2

I have the following dataset

df <- structure(list(RT = c(828L, 1430L, 963L, 2161L, 991L, 1353L, 
1025L, 1593L, 1031L, 1233L, 1334L, 1573L, 1284L, 1907L, 734L, 
1231L, 988L, 1695L, 1586L, 882L, 762L, 1201L, 1076L, 1024L, 638L, 
909L, 1066L, 610L, 947L, 1293L, 774L, 1512L, 888L, 1721L, 1745L, 
781L, 1084L, 1456L, 2687L, 1173L, 1854L, 1968L, 1305L, 1560L, 
2461L, 807L, 3851L, 2442L, 2864L, 933L, 2765L, 2308L, 1905L, 
1108L, 1204L, 1375L, 1087L, 821L, 1055L, 1068L, 1678L, 1039L, 
2673L, 1622L, 993L, 1390L, 922L, 1080L, 2747L, 863L, 1187L, 790L, 
805L, 666L, 810L, 1123L, 2171L, 790L, 769L, 557L, 952L, 955L, 
592L, 1045L, 946L, 643L, 677L, 1119L, 1486L, 890L, 1013L, 924L, 
1160L, 931L, 849L, 1663L, 1234L, 1398L, 1251L, 1275L, 1248L, 
1428L, 843L, 2815L, 907L, 993L, 1337L, 1027L, 1161L, 828L, 994L, 
764L, 1101L, 1012L, 951L, 906L, 786L, 724L, 910L, 1087L, 896L, 
1024L, 1691L, 1431L, 1367L, 1061L, 1048L, 2689L, 1261L, 1274L, 
1258L, 1192L, 937L, 1063L, 835L, 1074L, 1148L, 1702L, 1353L, 
1703L, 995L, 769L, 783L, 727L, 713L, 904L, 803L, 1492L, 704L, 
807L, 865L, 1092L, 928L, 1226L, 1108L, 734L, 745L, 917L, 818L, 
1188L, 1545L, 987L, 1917L, 928L, 708L, 1168L, 906L, 1256L, 952L, 
2363L, 985L, 1972L, 1423L, 1878L, 796L, 1694L, 1107L, 1982L, 
1170L, 1007L, 1798L, 1575L, 1263L, 944L, 966L, 1164L, 1266L, 
1138L, 1260L, 2664L, 1388L, 1530L, 1072L, 1137L, 1503L, 1325L, 
1256L, 1738L, 1139L, 718L, 1008L, 1202L, 1825L, 932L, 895L, 1106L, 
755L, 909L, 927L, 864L, 990L, 1964L, 976L, 751L, 1368L, 2593L, 
1097L, 1222L, 1587L, 1578L, 1719L, 3006L, 1014L, 2535L, 1772L, 
1172L, 1640L, 756L, 1245L, 908L, 962L, 1181L, 783L, 804L, 2069L, 
820L, 867L, 926L, 939L, 807L, 1100L, 762L, 1418L, 1480L, 1554L, 
942L, 2245L, 1523L, 1279L, 878L, 1166L, 1070L, 1041L, 645L, 753L, 
809L, 856L, 796L, 676L, 896L, 845L, 885L, 866L, 650L, 754L, 849L, 
786L, 1052L, 727L, 734L, 1586L, 820L, 874L, 1718L, 1281L, 996L, 
1024L, 1037L, 777L, 890L, 1189L, 891L, 782L, 974L, 895L, 940L, 
1126L, 1315L, 1747L, 1321L, 1326L, 1587L, 1326L, 3663L, 1219L, 
1066L, 1170L, 957L, 2827L, 1319L, 1283L, 1845L, 1576L, 1410L, 
1287L, 946L, 858L, 2158L, 1196L, 1029L, 1243L, 820L, 960L, 978L, 
1095L, 1928L, 862L, 1278L, 806L, 963L, 937L, 799L, 1529L, 762L, 
867L, 929L, 673L, 998L, 1091L, 1050L, 645L, 3445L, 1137L, 2342L, 
1014L, 990L, 2092L, 771L, 1062L, 3649L, 777L, 2460L, 1670L, 925L, 
1019L, 1713L, 922L, 1263L, 1005L, 1710L, 1367L, 812L, 935L, 746L, 
959L, 787L, 748L, 877L, 1153L, 915L, 1170L, 930L, 924L, 988L, 
