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Tschuprow's T

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1245: 966: 535: 1190: 765: 619: 89: 361: 291: 376: 689: 1023: 737: 1379: 1090: 189: 1055: 221: 1102: 961:{\displaystyle {\hat {T}}={\sqrt {\frac {\sum _{i=1}^{r}\sum _{j=1}^{c}{\frac {(p_{ij}-p_{i+}p_{+j})^{2}}{p_{i+}p_{+j}}}}{\sqrt {(r-1)(c-1)}}}},} 550: 23: 1328: 1262: 1221: 134: 1309: 1266: 1281: 1093: 530:{\displaystyle \phi ^{2}=\sum _{i=1}^{r}\sum _{j=1}^{c}{\frac {(\pi _{ij}-\pi _{i+}\pi _{+j})^{2}}{\pi _{i+}\pi _{+j}}},} 296: 229: 1288: 1366:
Liebetrau, A. (1983). Measures of Association (Quantitative Applications in the Social Sciences). Sage Publications
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equals one if and only there is perfect dependence in the table, i.e., if and only if for each
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and vice versa. Hence, it can only equal 1 for square tables. In this it differs from
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equals zero if and only if independence holds in the table, i.e., if and only if
1244: 1233: 107: 1185:{\displaystyle {\hat {T}}={\sqrt {\frac {\chi ^{2}/n}{\sqrt {(r-1)(c-1)}}}}.} 125:, giving a value between 0 and 1 (inclusive). It is closely related to 614:{\displaystyle T={\sqrt {\frac {\phi ^{2}}{\sqrt {(r-1)(c-1)}}}}.} 84:{\displaystyle T={\sqrt {\frac {\phi ^{2}}{\sqrt {(r-1)(c-1)}}}}} 1238: 15: 1105: 1071: 1031: 977: 768: 743:, which can be equal to 1 for any rectangular table. 709: 638: 553: 379: 299: 232: 197: 167: 26: 1358:; translated by M. Kantorowitsch. W. Hodge & Co. 1356:
Principles of the Mathematical Theory of Correlation
1269:. Unsourced material may be challenged and removed. 356:{\displaystyle \pi _{+j}=\sum _{i=1}^{r}\pi _{ij}.} 1184: 1084: 1049: 1017: 960: 731: 683: 613: 529: 355: 286:{\displaystyle \pi _{i+}=\sum _{j=1}^{c}\pi _{ij}} 285: 215: 183: 83: 1201:Other measures of correlation for nominal data: 8: 684:{\displaystyle \pi _{ij}=\pi _{i+}\pi _{+j}} 191:be the proportion of the population in cell 1380:Summary statistics for contingency tables 1329:Learn how and when to remove this message 1135: 1129: 1121: 1107: 1106: 1104: 1076: 1070: 1030: 1007: 998: 982: 976: 907: 894: 882: 869: 856: 840: 830: 824: 813: 803: 792: 784: 770: 769: 767: 714: 708: 672: 659: 643: 637: 567: 560: 552: 512: 499: 487: 474: 461: 445: 435: 429: 418: 408: 397: 384: 378: 341: 331: 320: 304: 298: 274: 264: 253: 237: 231: 196: 172: 166: 137:(alternative spelling: Chuprov) in 1939. 