 
4L, 0L, 8L, 4L, 4L, 4L, 4L, 8L, 8L, 8L, 0L, 8L, 4L, 4L, 8L, 4L, 
0L, 4L, 4L, 0L, 0L, 0L, 0L, 8L, 0L, 8L, 4L, 4L, 4L, 8L, 4L, 8L, 
0L, 0L, 0L, 4L, 0L, 8L, 0L, 8L, 0L, 4L, 8L, 8L, 0L, 8L, 0L, 8L, 
0L, 8L, 8L, 4L, 0L, 4L, 4L, 4L, 8L, 4L, 4L, 0L, 8L, 4L, 8L, 8L, 
0L, 0L, 0L, 0L, 4L, 8L, 4L, 4L, 4L, 4L, 8L, 4L, 0L, 8L, 4L, 0L, 
4L, 0L, 8L, 0L, 4L, 4L, 4L, 8L, 8L, 0L, 8L, 8L, 0L, 0L, 4L, 0L, 
4L, 4L, 8L), FirstSteeringTime = c(0.4333325988244, 0.33254630198401, 
0.400026140468498, 0.933583287728609, 0.366613128009007, 0.43335584150401, 
0.516691111726999, 0.383459543097199, 0.783127987777988, 0.283220203865, 
0.416811581253, 0.400114583392991, 0.416630167609981, 0.500080141967999, 
0.466509864102989, 0.183366330894984, 0.69996205708, 0.833361757822985, 
0.516727937792041, 0.38330197583997, 0.416698386385008, 0.46657234767099, 
0.433382404566999, 0.416618697195986, 0.399949469809997, 0.733300000000042, 
0.416799999999967, 0.36669999999998, 0.433299999999974, 0.616399999999999, 
0.398599999999988, 0.249961740134204, 0.466699999999946, 0.466800000000035, 
0.7166, 0.483280000000001, 0.383409999999998, 0.399969999999996, 
0.699960000000004, 0.61666, 0.582999999999998, 0.4495, 0.367099999999994, 
0.416699999999992, 0.399999999999977, 0.583399999999983, 0.516499999999979, 
0.449899999999985, 0.383399999999995, 0.282800000000009, 0.566699999999997, 
0.466700000000003, 0.483299999999986, 0.533299999999997, 0.433199999999999, 
0.61650000000003, 0.550000000000011, 0.683300000000031, 0.38330000000002, 
0.449999999999989, 0.433400000000006, 0.449999999999989, 0.399799999999999, 
0.583399999999997, 0.383279999999999, 0.450000000000003, 0.383200000000002, 
0.383399999999995, 0.549999999999983, 0.383299999999991, 0.716399999999993, 
0.566499999999991, 0.400000000000006, 0.616600000000005, 0.733399999999989, 
0.449899999999985, 0.566699999999997, 0.5, 0.600099999999998, 
0.5, 0.483439999999998, 0.483400000000017, 0.54989999999998, 
0.666699999999992, 0.716499999999996, 0.583300000000001, 0.433199999999999, 
0.633299999999963, 0.516599999999983, 0.499859999999998, 0.483400000000017, 
0.500099999999975, 0.416699999999992, 0.333399999999983, 0.483300000000042, 
0.550000000000011, 0.383319999999998, 0.599970000000001, 0.5, 
0.399969999999996, 0.483379999999997, 0.399969999999996, 0.683300000000003, 
0.599999999999994, 0.583400000000012, 0.566699999999997, 0.516099999999994, 
0.38330000000002, 0.383399999999995, 0.466100000000012, 0.800000000000011, 
0.616700000000009, 0.16670000000002, 0.599999999999994, 0.533299999999997, 
0.550100000000043, 0.38330000000002, 0.400000000000034, 0.466700000000003, 
0.666799999999967, 0.583399999999983, 0.716600000000028, 0.599899999999991, 
0.466600000000028, 0.599999999999966, 0.5, 0.449900000000014, 
0.550099999999986, 0.5, 0.583329999999997, 0.333350000000003, 
0.39996, 0.433340000000001, 0.450100000000006, 0.466790000000003, 
0.566599999999994, 0.666700000000006, 0.566099999999992, 0.383399999999995, 
0.433290000000003, 0.41640000000001, 0.350100000000026, 0.566699999999997, 
0.316900000000032, 0.400109999999998, 0.399900000000002, 0.333300000000008, 
0.45010000000002, 0.566800000000001, 0.433499999999981, 0.483299999999986, 
0.366199999999992, 0.433399999999949, 0.699999999999989, 0.45010000000002, 