842L, 2630L, 2646L, 2728L, 2743L, 2390L, 2652L, 701L, 782L, 1150L, 
901L, 2047L, 2520L, 827L, 1262L, 1826L, 947L, 699L, 2372L, 1431L, 
1789L, 1763L, 659L, 906L, 1393L, 784L, 1056L, 755L, 802L, 967L, 
724L, 830L, 902L, 796L, 1105L, 1191L, 1242L, 1483L, 922L, 875L, 
1285L, 1412L, 984L, 810L, 1838L, 1948L, 1753L, 1567L, 1081L, 
1827L, 1301L, 1958L, 1334L, 1482L, 1633L, 1591L, 957L, 1328L, 
1226L, 986L, 915L, 1342L, 1850L, 1422L, 1759L, 2504L, 1431L, 
2099L, 3188L, 1314L, 1314L, 3242L, 2042L, 2677L, 1345L, 2976L, 
1380L, 1268L, 699L, 1180L, 1022L, 1330L, 864L, 714L, 798L, 899L, 
1018L, 830L, 878L, 1287L, 721L, 743L, 681L, 828L, 689L, 807L, 
683L, 771L, 684L, 908L, 941L, 980L, 1233L, 1174L, 1005L, 1026L, 
810L, 1048L, 935L, 962L, 972L, 1295L, 1098L, 868L, 1049L, 984L, 
1413L, 1120L, 962L, 1215L, 1371L, 1715L, 1179L, 2814L, 1110L, 
2090L, 2022L, 1126L, 2058L, 1853L, 1469L, 2029L, 1973L, 1392L, 
1846L, 892L, 1253L, 1621L, 1487L, 1584L, 1706L, 1309L, 2392L, 
1046L, 1074L, 2327L, 1182L, 1842L, 1062L, 1546L, 768L, 986L, 
1813L, 2704L, 2328L, 1027L, 996L, 766L, 890L, 1864L, 861L, 1348L, 
1085L, 1272L, 1300L, 1252L, 1776L, 1974L, 963L, 2171L, 1610L, 
1194L, 1784L, 1268L, 1416L, 1484L, 1603L, 1092L, 1212L, 1248L, 
2485L, 2661L, 2472L, 936L, 3860L, 3875L, 3145L, 1800L, 986L, 
1428L, 703L, 1135L, 1178L, 1000L, 2497L, 1313L, 1179L, 1279L, 
1441L, 1456L, 999L, 1557L, 1334L, 1053L, 861L, 1236L, 648L, 658L, 
621L, 886L, 651L, 1386L, 601L, 549L, 732L, 573L, 935L, 707L, 
587L, 1503L, 1207L, 1637L, 2014L, 1908L, 2566L, 2007L, 1350L, 
1196L, 1382L, 1351L, 1347L, 2203L, 1693L, 3361L, 1485L, 2330L, 
1375L, 1672L, 1102L, 2367L, 3219L, 1109L, 977L, 1421L, 1341L, 
1082L, 2254L, 900L, 1450L, 1114L, 920L, 1414L, 1232L, 654L, 664L, 
912L, 1932L, 1335L, 1231L, 652L, 1170L, 1114L, 1333L, 831L, 3877L, 
939L, 1772L, 1584L, 1391L, 1111L, 776L, 1094L, 1811L, 1196L, 
1057L, 1420L, 1129L, 847L, 2818L, 862L, 2708L, 1928L, 1734L, 
1119L, 1757L, 928L, 1344L, 1012L, 2040L, 933L, 1406L, 1474L, 
793L, 1022L, 758L, 876L, 1904L, 1994L, 680L, 978L, 1208L, 1015L, 
840L, 1292L, 935L, 1064L, 3063L, 2166L, 1097L, 3711L, 1042L, 
1030L, 1233L, 960L, 1403L, 1330L, 976L, 1745L, 922L, 989L, 953L, 
787L, 1060L, 1024L, 1218L, 1159L, 1063L, 1130L, 1332L, 2422L, 
1423L, 927L, 1002L, 933L, 727L, 1935L, 1373L, 751L, 2000L, 2010L, 
845L, 934L, 1191L, 1462L, 705L, 703L, 911L, 1591L, 1319L, 833L, 
2421L, 1456L, 812L, 1256L, 1706L, 1557L, 1100L, 2586L, 1195L, 
1693L, 2377L, 2932L, 1331L, 948L, 911L, 1208L, 3297L, 3149L, 
