1015:
753:
1176:, and we refer to these as aa, Aa and AA. The distribution of genotype counts can be put in a 2 × 3 contingency table. For example, consider the following data, in which the genotype frequencies vary linearly in the cases and are constant in the controls:
1332:
and the
Pearson chi-squared test gives a standardized test statistic of 2. Thus, we obtain a stronger significance level if the weights corresponding to additive (codominant) inheritance are used. Note that for the significance level to give a
48:
categories of the second variable. For example, doses of a treatment can be ordered as 'low', 'medium', and 'high', and we may suspect that the treatment benefit cannot become smaller as the dose increases. The trend test is often used as a
768:
1093:
458:
1151:
than the chi-squared test when the suspected trend is correct, but the ability to detect unsuspected trends is sacrificed. This is an example of a general technique of directing hypothesis tests toward narrow
640:
625:
1570:
1010:{\displaystyle {\rm {Var}}(T)={\frac {R_{1}R_{2}}{N}}\left(\sum _{i=1}^{k}t_{i}^{2}C_{i}(N-C_{i})-2\sum _{i=1}^{k-1}\sum _{j=i+1}^{k}t_{i}t_{j}C_{i}C_{j}\right),}
1156:. The trend test exploits the suspected effect direction to increase power, but this does not affect the sampling distribution of the test statistic under the
1026:
1337:
with the usual probabilistic interpretation, the weights must be specified before examining the data, and only one set of weights may be used.
351:
1346:
1135: = (0,1,2) should be used. These weights are also often used when the frequencies are suspected to change monotonically with
1371:
748:{\displaystyle \operatorname {E} (T)=\operatorname {E} \left(\operatorname {E} (T|R_{1},R_{2})\right)=\operatorname {E} (0)=0.}
1160:. Thus, the suspected trend in effects is not an assumption that must hold in order for the test results to be meaningful.
1287:
25:
537:
29:
41:
33:
21:
1259:
1153:
1131: = (1,1,0) should be used. If we suspect a linear trend in the frequencies, then the weights
1267:
60:
1498:
1463:
1398:
54:
1547:
1406:
1367:
1148:
1108:
75:
1419:
1537:
1529:
1490:
1455:
1390:
1173:
1293:
In the numerical example, the standardized test statistics for various weight vectors are
1283:
1157:
528:
1542:
1517:
631:
342:
1518:"PLINK: a tool set for whole-genome association and population-based linkage analyses"
1564:
1446:
Cochran, WG (1954). "Some methods for strengthening the common chi-squared tests".
1359:
1420:"A derivation for Armitage's trend test for the 2 × 3 genotype table"
1481:
Armitage, P (1955). "Tests for Linear Trends in
Proportions and Frequencies".
1275:
1250:
In genetics applications, the weights are selected according to the suspected
1088:{\displaystyle {\frac {T}{\sqrt {\mathrm {Var} (T)}}}\sim \mathrm {N} (0,1).}
1551:
1410:
160:
This table can be completed with the marginal totals of the two variables
1251:
1169:
759:
71:
The trend test is applied when the data take the form of a 2 ×
57:
50:
1381:
Sasieni, P (1997). "From genotypes to genes: doubling the sample size".
1502:
1467:
1402:
1334:
1266: = (1, 1, 0) is locally optimal. To test whether allele a is
453:{\displaystyle T\equiv \sum _{i=1}^{k}t_{i}(N_{1i}R_{2}-N_{2i}R_{1}),}
1255:
1494:
1459:
1394:
1123: = 2 have similar frequencies (within each row), but that
36:
between a variable with two categories and an ordinal variable with
1533:
1290:, the additive (or codominant) version of the test is often used.
1111:
for detecting particular types of associations. For example, if
1516:
Purcell S, Neale B, Todd-Brown K, et al. (September 2007).
1274: = (0, 1, 1). To test whether alleles a and A are
1107:
can be chosen such that the trend test becomes locally most
1127: = 3 has a different frequency, then the weights
44:
to incorporate a suspected ordering in the effects of the
1029:
771:
643:
540:
354:
1286:, the underlying genetic model is often unknown. In
524:after reweighting the rows to have the same total.
1087:
1009:
747:
620:{\displaystyle \Pr(A=1|B=1)=\cdots =\Pr(A=1|B=k).}
619:
452:
32:when the aim is to assess for the presence of an
582:
541:
1489:(3). International Biometric Society: 375–386.
1454:(4). International Biometric Society: 417–451.
1389:(4). International Biometric Society: 1253–61.
1282: = (0, 1, 2) is locally optimal. For
1139:, even if the trend is not necessarily linear.