0.333499999999958, 0.38344, 0.266697979166521, 0.566645063859099, 
0.483681528935506, 0.516593159768306, 0.499657544687992, 0.566733809440009, 
0.533366376472998, 0.549978704078995, 0.700037822520983, 0.616626333711991, 
0.416589265829003, 0.466635434660986, 0.549884525094996, 0.400077758033007, 
1.70000789965499, 0.600099999999998, 0.599900000000019, 0.4666, 
0.583300000000008, 0.483400000000017, 0.483400000000017, 0.516699999999958, 
0.433400000000006, 0.566800000000001, 0.483400000000017, 0.533299999999997, 
0.566700000000026, 0.500099999999975, 0.533400000000029, 0.566700000000026, 
0.466700000000003, 0.666700000000048, 0.61669999999998, 0.44997, 
0.5167, 0.732889999999998, 0.516690000000001, 0.516800000000003, 
0.700000000000003, 0.433400000000006, 0.4161, 0.666599999999988, 
0.583300000000008, 0.466699999999999, 0.433300000000003, 0.433500000000009, 
0.466499999999996, 0.483399999999989, 0.516600000000011, 0.516899999999993, 
0.61669999999998, 0.599999999999994, 0.533500000000004, 0.399999999999999, 
0.550000000000011, 0.550099999999986, 0.566700000000026, 0.5, 
0.566800000000001, 0.816599999999994, 0.516599999999983, 0.399499999999989, 
0.532940000000004, 0.583300000000008, 0.5, 0.466599999999971, 
0.5, 0.650000000000034, 0.383309999999994, 0.533330000000007, 
0.483360000000005, 0.616789999999995, 0.5, 0.400099999999995, 
0.5501, 0.599599999999995, 0.433199999999999, 0.483400000000017, 
0.800000000000011, 0.616669999999999, 0.316700000000026, 0.549800000000005, 
0.450000000000017, 0.383400000000023, 0.4666, 0.4666, 0.35004, 
0.4666, 0.483400000000017, 0.499900000000025, 0.416679999999999, 
0.533400000000029, 0.416800000000023, 0.566599999999994, 0.450000000000045, 
0.516599999999983, 0.550000000000011, 0.816700000000026, 0.400010000000002, 
0.316699999999969, 0.516399999999976, 0.433259999999997, 0.733278, 
0.55003, 0.483340000000013, 0.783349999999999, 0.533460000000005, 
0.566699999999997, 0.716700000000003, 0.716700000000003, 0.516599999999983, 
0.433399999999978, 0.533299999999997, 0.666699999999992, 0.433299999999974, 
0.516699999999986, 0.466639999999998, 0.650000000000006, 0.566300000000012, 
0.416699999999992, 0.416600000000017, 0.349899999999991, 0.449999999999989, 
0.483290000000004, 0.733200000000011, 0.466700000000003, 0.583300000000008, 
0.38330000000002, 0.483299999999986, 0.816600000000001, 0.433400000000006, 
0.550000000000011, 0.616100000000017, 0.483299999999986, 0.583330000000004, 
0.449950000000001, 0.566629999999989, 0.483359999999999, 0.816699999999997, 
0.783299999999997, 0.5334, 0.299900000000008, 0.400100000000009, 
0.600099999999998, 0.500100000000003, 0.449900000000014, 0.63330000000002, 
0.533199999999994, 0.683399999999978, 0.516799999999989, 0.566599999999994, 
0.650100000000009, 0.883369999999999, 0.399999999999977, 0.433300000000031, 
0.54989999999998, 0.45010000000002, 0.466600000000028, 0.383399999999995, 
0.63330000000002, 0.549909999999997, 0.233300000000042, 0.633299999999963, 
0.466700000000003, 0.632779999999997, 0.48338460869733, 0.583214904951404, 
0.466673227015406, 0.416451543573601, 0.36661533093141, 0.583078163940002, 
0.616621669918999, 0.533258553406995, 0.500063088017015, 0.433389619515992, 
0.549822629098003, 0.449958223055006, 0.516826992736014, 0.532235259741014, 