1485L, 1335L, 687L, 2084L, 1490L, 2774L, 2247L, 2258L, 1425L, 
1221L, 1653L, 1326L, 935L, 1352L, 963L, 1157L, 1987L, 943L, 1111L, 
779L, 1202L, 784L, 1525L, 1636L, 1174L, 1039L, 880L, 979L, 1305L, 
1742L, 1285L, 1061L, 1007L, 2175L, 770L, 1061L, 1571L, 1648L, 
792L, 1148L, 772L, 1139L, 885L, 1107L, 1585L, 1081L, 603L, 644L, 
2780L, 1321L, 1951L, 1689L, 2102L, 2795L, 2591L, 2410L, 1504L, 
1908L, 3877L, 1607L, 1209L, 1744L, 1209L, 1399L, 2928L, 1885L, 
3645L, 2203L, 1533L, 1623L, 3010L, 2569L, 1641L, 1053L, 932L, 
1276L, 1361L, 1380L, 1028L, 771L, 770L, 1270L, 1038L, 1154L, 
1195L, 658L, 588L, 943L, 829L, 1470L, 772L, 832L, 907L, 1193L, 
885L, 797L, 770L, 1294L, 945L, 735L, 1368L, 2600L, 2005L, 2384L, 
2750L, 1265L, 1435L, 1144L, 1057L, 951L, 769L, 1525L, 727L, 809L, 
617L, 697L, 695L, 1058L, 683L, 671L, 690L, 679L, 1123L, 1056L, 
739L, 762L, 787L, 572L, 2902L, 2589L, 3710L, 3940L, 2483L, 3280L, 
2415L, 1873L, 3149L, 1367L, 2735L, 3680L, 2451L, 1679L, 733L, 
699L, 1043L, 663L, 960L, 817L, 1373L, 1332L, 1092L, 739L, 1608L, 
860L, 1622L, 1531L, 924L, 1570L, 737L, 813L, 711L, 930L, 901L, 
860L, 1169L, 764L, 629L, 717L, 1202L, 1071L, 1415L, 1112L, 1086L, 
1140L, 1553L, 1172L, 874L, 1258L, 803L, 2554L, 1131L, 833L, 1271L, 
1342L, 755L, 1369L, 1474L, 1215L, 917L, 1280L, 847L, 816L, 993L, 
1916L, 904L, 1951L, 1546L, 1805L, 3134L, 1741L, 2998L, 2243L, 
1324L, 1661L, 1866L, 1344L, 1323L, 802L, 813L, 1610L, 974L, 685L, 
1421L, 1123L, 852L, 787L, 793L, 845L, 1834L, 914L, 718L, 1206L, 
762L, 1038L, 851L, 823L, 802L, 866L, 818L, 726L, 767L, 1099L, 
1164L, 1028L, 1188L, 842L, 1188L, 987L, 1093L, 989L, 1574L, 1938L, 
1000L, 1353L, 1083L, 1316L, 1130L, 1154L, 1178L, 665L, 833L, 
1223L, 2246L, 1493L, 1896L, 3150L, 1777L, 1054L, 1361L, 1504L, 
1993L, 1428L, 2794L, 1240L, 1613L, 1125L, 1240L, 1930L, 1399L, 
1170L, 2136L, 1419L, 1799L, 972L, 2032L, 1345L, 983L, 917L, 2747L, 
709L, 2196L, 1022L, 1245L, 853L, 878L, 891L, 1121L, 2461L, 1176L, 
819L, 752L, 1537L, 1429L, 1892L, 1422L, 2077L, 1430L, 1485L, 
3147L, 1447L, 1133L, 1217L, 1376L, 2337L, 2438L, 1145L, 1410L, 
1212L, 1886L, 3741L, 1732L, 2892L, 870L, 1789L, 2430L, 936L, 
957L, 1480L, 1802L, 978L, 846L, 876L, 1883L, 1242L, 1221L, 978L, 
1025L, 921L, 1074L, 801L, 717L, 970L, 913L, 782L, 686L, 719L, 
1246L, 638L, 1261L, 1076L, 639L, 818L, 541L, 677L, 672L, 799L, 
805L, 619L, 569L, 749L, 1002L, 1778L, 1256L, 2191L, 2524L, 1499L, 
1483L, 1659L, 1936L, 2404L, 1320L, 1200L, 1216L, 1257L, 1234L, 