40: 33: 25: 1025:is the proportion of the sample in cell 751:If we have a multinomial sample of size 1347: 1096:, this formula can also be written as 7: 1267:adding citations to reliable sources 14: 759:from the data is via the formula 1243: 129:, coinciding with it for square 1254:needs additional citations for 1018:{\displaystyle p_{ij}=n_{ij}/n} 1172: 1160: 1157: 1145: 1112: 1044: 1032: 948: 936: 933: 921: 879: 833: 775: 732:{\displaystyle \pi _{ij}>0} 601: 589: 586: 574: 484: 438: 210: 198: 74: 62: 59: 47: 1: 1094:Pearson chi-square statistic 755:, the usual way to estimate 18: 1396: 1085:{\displaystyle \chi ^{2}} 184:{\displaystyle \pi _{ij}} 1354:Tschuprow, A. A. (1939) 1228:Other related articles: 1217:Uncertainty coefficient 368:mean square contingency 153:contingency table with 1186: 1086: 1051: 1019: 962: 829: 808: 733: 685: 615: 531: 434: 413: 357: 336: 287: 269: 217: 185: 133:. It was published by 85: 1187: 1087: 1052: 1050:{\displaystyle (i,j)} 1020: 963: 809: 788: 734: 686: 616: 532: 414: 393: 358: 316: 288: 249: 218: 216:{\displaystyle (i,j)} 186: 86: 1263:improve this article 1103: 1069: 1029: 975: 766: 707: 636: 551: 377: 297: 230: 195: 165: 24: 135:Alexander Tschuprow 1222:Lambda coefficient 1182: 1082: 1047: 1015: 958: 729: 699:there is only one 681: 611: 527: 353: 283: 213: 181: 131:contingency tables 81: 1339: 1338: 1331: 1313: 1177: 1176: 1175: 1115: 953: 952: 951: 917: 778: 606: 605: 604: 522: 123:nominal variables 104: 103: 79: 78: 77: 1387: 1359: 1352: 1334: 1327: 1323: 1320: 1314: 1312: 1271: 1247: 1239: 1191: 1189: 1188: 1183: 1178: 1144: 1143: 1139: 1134: 1133: 1123: 1122: 1117: 1116: 1108: 1091: 1089: 1088: 1083: 1081: 1080: 1056: 1054: 1053: 1048: 1024: 1022: 1021: 1016: 1011: 1006: 1005: 990: 989: 967: 965: 964: 959: 954: 920: 919: 918: 916: 915: 914: 902: 901: 888: 887: 886: 877: 876: 864: 863: 848: 847: 831: 828: 823: 807: 802: 786: 785: 780: 779: 771: 738: 736: 735: 730: 722: 721: 690: 688: 687: 682: 680: 679: 667: 666: 651: 650: 620: 618: 617: 612: 607: 573: 572: 571: 562: 561: 540:and Tschuprow's 536: 534: 533: 528: 523: 521: 520: 519: 507: 506: 493: 492: 491: 482: 481: 469: 468: 453: 452: 436: 433: 428: 412: 407: 389: 388: 362: 360: 359: 354: 349: 348: 335: 330: 312: 311: 292: 290: 289: 284: 282: 281: 268: 263: 245: 244: 222: 220: 219: 214: 190: 188: 187: 182: 180: 179: 117:is a measure of 90: 88: 87: 82: 80: 46: 45: 44: 35: 34: 16: 1395: 1394: 1390: 1389: 1388: 1386: 1385: 1384: 1370: 1369: 1363: 1362: 1353: 1349: 1344: 1335: 1324: 1318: 1315: 1278:"Tschuprow's T" 1272: 1270: 1260: 1248: 1212:Phi coefficient 1198: 1125: 1124: 1101: 1100: 1072: 1067: 1066: 1059:empirical value 1027: 1026: 994: 978: 973: 972: 903: 890: 889: 878: 865: 852: 836: 832: 787: 764: 763: 749: 710: 705: 704: 668: 655: 639: 634: 633: 627: 563: 549: 548: 508: 495: 494: 483: 470: 457: 441: 