0.48334687662998, 0.483007589880998, 0.599507007900996, 0.549939447577998, 
0.400010323494001, 0.83332719558399, 0.466675340011989, 0.516813107334997, 
0.833300000000008, 0.533299999999997, 0.550099999999986, 0.516699999999958, 
0.466599999999971, 0.533299999999997, 0.498899999999992, 0.483300000000042, 
0.600000000000023, 0.466600000000028, 0.483399999999961, 0.466700000000003, 
0.683299999999974, 0.333349, 0.416550000000001, 0.516599999999997, 
0.566729999999993, 0.416650000000004, 0.583269999999999, 0.54965, 
0.399899999999988, 0.500099999999989, 0.682200000000009, 0.29989999999998, 
0.46669, 0.633499999999998, 0.316800000000001, 0.449999999999989, 
0.350070000000002, 0.683399999999978, 0.4666, 0.5, 0.616800000000012, 
0.449999999999989, 0.483399999999961, 0.683399999999949, 0.649999999999977, 
0.433199999999999, 0.583300000000008, 0.599899999999991, 0.516599999999983, 
0.449999999999989, 0.833399999999983, 0.599899999999991, 0.61562, 
0.600099999999998, 0.433499999999981, 0.433400000000006, 0.466679999999997, 
0.516559999999998, 0.516640000000002, 0.566780000000008, 0.533420000000007, 
0.51671, 0.4833, 0.465599999999995, 0.416700000000006, 0.449949999999998, 
0.650099999999981, 0.383299999999991, 0.416600000000017, 0.399999999999999, 
0.566800000000001, 0.800000000000011, 0.350099999999998, 0.5822, 
0.550000000000011, 0.433400000000006, 0.483200000000011, 0.366700000000037, 
0.316699999999969, 0.45010000000002, 0.45010000000002, 0.649999999999977, 
0.466769999999997, 0.516599999999983, 0.600099999999998, 0.499899999999968, 
0.549800000000005, 0.583300000000008, 0.400009999999995, 0.49995, 
0.48319, 0.516689999999997, 0.432280000000006, 0.400030000000001, 
0.500020000000006, 0.38336000000001, 0.53308, 0.533500000000004, 
0.666600000000017, 0.516700000000014, 0.366700000000009, 0.650020000000001, 
0.38330000000002, 0.616700000000009, 0.483399999999989, 0.45010000000002, 
0.66670000000002, 0.449900000000014, 0.483299999999986, 0.783299999999997, 
0.399999999999977, 0.366669999999999, 0.400100000000009, 0.800000000000011, 
0.38349999999997, 0.433369999999996, 0.399900000000002, 0.383299999999998, 
0.350099999999998, 0.450099999999964, 0.366499999999974, 0.550099999999986, 
0.416600000000017, 0.450049999999997, 0.516333, 0.449950000000001, 
0.550070000000005, 0.400019999999998, 0.383349999999993, 0.516569999999987, 
0.599890000000002, 0.533349999999999, 0.549999999999997, 0.482399999999998, 
0.416799999999995, 0.649100000000004, 0.449999999999989, 0.5334, 
0.450000000000017, 0.383399999999995, 0.515500000000003, 0.733500000000021, 
0.41670000000002, 0.75, 0.649999999999977, 0.633399999999995, 
0.51662, 0.36669999999998, 0.483299999999986, 0.466700000000003, 
0.483100000000007, 0.45010000000002, 0.416600000000017, 0.399999999999977, 
0.533299999999997, 0.533299999999997, 0.382300000000001, 0.533299999999997, 
0.516600000000039, 0.448603033908896, 0.349815655656002, 0.383307041550012, 
0.399893806548988, 0.383485741159006), pNum = c(1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 10L, 
10L, 10L, 10L, 10L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 
11L, 11L, 11L, 11L, 11L, 11L, 11L, 11L, 12L, 12L, 12L, 12L, 12L
)), row.names = c(NA, 500L), class = "data.frame")