1011L, 1930L, 1287L, 2922L, 1993L, 1431L, 2537L, 1004L, 2096L, 
3900L, 1112L, 1424L, 1238L, 1176L, 852L, 1911L, 855L, 665L, 671L, 
736L, 1021L, 1415L, 1139L, 1231L, 690L, 1006L, 1144L, 1064L, 
775L, 729L, 627L, 899L, 608L, 734L, 570L, 1057L, 601L, 1215L, 
430L, 572L, 1219L, 1189L, 677L, 1439L, 2591L, 1025L, 2357L, 3083L, 
1210L, 819L, 928L, 854L, 1166L, 1047L, 1280L, 1325L, 982L, 1089L, 
708L, 774L, 1959L, 1351L, 985L, 978L, 713L, 813L, 904L, 912L, 
1328L, 896L, 2943L, 1635L, 1984L, 1418L, 1155L, 1764L, 1636L, 
1946L, 1376L, 1526L, 1229L, 1227L, 1140L, 1870L, 973L, 664L, 
779L, 677L, 1306L, 671L, 678L, 938L, 620L, 1459L, 883L, 991L, 
810L, 705L, 985L, 1099L, 1061L, 2261L, 1628L, 3726L, 1486L, 1220L, 
1367L, 1075L, 973L, 1625L, 829L, 1501L, 1031L, 806L, 831L, 1603L, 
1181L, 1584L, 2327L, 1831L, 1370L, 951L, 812L, 1601L, 996L, 1721L, 
1999L, 1253L, 1920L, 1279L, 1451L, 918L, 2574L, 1336L, 749L, 
1262L, 1142L, 917L, 1398L, 1170L, 883L, 2158L, 1738L, 1177L, 
1848L, 1483L, 1333L, 1482L, 1287L, 927L, 858L, 981L, 825L, 1944L, 
1220L, 751L, 1787L, 1006L, 1047L, 842L, 850L, 1494L, 1026L, 1084L, 
837L, 805L, 702L, 765L, 559L, 857L, 1055L, 855L, 1482L, 1613L, 
1141L, 1589L, 1028L, 1329L, 1015L, 1103L, 919L, 823L, 935L, 891L, 
1149L, 1123L, 958L, 924L, 1209L, 1807L, 1551L, 1976L, 1378L, 
3033L, 1421L, 1508L, 3128L, 2204L, 1862L, 1355L, 2013L, 3008L, 
1091L, 1245L, 2086L, 902L, 2955L, 1364L, 1902L, 1756L, 846L, 
904L, 1011L, 2116L, 2054L, 1086L, 1955L, 1338L, 1505L, 1690L, 
1314L, 1225L, 1234L, 730L, 1819L, 957L, 1078L, 678L, 763L, 751L, 
1655L, 863L, 1129L, 1341L, 654L, 542L, 965L, 823L, 686L, 1167L, 
929L, 1023L, 1204L, 3396L, 827L, 1035L, 1310L, 1228L, 1036L, 
893L, 1082L, 1289L, 940L, 959L, 1062L, 958L, 1279L, 1105L, 627L, 
1038L, 733L, 882L, 766L, 811L, 623L, 919L, 803L, 2573L, 3539L, 
2340L, 2702L, 2278L, 1403L, 1644L, 1916L, 1722L, 3382L, 2704L, 
1654L, 1491L, 1127L, 708L, 712L, 709L, 1220L, 635L, 977L, 928L, 
806L, 766L, 1606L, 884L, 1072L, 1214L, 705L, 826L, 1315L, 746L, 
644L, 858L, 1244L, 788L, 650L, 1139L, 731L, 2042L, 1500L, 1658L, 
1370L, 1762L, 1207L, 1171L, 863L, 1048L, 847L, 1840L, 861L, 1188L, 
1053L, 1559L, 1035L, 1427L, 1209L, 999L, 1042L, 2082L, 929L, 
2476L, 1194L, 1686L, 795L, 1514L, 1496L, 1491L, 1743L, 2374L, 
3245L, 2473L, 1469L, 1998L, 2736L, 1317L, 1148L, 3011L, 1296L, 
1277L, 743L, 879L, 719L, 2158L, 898L, 1201L, 731L, 869L, 1054L, 
1696L, 831L, 890L, 775L, 1011L, 1159L, 819L, 1183L, 694L, 953L, 
