simulated_scatter <-
structure(list(group = c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 
2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 
3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 
5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 
6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 
6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 
7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 
8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 
8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 
9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 
9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 
10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 
10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 
10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 
10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 
11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 
11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 
11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 
12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 
12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 
12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 
13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 
13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 
13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 
14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 
14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 
14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 
15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 
15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 
15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 
16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 
16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 
16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 
17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 
17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 
17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 
18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 
18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 
18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 
18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 
19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 
19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 
19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20, 20, 
20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 
20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 
20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 
20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 
20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 
20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 
21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 
21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 
21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 
21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 
21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 
21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 
22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 
22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 
22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 
22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 
22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 
22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 
22, 22, 22, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 
23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 
23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 
23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 24, 
24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 
24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 
24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 
24, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 
25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 
26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 
26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 
26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 
26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 
26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 27, 27, 27, 27, 27, 
27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 
27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 
27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 
27, 27, 27, 27, 27, 27, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 
28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 
28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 
28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 
28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 29, 29, 29, 
29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 
29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 
29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 
29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 
29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 
29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 
29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 
29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 
29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 
29, 29, 29, 29, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 
30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 
30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 
30, 30, 30, 30, 30, 30), x = c(-15.743802725356129, 6.346651145010604, 
24.540011135961301, -22.020352235199375, 22.46083326745039, 16.994090171853841, 
18.818785143204966, 52.848137019603769, -24.222294714073598, 
57.823167828966291, -8.5777926724596476, -21.330212837579289, 
-7.6398301710882386, 24.337528198325089, 21.017508319634658, 
5.8473378206121431, -15.449571919968033, -5.3920127777995184, 
56.40234960125931, 22.593783063437996, -3.0899628213790677, -15.095924204758788, 
9.2769700711541763, -38.99366972099525, 0.012981398017300805, 
-8.4553978038126445, 54.300320714964066, 49.398215122829711, 
13.621410355143528, -21.513815837277626, 45.720616056428113, 
44.108424754043298, 40.01460074709783, 40.304570807724126, 4.3797226407921706, 
39.282011945021942, 58.911813628158583, 17.262316466760947, -16.316364409877188, 
42.194120618793612, 41.954726776311823, 5.7547166783430868, 72.063199903450197, 
-10.221592382126271, 27.498444633940917, -58.758235426086756, 
10.647225792067887, 53.318544718143308, 0.96698278340492294, 
-13.672488412782783, 39.985683757131191, -10.71154976155275, 