I've read online many ways to fit a quadratic regression. Some examples include:

m1 <- glmer(formula = FirstSteeringTime ~ stats::poly(startingpos, 2) + (1 |pNum),
            family = Gamma(link = "identity"),
            data = dat)

m2 <- glmer(formula = FirstSteeringTime ~ startingpos^2 + (1 | pNum),
            family = Gamma(link = "identity"),
            data = dat)

However I am yet to find a consensus. Are these 2 models equivalent? The output I get is very similar so I imagine they are doing very similar things?

Also side note: when I run a summary on the first model (m1) I get the following output:

    AIC      BIC   logLik deviance df.resid 
  -777.2   -756.1    393.6   -787.2      495 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.7497 -0.6122 -0.1233  0.5585 10.3273 

Random effects:
 Groups   Name        Variance  Std.Dev.
 pNum     (Intercept) 0.0003794 0.01948 
 Residual             0.0547070 0.23390 
Number of obs: 500, groups:  pNum, 4

Fixed effects:
                             Estimate Std. Error t value Pr(>|z|)    
(Intercept)                   0.49196    0.03562  13.810  < 2e-16 ***
stats::poly(startingpos, 2)1 -0.79182    0.11437  -6.923 4.41e-12 ***
stats::poly(startingpos, 2)2  0.35565    0.10966   3.243  0.00118 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr) s::(,2)1
stts::(,2)1 -0.018         
stts::(,2)2  0.058 -0.103

What is the difference between stats::poly(startingpos, 2)1 and stats::poly(startingpos, 2)2?

Any help is most appreciated, thank you!

Andrew M · Accepted Answer · 2020-01-27 19:04:17Z

9

Function poly() in R is used to construct orthogonal polynomials. These are equivalent to standard polynomials but are numerically more stable. That is, the two models

m1 <- glmer(FirstSteeringTime ~ poly(startingpos, 2) + (1 | pNum), 
            family = Gamma(link = "identity"), data = data)

and

m2 <- glmer(FirstSteeringTime ~ startingpos + I(startingpos^2) + (1 | pNum), 
            family = Gamma(link = "identity"), data = data)

are equivalent, but m1 is preferable.

From both models, you will get the corresponding terms for the linear and quadratic terms for startingpos. For both simple and orthogonal polynomials, the interpretation of these coefficients is not straighforward. That is, you cannot interpret one of them in isolation. What you could perhaps look at is the magnitude and statistical significance of the second coefficient. This would tell you if you simplify the model and only use the linear term.

edited Jan 27, 2020 at 19:04

Andrew M

3,00419 silver badges25 bronze badges

answered Jan 27, 2020 at 9:57

Dimitris Rizopoulos

21.5k2 gold badges25 silver badges51 bronze badges

4

$\begingroup$ +1. Just as an observation, OPs second model will only include the linear term, afaik. Without the I(), R will ignore the ^2, I think. $\endgroup$
– COOLSerdash
Commented Jan 27, 2020 at 10:01
1

$\begingroup$ @COOL That's an excellent observation. Because ^ is one of the "symbolic" operators used by formula, it will not be interpreted numerically without the protection by I. It's not that R "ignores" it, though: it interprets it as a "crossing" of its argument with itself. That's why you get no error message or warning. $\endgroup$
– whuber ♦
Commented Jan 27, 2020 at 15:43

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