1329L, 973L, 842L, 840L, 768L, 962L, 1013L, 792L, 750L, 1123L, 
814L, 938L, 1829L, 843L, 1022L, 2177L, 1033L, 1626L, 610L, 743L, 
1078L, 887L, 962L, 812L, 1576L, 1231L, 1246L, 1821L, 1479L, 1957L, 
2171L, 1698L, 1624L, 1579L, 2830L, 896L, 1466L, 900L, 982L, 1901L, 
875L, 1443L, 1418L, 1637L, 1627L, 1411L, 3675L, 1987L, 824L, 
835L, 1243L, 1635L, 1180L, 2415L, 1156L, 1034L, 1794L, 985L, 
2188L, 1722L, 1245L, 4410L, 2778L, 1425L, 2757L, 3464L, 2066L, 
5178L, 1770L, 3148L, 3862L, 2190L, 2601L, 847L, 912L, 1196L, 
751L, 1204L, 792L, 788L, 840L, 872L, 984L, 867L, 1003L, 704L, 
1193L, 921L, 921L, 2703L, 2549L, 2670L, 4433L, 5575L, 2938L, 
2849L, 3189L, 2439L, 3861L, 2668L, 2337L, 3521L, 4095L, 3214L, 
3807L, 620L, 751L, 872L, 714L, 1032L, 770L, 692L, 662L, 1002L, 
746L, 837L, 858L, 1319L, 1028L, 1214L, 678L, 1562L, 1257L, 1295L, 
1110L, 1025L, 1641L, 933L, 1282L, 1144L, 974L, 916L, 1152L, 1154L, 
1215L, 764L, 872L, 1961L, 888L, 1084L, 1496L, 1486L, 3219L, 849L, 
1002L, 896L, 900L, 1304L, 1657L, 2398L, 1797L, 1669L, 943L, 718L, 
1251L, 1183L, 766L, 944L, 924L, 731L, 939L, 891L, 850L, 1228L, 
940L, 774L, 1089L, 888L, 732L, 747L, 732L, 895L, 762L, 727L, 
808L, 747L, 695L, 1021L, 741L, 996L, 715L, 684L, 941L, 869L, 
857L, 1210L, 928L, 778L, 832L, 2041L, 1270L, 813L, 867L, 812L, 
964L, 1784L, 733L, 915L, 1088L, 867L, 699L, 734L, 832L, 969L, 
779L, 938L, 1328L, 611L, 687L, 1088L, 871L, 1301L, 1131L, 779L, 
1060L, 934L, 661L, 3351L, 963L, 4323L, 1974L, 3932L, 2279L, 1534L, 
979L, 1078L, 1089L, 1521L, 2783L, 2099L, 2500L, 1040L, 1118L, 
823L, 1667L, 3055L, 1242L, 2068L, 1129L, 1331L, 1657L, 1168L, 
1470L, 1775L, 1554L, 2319L, 4628L, 1407L, 1441L, 1208L, 1533L, 
1072L, 1175L, 821L, 895L, 1280L, 938L, 763L, 861L, 1674L, 1002L, 
1319L, 1276L, 1109L, 926L, 845L, 922L, 1574L, 3261L, 4930L, 2349L, 
4168L, 1562L, 669L, 1679L, 963L, 2081L, 1693L, 1487L, 634L, 4017L, 
4736L, 4249L, 3758L, 5104L, 6543L, 6710L, 5141L, 3891L, 4368L, 
5538L, 7274L, 3858L, 4582L, 5434L, 4055L, 5092L, 1056L, 1207L, 
1961L, 1146L, 1203L, 1593L, 857L, 1133L, 1122L, 1314L, 980L, 
1211L, 1034L, 1341L, 1934L, 3318L, 1163L, 819L, 1126L, 1013L, 
2814L, 1302L, 985L, 1704L, 736L, 779L, 1058L, 809L, 776L, 1410L, 
765L, 1341L, 1377L, 793L, 1214L, 1453L, 2450L, 1942L, 1054L, 
991L, 1042L, 981L, 1030L, 878L, 1133L, 2204L, 841L, 1714L, 1140L, 
983L, 777L, 802L, 1129L, 1410L, 759L, 868L, 852L, 670L, 1345L, 
972L, 900L, 863L, 679L, 967L, 746L, 737L, 929L, 874L, 764L, 1619L, 
761L, 747L, 705L, 847L, 1358L, 957L, 1344L, 1301L, 880L, 899L, 