24.813808826522674, -41.329702447608597, -30.57702948169284, 
1.0328095163099249, -18.171212087145989, 60.9507154775564, 46.276072320175714, 
-9.909691315535472, 34.926099713859735, 46.300474855346479, 24.457480008528911, 
27.614896345939833, 54.753132784154694, 0.1320696154562917, 2.1786191695251738, 
47.518722506271594, -26.53721816127959, 22.144515291476424, 14.966258435741485, 
31.414211224936818, 49.798523248750428, 24.822828923735045, 23.650261394386586, 
664.59520370848657, 1842.5136124295841, 1198.1216864690741, 1308.1676953837621, 
1378.5328116353801, 1119.8041871266498, 1072.4596512686187, 756.55534020319362, 
658.7436734468771, 1791.2166005690544, 852.37157904167941, 1325.8587944067876, 
1341.3992678868092, 1534.4008811395088, 1602.6198574706464, 1262.7388199733607, 
1417.3562236261878, 714.90385125612511, 1252.4990346617967, 1416.9234465009574, 
1122.3726061844509, 926.39102374928393, 1123.466026988536, 886.82354237887625, 
901.81916944341754, 1401.1692476125706, 1698.6610863277735, 539.25289726924939, 
571.7532154629655, 740.43397596118928, 1148.9282661833861, 785.34585387944867, 
1170.1036945655298, 1122.9624404200824, 922.85095595391681, 887.62006209946469, 
655.32450234934345, 1256.638024814358, 1700.8368444596078, 926.99910521217919, 
644.8046309220324, 850.63206142413662, 1246.9754730358738, 1283.3873524313235, 
1854.9895781551988, 640.73857362385775, 1069.3277436487397, 992.87974754970514, 
673.5447123433396, 23.828846878498979, 40.28988635796803, 15.719495317143004, 
30.944849776702505, 21.134754321026069, 13.737480248489936, 35.052149863140009, 
24.423382917177793, 5.2411473325037008, 15.597536690451747, 19.791881930995604, 
2.7259971073582143, 34.164800643436621, 26.08378785807907, 16.123657231909316, 
16.217888805839099, 22.653376597084826, 12.221593619458252, 15.75004632484197, 
20.496293921220214, 20.828239903969173, 29.358394630802604, 23.153513608958828, 
18.927181108892846, 16.960995473476082, 30.229332652350276, 14.405341524279311, 
24.514004205081321, 22.42353564656576, 21.723529254184839, 27.368023994574273, 
12.899584861265453, 19.97685588824444, 18.438611006187028, 2.1225687900429335, 
23.826583879752278, 32.981536123212599, 10.431407315649512, 33.276073297322704, 
21.636363471062992, 31.087806821668714, 13.137505641600082, 25.486896606948989, 
15.346699683642917, 14.52357672304894, 23.027783376765743, 30.503659276920978, 
22.512172166842898, 15.982518608635729, 30.016916828165989, 17.574756975754514, 
26.489854915558364, 9.9249696516729298, 26.103898313564208, 13.572371677375665, 
16.964928665580423, 9.0680457164247663, 48.154392519709091, 10.294812023817066, 
21.55485853838216, 20.754925097408464, 12.972068651136706, 22.083486870046983, 
25.261656503023328, 26.126372236120655, 2.1171149911426319, 12.777391064764082, 
18.668929321349079, 19.476004666463396, 15.954895832192406, 24.16229371732129, 
20.285611864598767, 12.714483487941347, 27.314744992118968, 17.306327033077654, 
20.758102853970833, 37.017765731106657, 21.57267582792646, 12.259162083569898, 
17.720822041253356, 21.315567976208914, 11.105033515805301, 4.5650814459484526, 
16.237913146444725, 24.275102671709828, 12.631600458332333, 25.474994821001232, 
32.176630160888571, 21.556906439494291, 33.577797740067396, 22.224545489739768, 
15.43191629721208, 18.736964480073457, 39.906714827916254, 28.570952752297224, 
22.737752612348345, 21.770619413166717, 20.546901936480189, 22.459372819283956, 
32.069070908149158, 35.322041900828481, 51.406115984078497, 23.21846523694694, 
29.052811475005001, -9.4896008395589888, 51.142441199161112, 
22.182035131845623, 31.050245909485966, 29.160354303661734, 35.531428191810846, 
31.426093608606607, 29.779939254745841, 26.181066029705107, 35.925971321295947, 
23.734816500451416, 33.104236654937267, -8.5669907536357641, 
-9.6752353776246309, 47.89807892171666, 20.686074344906956, 29.580144848208874, 
-8.4859268181025982, 31.425914666149765, -5.3061217274516821, 
38.287389220669866, 24.07169588515535, 30.856928567402065, 4.664987241383642, 
-8.5094883688725531, 50.478457702789456, -1.4197934833355248, 
-4.447792561724782, 43.104937590193003, 44.844307163730264, 50.946341717615724, 
47.670702504459769, 9.0020915996283293, 41.555707672145218, 30.795458311215043, 
-0.30272375978529453, 50.718219187110662, 36.517546133603901, 
16.233021294232458, 1.2923486866056919, 46.71086280234158, 25.707361765671521, 
8.7954031880944967, 41.191675776150078, 4.6510264240205288, -4.3088333937339485, 
16.638860592152923, 10.339313682168722, 11.327868235297501, 22.507822884712368, 
39.395256861578673, -4.1112663703970611, 3.3914048685692251, 
-0.59019019734114408, 0.78759392211213708, -9.165778826456517, 
31.041259392164648, 12.879235531203449, 50.887208410073072, 18.569026852957904, 
10.259459744673222, 23.58107620337978, -1.5248962217010558, -8.3263417747803032, 
8.5015549603849649, 11.317855224013329, 9.4624980832450092, 10.215069947298616, 
24.151421332266182, 17.142977610696107, 19.154718621633947, 40.956115166656673, 
12.303687674924731, 17.878761363681406, -8.7440985883586109, 
-6.7576695769093931, 4.0288742491975427, 14.645688700955361, 
-6.472494731657207, 30.620969725772738, 13.931614539586008, 24.315287482924759, 
5.1579427747055888, -2.6846914016641676, 16.766857167240232, 
31.977919838856906, 15.540120301302522, 4.3076404687017202, 40.902949003968388, 
6.0948069640435278, -1.7239051060751081, 42.824651269242167, 
5.7691888716071844, -9.8919325997121632, 1.8494659750722349, 
46.357358586508781, 17.724660135805607, 48.403695398941636, 28.440196488518268, 
9.6180127197876573, 51.055230151396245, 34.149633731693029, 40.261716793756932, 
7.0993854273110628, 32.982207109220326, 46.041913927067071, 23.729819377884269, 
10.011116673238575, 13.858537419699132, 29.210685695055872, 38.362327206414193, 
43.336531627457589, 50.96859021531418, 28.328518755268306, 33.904868975281715, 
23.550875564105809, 48.109792994800955, 43.615577332675457, 49.035543682519346, 
14.78945370670408, 9.6121553713455796, 3.5535783846862614, 12.083316863514483, 
-9.3090326855890453, 8.7778896065428853, 49.337298565544188, 
30.566184203140438, 19.39386588986963, 13.338629523292184, 42.971213975921273, 
5.2997302324511111, 33.104380735661834, 48.81855363631621, 31.584442453458905, 
5.3027837160043418, 11.363101753871888, 13.38088789023459, -4.1725726393051445, 
6.1866927752271295, 13.44078132789582, 35.717872128356248, 47.228031634818763, 
21.614546441938728, 51.509039236232638, 26.581861563958228, 11.733377205673605, 
14.974511503241956, -6.8194491136819124, 21.599103964865208, 
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