437: 380: 375: 374: 337: 300: 295: 294: 270: 233: 228: 227: 193: 192: 168: 163: 162: 143: 36: 22: 21: 12: 11: 5: 1393: 1391: 1383: 1382: 1372: 1371: 1368: 1367: 1361: 1360: 1346: 1345: 1343: 1340: 1337: 1336: 1251: 1249: 1242: 1237: 1236: 1225: 1224: 1219: 1214: 1209: 1197: 1194: 1193: 1192: 1181: 1174: 1171: 1168: 1165: 1162: 1159: 1156: 1153: 1150: 1147: 1142: 1138: 1132: 1128: 1120: 1114: 1111: 1079: 1075: 1057:. This is the 1046: 1043: 1040: 1037: 1034: 1014: 1010: 1004: 1001: 997: 993: 988: 985: 981: 969: 968: 957: 950: 947: 944: 941: 938: 935: 932: 929: 926: 923: 913: 910: 906: 900: 897: 893: 885: 881: 875: 872: 868: 862: 859: 855: 851: 846: 843: 839: 835: 827: 822: 819: 816: 812: 806: 801: 798: 795: 791: 783: 777: 774: 748: 745: 728: 725: 720: 717: 713: 678: 675: 671: 665: 662: 658: 654: 649: 646: 642: 626: 623: 622: 621: 610: 603: 600: 597: 594: 591: 588: 585: 582: 579: 576: 570: 566: 559: 556: 538: 537: 526: 518: 515: 511: 505: 502: 498: 490: 486: 480: 477: 473: 467: 464: 460: 456: 451: 448: 444: 440: 432: 427: 424: 421: 417: 411: 406: 403: 400: 396: 392: 387: 383: 364: 363: 352: 347: 344: 340: 334: 329: 326: 323: 319: 315: 310: 307: 303: 280: 277: 273: 267: 262: 259: 256: 252: 248: 243: 240: 236: 212: 209: 206: 203: 200: 178: 175: 171: 142: 139: 102: 101: 94: 93: 76: 73: 70: 67: 64: 61: 58: 55: 52: 49: 43: 39: 32: 29: 13: 10: 9: 6: 4: 3: 2: 1392: 1381: 1378: 1377: 1375: 1365: 1364: 1357: 1351: 1348: 1341: 1333: 1330: 1322: 1311: 1308: 1304: 1301: 1297: 1294: 1290: 1287: 1283: 1280: â€“  1279: 1275: 1274:Find sources: 1268: 1264: 1258: 1257: 1252:This article 1250: 1246: 1241: 1240: 1235: 1232: 1231: 1230: 1229: 1223: 1220: 1218: 1215: 1213: 1210: 1208: 1205: 1204: 1203: 1202: 1195: 1179: 1169: 1166: 1163: 1154: 1151: 1148: 1140: 1136: 1130: 1126: 1118: 1109: 1099: 1098: 1097: 1095: 1077: 1073: 1064: 1060: 1041: 1038: 1035: 1012: 1008: 1002: 999: 995: 991: 986: 983: 979: 955: 945: 942: 939: 930: 927: 924: 911: 908: 904: 898: 895: 891: 883: 873: 870: 866: 860: 857: 853: 849: 844: 841: 837: 825: 820: 817: 814: 810: 804: 799: 796: 793: 789: 781: 772: 762: 761: 760: 758: 754: 746: 744: 742: 726: 723: 718: 715: 711: 702: 698: 694: 676: 673: 669: 663: 660: 656: 652: 647: 644: 640: 631: 624: 608: 598: 595: 592: 583: 580: 577: 568: 564: 557: 554: 547: 546: 545: 543: 524: 516: 513: 509: 503: 500: 496: 488: 478: 475: 471: 465: 462: 458: 454: 449: 446: 442: 430: 425: 422: 419: 415: 409: 404: 401: 398: 394: 390: 385: 381: 373: 372: 371: 369: 350: 345: 342: 338: 332: 327: 324: 321: 317: 313: 308: 305: 301: 278: 275: 271: 265: 260: 257: 254: 250: 246: 241: 238: 234: 226: 225: 224: 207: 204: 201: 176: 173: 169: 161:columns, let 160: 156: 152: 149: Ă—  148: 140: 138: 136: 132: 128: 124: 120: 116: 115: 109: 100: 96: 95: 92: 71: 68: 65: 56: 53: 50: 41: 37: 30: 27: 17: 1355: 1350: 1325: 1319:October 2011 1316: 1306: 1299: 1292: 1285: 1273: 1261:Please help 1256:verification 1253: 1227: 1226: 1200: 1199: 1062: 970: 756: 752: 750: 700: 696: 692: 629: 628: 541: 539: 370:is given as 365: 158: 154: 150: 146: 144: 121:between two 113: 112:Tschuprow's 111: 105: 98: 97:Tschuprow's 19: 1234:Effect size 119:association 1342:References 1289:newspapers 1207:CramĂ©r's V 747:Estimation 741:CramĂ©r's V 703:such that 625:Properties 141:Definition 127:CramĂ©r's V 108:statistics 1167:− 1152:− 1127:χ 1113:^ 1074:χ 943:− 928:− 850:− 811:∑ 790:∑ 776:^ 712:π 670:π 657:π 641:π 596:− 581:− 565:ϕ 510:π 497:π 472:π 459:π 455:− 443:π 416:∑ 395:∑ 382:ϕ 366:Then the 339:π 318:∑ 302:π 272:π 251:∑ 235:π 170:π 157:rows and 69:− 54:− 38:ϕ 1374:Category 1196:See also 223:and let 1303:scholar 1065:. With 145:For an 91:  1305:  1298:  1291:  1284:  1276:  971:where 20:  1310:JSTOR 1296:books 1282:news 1092:the 724:> 293:and 1265:by 1061:of 544:as 106:In 1376:: 691:. 110:, 1332:) 1326:( 1321:) 1317:( 1307:· 1300:· 1293:· 1286:· 1259:. 1180:. 1173:) 1170:1 1164:c 1161:( 1158:) 1155:1 1149:r 1146:( 1141:n 1137:/ 1131:2 1119:= 1110:T 1078:2 1063:T 1045:) 1042:j 1039:, 1036:i 1033:( 1013:n 1009:/ 1003:j 1000:i 996:n 992:= 987:j 984:i 980:p 956:, 949:) 946:1 940:c 937:( 934:) 931:1 925:r 922:( 912:j 909:+ 905:p 899:+ 896:i 892:p 884:2 880:) 874:j 871:+ 867:p 861:+ 858:i 854:p 845:j 842:i 838:p 834:( 826:c 821:1 818:= 815:j 805:r 800:1 797:= 794:i 782:= 773:T 757:T 753:n 727:0 719:j 716:i 701:j 697:i 693:T 677:j 674:+ 664:+ 661:i 653:= 648:j 645:i 630:T 609:. 602:) 599:1 593:c 590:( 587:) 584:1 578:r 575:( 569:2 558:= 555:T 542:T 525:, 517:j 514:+ 504:+ 501:i 489:2 485:) 479:j 476:+ 466:+ 463:i 450:j 447:i 439:( 431:c 426:1 423:= 420:j 410:r 405:1 402:= 399:i 391:= 386:2 351:. 346:j 343:i 333:r 328:1 325:= 322:i 314:= 309:j 306:+ 279:j 276:i 266:c 261:1 258:= 255:j 247:= 242:+ 239:i 211:) 208:j 205:, 202:i 199:( 177:j 174:i 159:c 155:r 151:c 147:r 114:T 99:T 75:) 72:1 66:c 63:( 60:) 57:1 51:r 48:( 42:2 31:= 28:T

Index

statistics
association
nominal variables
Cramér's V
contingency tables
Alexander Tschuprow
mean square contingency
Cramér's V
empirical value
Pearson chi-square statistic
Cramér's V
Phi coefficient
Uncertainty coefficient
Lambda coefficient
Effect size

verification
improve this article
adding citations to reliable sources
"Tschuprow's T"
news
newspapers
books
scholar
JSTOR
Learn how and when to remove this message
Category
Summary statistics for contingency tables

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