1915L, 864L, 1081L, 1307L, 783L, 897L, 1076L, 722L, 709L, 997L, 
770L, 700L, 1391L, 954L, 725L, 685L, 1168L, 1278L, 3040L, 1136L, 
1225L, 1674L, 1528L, 1082L, 1268L, 1803L, 1118L, 1048L, 1509L, 
1115L, 864L, 1352L, 1040L, 3815L, 1127L, 1464L, 3160L, 3412L, 
1221L, 1468L, 3196L, 1510L, 1992L, 1727L, 2488L, 2814L, 3158L, 
1811L, 921L, 844L, 806L, 808L, 1440L, 1316L, 836L, 1598L, 968L, 
968L, 1061L, 1068L, 952L, 2306L, 886L, 766L, 3891L, 3452L, 3095L, 
3555L, 2143L, 1890L, 2886L, 5629L, 2895L, 4380L, 2583L, 2113L, 
8157L, 2814L, 2908L, 6832L, 714L, 989L, 877L, 823L, 1032L, 878L, 
1132L, 807L, 1383L, 1378L, 1971L, 936L, 1586L, 1072L, 1351L, 
773L, 1017L, 1126L, 1239L, 883L, 794L, 1498L, 1284L, 874L, 734L, 
817L, 1249L, 940L, 943L, 702L, 641L, 877L, 1786L, 807L, 1769L, 
1079L, 1477L, 1239L, 1845L, 835L, 1235L, 889L, 969L, 1105L, 1155L, 
765L, 980L, 1194L, 809L, 1137L, 865L, 847L, 898L, 1290L, 1110L, 
881L, 1198L, 947L, 760L, 929L, 848L, 1054L, 908L, 971L, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 1436L, 
1206L, 1166L, 5552L, 3010L, 1617L, 1148L, 1117L, 1642L, 976L, 
2586L, 1375L, 2983L, 1154L, 2185L, 922L, 1188L, 743L, 992L, 925L, 
2946L, 1433L, 810L, 916L, 715L, 778L, 1204L, 843L, 1588L, 943L, 
975L, 1314L, 2806L, 1029L, 3876L, 1689L, 5250L, 1119L, 1680L, 
1060L, 897L, 711L, 1266L, 935L, 992L, 1146L, 674L, 998L, 1177L, 
1414L, 1061L, 911L, 4131L, 1553L, 1990L, 1396L, 4544L, 1147L, 
3629L, 1380L, 1893L, 2686L, 2348L, 1000L, 1373L, 1785L, 1356L, 
1676L, 1650L, 1527L, 1091L, 1184L, 5865L, 807L, 1159L, 916L, 
1359L, 1402L, 824L, 981L, 984L, 1730L, 1299L, 1087L, 1440L, 1199L, 
3112L, 2332L, 1072L, 1059L, 1577L, 1243L, 1032L, 868L, 1093L, 
970L, 2379L, 4909L, 6407L, 5135L, 11244L, 3931L, 4331L, 6399L, 
5346L, 5452L, 6768L, 6934L, 1252L, 2864L, 6068L, 3996L, 908L, 
740L, 1740L, 939L, 1892L, 1169L, 843L, 1255L, 1416L, 908L, 847L, 
905L, 816L, 1165L, 1888L, 953L, 1048L, 646L, 1263L, 1074L, 2067L, 
1828L, 1175L, 803L, 1354L, 749L, 1506L, 633L, 1008L, 1246L, 708L, 
916L, 2795L, 934L, 3660L, 964L, 2519L, 2572L, 1412L, 937L, 1215L, 
939L, 1320L, 937L, 921L, 1074L, 1208L, 1368L, 1179L, 675L, 1386L, 
799L, 1072L, 1250L, 785L, 750L, 873L, 781L, 754L, 809L, 1057L, 
925L, 933L, 836L, 558L, 523L, 802L, 577L, 664L, 1357L, 765L, 
590L, 833L, 866L, 830L, 799L, 609L, 1067L, 684L, 652L, 1713L, 
1042L, 1305L, 1134L, 577L, 712L, 903L, 631L, 782L, 754L, 734L, 
825L, 894L, 974L, 793L, 824L, 1632L, 3015L, 3626L, 1442L, 804L, 
2086L, 1108L, 1777L, 999L, 1099L, 1030L, 1386L, 2945L, 1341L, 
730L, 1242L, 1161L, 826L, 1304L, 1580L, 2979L, 2258L, 1180L, 
2051L, 6211L, 122L, 2947L, 1328L, 2335L, 1879L, 2104L, 2126L, 
796L, 1082L, 799L, 805L, 1550L, 1010L, 1153L, 1089L, 1004L, 889L, 
1043L, 1061L, 843L, 1414L, 883L, 727L, 3621L, 2941L, 3262L, 2608L, 
3663L, 3444L, 4073L, 3057L, 2572L, 1988L, 2243L, 1974L, 3512L, 
5119L, 1948L, 3885L, 1055L, 1014L, 1291L, 868L, 919L, 1056L, 
1097L, 1092L, 1239L, 938L, 1915L, 942L, 1123L, 955L, 977L, 765L, 
933L, 791L, 1627L, 1054L, 1005L, 1331L, 622L, 1127L, 736L, 937L, 
1190L, 1345L, 995L, 1667L, 836L, 847L, 1528L, 1279L, 1414L, 1131L, 
1178L, 1434L, 1227L, 1385L, 851L, 1030L, 708L, 823L, 1176L, 767L, 
714L, 1322L, 984L, 1028L, 830L, 799L, 1188L, 1049L, 961L, 928L, 
1052L, 644L, 873L, 631L, 779L, 755L, 1022L, 787L, 739L, 905L, 
900L, 698L, 710L, 1084L, 1074L, 787L, 818L, 724L, 664L, 916L, 
746L, 724L, 852L, 815L, 1064L, 890L, 1207L, 1454L, 1344L, 1105L, 
911L, 812L, 700L, 712L, 863L, 769L, 675L, 679L, 622L, 807L, 1062L, 
932L, 838L, 718L, 2526L, 1676L, 2115L, 1293L, 2430L, 1013L, 744L, 
869L, 1262L, 1144L, 1283L, 1855L, 988L, 1306L, 1187L, 2189L, 
1425L, 2255L, 1207L, 703L, 848L, 952L, 3522L, 933L, 954L, 2031L, 
609L, 671L, 938L, 1689L, 1344L, 1156L, 1263L, 526L, 1452L, 867L, 
961L, 1082L, 1571L, 1523L, 921L, 1885L, 1153L, 1910L, 1547L, 
1231L, 1748L, 1717L, 2328L, 1427L, 2721L, 893L, 822L, 1364L, 
922L, 858L, 1044L, 1285L, 961L, 776L, 986L, 1462L, 1078L, 2385L, 
1845L, 1154L, 1713L, 1033L, 761L, 906L, 1172L, 1011L, 1084L, 
678L, 631L, 908L, 5573L, 3652L, 3214L, 3195L, 5070L, 7386L, 1848L, 
3624L, 6290L, 4809L, 4067L, 1651L, 1961L, 2212L, 6184L, 4887L, 
987L, 1284L, 918L, 1479L, 1337L, 1461L, 1030L, 1029L, 1102L, 
875L, 1068L, 890L, 1046L, 842L, 1195L, 1442L, 856L, 1150L, 1123L, 
909L, 2076L, 2208L, 1146L, 836L, 947L, 752L, 1185L, 763L, 617L, 
876L, 847L, 1000L, 1092L, 1386L, 2254L, 1096L, 2142L, 2617L, 
1215L, 1208L, 756L, 895L, 1031L, 798L, 1288L, 1043L, 718L, 806L, 
1476L, 659L, 1614L, 694L, 1500L, 1070L, 715L, 1865L, 544L, 991L, 
619L, 656L, 611L, 683L, 725L, 943L, 779L, 691L, 829L, 813L, 1227L, 
1099L, 895L, 634L, 593L, 595L, 779L, 743L, 806L, 1102L, 883L, 
615L, 937L, 1057L, 881L, 1655L, 2067L, 1144L, 905L, 1008L, 748L, 
1227L, 942L, 716L, 669L, 695L, 749L, 1409L), test.time = structure(c(3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L), .Label = c("delayed", 
"immediate", "pretest"), class = "factor")), class = "data.frame", row.names = c(NA, 
-2559L))

These are my default contrasts for my factors

contrasts(df$test.time)

         immediate pretest
delayed           0       0
immediate         1       0
pretest           0       1

I define a contrast matrix so that I compare pretest vs immediate and pretest vs delayed and perform a regression:

contrastmatrix <- cbind(c(0, -1, 1), c(-1, 0, 1))

contrasts(df$test.time) <- contrastmatrix

contrasts(df$test.time)

          [,1] [,2]
delayed      0   -1
immediate   -1    0
pretest      1    1

summary(lm(RT ~ test.time, data = df))

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1403.25      17.72  79.194  < 2e-16 ***
test.time1    -80.59      25.16  -3.203  0.00138 ** 
test.time2     63.63      24.97   2.548  0.01090 *  

I am aware that those contrasts also provide information on the immediate versus delayed contrasts due to being orthogonal. However suppose I want to assess that contrast. If I specify this, leaving my other contrast the same (test.time1), why do the results for my first contrast change? Also, the results for my immediate versus delayed contrast are the exact same as my pre-test versus delayed - I assume this is not a coincidence.

contrastmatrix <- cbind(c(0, -1, 1), c(-1, 1, 0))

contrasts(df$test.time) <- contrastmatrix

contrasts(df$test.time) 

          [,1] [,2]
delayed      0   -1
immediate   -1    1
pretest      1    0

summary(lm(RT ~ test.time, data = df))

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1403.25      17.72  79.194   <2e-16 ***
test.time1    -16.96      25.04  -0.677   0.4982    
test.time2     63.63      24.97   2.548   0.0109 *  
---

Any help is much appreciated!

kjetil b halvorsen
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1 Answers1

2

As this answer puts it:

Testing contrasts of factorial variables is notoriously difficult in R. Most of the things that go on beneath the surface in friendlier programs like SPSS must be spelled out specifically in R. This is both a bad and a good thing. Bad because it takes you a long time to learn how, but good because your understanding is vastly improved as a result.

In particular when you write

contrastmatrix <- cbind(c(0, -1, 1), c(-1, 0, 1))

you are not accomplishing what you intend, which is to

define a contrast matrix so that I compare pretest vs immediate and pretest vs delayed.

What you have done is to define two new variables (called test.time1 and test.time2 by R) that will be used to get around the problem that the 3 levels of your factor must be reduced to 2 linearly independent variables to get a unique solution. That "contrast matrix" is combined with a leading column of 1's (representing the intercept) to set the form of the design matrix for the regression.

This is easy to see with the default treatment contrasts matrix, with its prepended column of 1's:

Treatment contrasts:
          Intercept test.time1 test.time2
delayed           1          0          0
immediate         1          1          0
pretest           1          0          1

As you can see from the extended discussion on this page, that matrix must be inverted to show how the coefficients for the Intercept, test.time1, and test.time2 are estimated from the data. That inverse is:

Inverse, treatment contrasts:
           delayed immediate pretest
Intercept        1         0       0
test.time1      -1         1       0
test.time2      -1         0       1

This provides the well-known result with treatment contrasts: the intercept is the reference level of the factor (in R the first level, "delayed") and the linearly independent constructed variables represent the differences of each other level from that reference.

Here are the inverses of the two contrast matrices (including the leading columns of 1's) that you proposed, called CM1 and CM2.

Inverse with CM1:
              delayed  immediate   pretest
Intercept       1/3       1/3        1/3
test.time1      1/3      -2/3        1/3  
test.time2     -2/3       1/3        1/3  


Inverse with CM2:
              delayed  immediate   pretest
Intercept       1/3       1/3        1/3
test.time1     -1/3      -1/3        2/3  
test.time2     -2/3       1/3        1/3  

In both cases, the intercept is estimated as the mean over the 3 levels of the factor. In this respect, the intercept is what you get with deviation coding or contr.sum in R, in which 2 of the levels are expressed as differences from the mean, with the mean as the intercept. The estimation of the coefficient for test.time2 is identical for both of your proposed contrast matrices, as you discovered. It represents the negative of the difference between "delayed" and the mean. For CM1, test.time1 represents the negative of the difference between "immediate" and the mean; for CM2, it's the difference between "pretest" and the mean.

For comparison, here's what you would get with sum contrasts in R:

Sum contrast matrix with intercept column
          Intercept test.time1 test.time2
delayed           1          1          0
immediate         1          0          1
pretest           1         -1         -1

Inverse with sum contrasts:
              delayed  immediate    pretest
Intercept       1/3       1/3        1/3
test.time1      2/3      -1/3       -1/3
test.time2     -1/3       2/3       -1/3

which estimates coefficients for test.time1 and test.time2 as differences of "delayed" and "immediate" from the mean (intercept), respectively.

There are two simple solutions if your interest is in an application instead of in figuring out how to code your own contrast matrices.

First, accept the default treatment contrast matrix and get the coefficients and the coefficient variance-covariance matrix returned by the regression. Together with the formula for the variance of a sum of correlated variables you can reconstruct any desired comparisons among factor levels. This can be combined with a judicious choice of reference level. For example, to compare "pretest vs immediate and pretest vs delayed" just set the reference level to "pretest" and you get those comparisons automatically.

Second, use a program that provides a more direct way to translate your intentions into the correct contrast matrix. The answer linked at the beginning of my answer recommended the lsmeans package in R, which I understand is now deprecated in favor of the emmeans package.

EdM
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