8:
1541:
1062:
1036:
1030:
1028:
993:
983:
973:
963:
953:
936:
920:
909:
890:
871:
861:
856:
846:
835:
814:
804:
797:
773:
772:
770:
707:
694:
685:
642:
597:
556:
539:
438:
425:
412:
399:
386:
376:
365:
353:
1571:Statistical tests for contingency tables
1295:
1254:. For example, in order to test whether
1178:
162:
84:
1438:
1168:Suppose that there are three possible
527:The hypothesis of no association (the
504:can be seen as the difference between
1020:and as a large sample approximation,
7:
1347:List of analyses of categorical data
1270:to allele A, the optimal choice is
1115: = 3 and we suspect that
1063:
1043:
1040:
1037:
780:
777:
774:
724:
673:
662:
644:
14:
630:Assuming this holds, then, using
1147:The trend test will have higher
758:The variance can be computed by
472:are weights, and the difference
1288:genome-wide association studies
18:Cochran–Armitage test for trend
1079:
1067:
1053:
1047:
896:
877:
791:
785:
736:
730:
713:
686:
679:
656:
650:
611:
598:
585:
570:
557:
544:
444:
392:
1:
40:categories. It modifies the
1302:Standardized test statistic
1587:
1366:(Second ed.). Wiley.
1262:over allele A, the choice
1364:Categorical Data Analysis
30:categorical data analysis
42:Pearson chi-squared test
1164:Application to genetics
1143:Interpretation and role
531:) can be expressed as:
82: = 3 we have
1089:
1011:
958:
931:
851:
749:
621:
454:
381:
1090:
1012:
932:
905:
831:
750:
622:
455:
361:
1418:statgen.org (2007).
1027:
769:
641:
632:iterated expectation
538:
352:
1252:mode of inheritance
1119: = 1 and
866:
78:. For example, if
61:association studies
1085:
1007:
852:
745:
617:
450:
1522:Am. J. Hum. Genet
1330:
1329:
1248:
1247:
1057:
1056:
824:
287:
286:
158:
157:
76:contingency table
1578:
1556:
1555:
1545:
1513:
1507:
1506:
1478:
1472:
1471:
1443:
1433:
1431:
1429:
1424:
1414:
1377:
1296:
1284:complex diseases
1179:
1094:
1092:
1091:
1086:
1066:
1058:
1046:
1035:
1031:
1016:
1014:
1013:
1008:
1003:
999:
998:
997:
988:
987:
978:
977:
968:
967:
957:
952:
930:
919:
895:
894:
876:
875:
865:
860:
850:
845:
825:
820:
819:
818:
809:
808:
798:
784:
783:
754:
752:
751:
746:
720:
716:
712:
711:
699:
698:
689:
626:
624:
623:
618:
601:
560:
459:
457:
456:
451:
443:
442:
433:
432:
417:
416:
407:
406:
391:
390:
380:
375:
163:
85:
53:-based test for
1586:
1585:
1581:
1580:
1579:
1577:
1576:
1575:
1561:
1560:
1559:
1515:
1514:
1510:
1495:10.2307/3001775
1480:
1479:
1475:
1460:10.2307/3001616
1445:
1444:
1440:
1427:
1425:
1422:
1417:
1395:10.2307/2533494
1380:
1374:
1358:
1355:
1343:
1166:
1158:null hypothesis
1145:
1106:
1025:
1024:
989:
979:
969:
959:
886:
867:
830:
826:
810:
800:
799:
767:
766:
703:
690:
672:
668:
639:
638:
536:
535:
529:null hypothesis
523:
513:
503:
497:
487:
481:
471:
434:
421:
408:
395:
382:
350:
349:
337:
330:
323:
316:
309:
302:
295:
280:
272:
264:
251:
243:
235:
227:
214:
206:
198:
190:
154:
146:
138:
125:
117:
109:
69:
22:William Cochran
12:
11:
5:
1584:
1582:
1574:
1573:
1563:
1562:
1558:
1557:
1534:10.1086/519795
1508:
1473:
1437:
1436:
1435:
1415:
1378:
1372:
1354:
1351:
1350:
1349:
1342:
1339:
1328:
1327:
1324:
1320:
1319:
1316:
1312:
1311:
1308:
1304:
1303:
1300:
1246:
1245:
1242:
1239:
1236:
1233:
1229:
1228:
1225:
1222:
1219:
1216:
1212:
1211:
1208:
1205:
1202:
1199:
1195:
1194:
1191:
1188:
1185:
1182:
1165:
1162:
1144:
1141:
1102:
1096:
1095:
1084:
1081:
1078:
1075:
1072:
1069:
1065:
1061:
1055:
1052:
1049:
1045:
1042:
1039:
1034:
1018:
1017:
1006:
1002:
996:
992:
986:
982:
976:
972:
966:
962:
956:
951:
948:
945:
942:
939:
935:
929:
926:
923:
918:
915:
912:
908:
904:
901:
898:
893:
889:
885:
882:
879:
874:
870:
864:
859:
855:
849:
844:
841:
838:
834:
829:
823:
817:
813:
807:
803:
796:
793:
790:
787:
782:
779:
776:
756:
755:
744:
741:
738:
735:
732:
729:
726:
723:
719:
715:
710:
706:
702:
697:
693:
688:
684:
681:
678:
675:
671:
667:
664:
661:
658:
655:
652:
649:
646:
628:
627:
616:
613:
610:
607:
604:
600:
596:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
559:
555:
552:
549:
546:
543:
518:
508:
501:
492:
485:
476:
467:
461:
460:
449:
446:
441:
437:
431:
428:
424:
420:
415:
411:
405:
402:
398:
394:
389:
385:
379:
374:
371:
368:
364:
360:
357:
343:test statistic
335:
328:
321:
314:
307:
300:
293:
285:
284:
281:
278:
273:
270:
265:
262:
257:
253:
252:
249:
244:
241:
236:
233:
228:
225:
220:
216:
215:
212:
207:
204:
199:
196:
191:
188:
183:
179:
178:
175:
172:
169:
166:
156:
155:
152:
147:
144:
139:
136:
131:
127:
126:
123:
118:
115:
110:
107:
102:
98:
97:
94:
91:
88:
68:
65:
26:Peter Armitage
13:
10:
9:
6:
4:
3:
2:
1583:
1572:
1569:
1568:
1566:
1553:
1549:
1544:
1539:
1535:
1531:
1528:(3): 559–75.
1527:
1523:
1519:
1512:
1509:
1504:
1500:
1496:
1492:
1488:
1484:
1477:
1474:
1469:
1465:
1461:
1457:
1453:
1449:
1442:
1439:
1421:
1416:
1412:
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1379:
1375:
1373:0-471-36093-7
1369:
1365:
1361:
1360:Agresti, Alan
1357:
1356:
1352:
1348:
1345:
1344:
1340:
1338:
1336:
1325:
1322:
1321:
1317:
1314:
1313:
1309:
1306:
1305:
1301:
1298:
1297:
1294:
1291:
1289:
1285:
1281:
1278:, the choice
1277:
1273:
1269:
1265:
1261:
1257:
1253:
1243:
1240:
1237:
1234:
1231:
1230:
1226:
1223:
1220:
1217:
1214:
1213:
1209:
1206:
1203:
1200:
1197:
1196:
1192:
1189:
1186:
1183:
1181:
1180:
1177:
1175:
1171:
1163:
1161:
1159:
1155:
1150:
1142:
1140:
1138:
1134:
1130:
1126:
1122:
1118:
1114:
1110:
1105:
1101:
1082:
1076:
1073:
1070:
1059:
1050:
1032:
1023:
1022:
1021:
1004:
1000:
994:
990:
984:
980:
974:
970:
964:
960:
954:
949:
946:
943:
940:
937:
933:
927:
924:
921:
916:
913:
910:
906:
902:
899:
891:
887:
883:
880:
872:
868:
862:
857:
853:
847:
842:
839:
836:
832:
827:
821:
815:
811:
805:
801:
794:
788:
765:
764:
763:
761:
760:decomposition
742:
739:
733:
727:
721:
717:
708:
704:
700:
695:
691:
682:
676:
669:
665:
659:
653:
647:
637:
636:
635:
633:
614:
608:
605:
602:
594:
591:
588:
579:
576:
573:
567:
564:
561:
553:
550:
547:
534:
533:
532:
530:
525:
522:
517:
512:
507:
500:
496:
491:
484:
480:
475:
470:
466:
447:
439:
435:
429:
426:
422:
418:
413:
409:
403:
400:
396:
387:
383:
377:
372:
369:
366:
362:
358:
355:
348:
347:
346:
344:
339:
334:
331: +
327:
324: =
320:
313:
310: +
306:
303: +
299:
296: =
292:
282:
277:
274:
269:
266:
261:
258:
255:
254:
248:
245:
240:
237:
232:
229:
224:
221:
218:
217:
211:
208:
203:
200:
195:
192:
187:
184:
181:
180:
176:
173:
170:
167:
165:
164:
161:
151:
148:
143:
140:
135:
132:
129:
128:
122:
119:
114:
111:
106:
103:
100:
99:
95:
92:
89:
87:
86:
83:
81:
77:
74:
66:
64:
62:
59:
56:
52:
47:
43:
39:
35:
31:
28:, is used in
27:
23:
19:
1525:
1521:
1511:
1486:
1482:
1476:
1451:
1447:
1441:
1426:. Retrieved
1386:
1382:
1363:
1331:
1292:
1279:
1271:
1263:
1249:
1167:
1154:alternatives
1146:
1136:
1132:
1128:
1124:
1120:
1116:
1112:
1103:
1099:
1098:The weights
1097:
1019:
757:
629:
526:
520:
515:
510:
505:
498:
494:
489:
482:
478:
473:
468:
464:
462:
340:
332:
325:
318:
311:
304:
297:
290:
288:
275:
267:
259:
246:
238:
230:
222:
209:
201:
193:
185:
159:
149:
141:
133:
120:
112:
104:
79:
72:
70:
67:Introduction
55:case-control
45:
37:
20:, named for
17:
15:
1190:Genotype AA
1187:Genotype Aa
1184:Genotype aa
762:, yielding
34:association
1483:Biometrics
1448:Biometrics
1428:6 February
1383:Biometrics
1353:References
1276:codominant
463:where the
341:The trend
1268:recessive
1198:Controls
1170:genotypes
1060:∼
934:∑
925:−
907:∑
900:−
884:−
833:∑
728:
677:
666:
648:
577:⋯
419:−
363:∑
359:≡
1565:Category
1552:17701901
1362:(2002).
1341:See also
1260:dominant
1172:at some
1109:powerful
51:genotype
1543:1950838
1503:3001775
1468:3001616
1411:9423247
1403:2533494
1335:p-value
1299:Weights
488: −
338:, etc.
317:, and
58:genetic
1550:
1540:
1501:
1466:
1409:
1401:
1370:
1326:−4.67
1323:0,1,2
1315:0,1,1
1307:1,1,0
1256:allele
1215:Cases
289:where
219:A = 2
182:A = 1
130:A = 2
101:A = 1
96:B = 3
1499:JSTOR
1464:JSTOR
1423:(PDF)
1399:JSTOR
1318:−2.1
1310:1.85
1258:a is
1174:locus
1149:power
174:B = 3
171:B = 2
168:B = 1
93:B = 2
90:B = 1
1548:PMID
1430:2009
1407:PMID
1368:ISBN
1244:120
1232:Sum
1193:Sum
514:and
256:Sum
177:Sum
24:and
16:The
1538:PMC
1530:doi
1491:doi
1456:doi
1391:doi
1227:60
1210:60
345:is
1567::
1546:.
1536:.
1526:81
1524:.
1520:.
1497:.
1487:11
1485:.
1462:.
1452:10
1450:.
1405:.
1397:.
1387:53
1385:.
1241:50
1238:40
1235:30
1224:30
1221:20
1218:10
1207:20
1204:20
1201:20
743:0.
634:,
583:Pr
542:Pr
336:21
329:11
315:13
308:12
301:11
283:N
242:23
234:22
226:21
205:13
197:12
189:11
153:23
145:22
137:21
124:13
116:12
108:11
63:.
1554:.
1532::
1505:.
1493::
1470:.
1458::
1434:–
1432:.
1413:.
1393::
1376:.
1280:t
1272:t
1264:t
1137:B
1133:t
1129:t
1125:B
1121:B
1117:B
1113:k
1104:i
1100:t
1083:.
1080:)
1077:1
1074:,
1071:0
1068:(
1064:N
1054:)
1051:T
1048:(
1044:r
1041:a
1038:V
1033:T
1005:,
1001:)
995:j
991:C
985:i
981:C
975:j
971:t
965:i
961:t
955:k
950:1
947:+
944:i
941:=
938:j
928:1
922:k
917:1
914:=
911:i
903:2
897:)
892:i
888:C
881:N
878:(
873:i
869:C
863:2
858:i
854:t
848:k
843:1
840:=
837:i
828:(
822:N
816:2
812:R
806:1
802:R
795:=
792:)
789:T
786:(
781:r
778:a
775:V
740:=
737:)
734:0
731:(
725:E
722:=
718:)
714:)
709:2
705:R
701:,
696:1
692:R
687:|
683:T
680:(
674:E
670:(
663:E
660:=
657:)
654:T
651:(
645:E
615:.
612:)
609:k
606:=
603:B
599:|
595:1
592:=
589:A
586:(
580:=
574:=
571:)
568:1
565:=
562:B
558:|
554:1
551:=
548:A
545:(
521:i
519:2
516:N
511:i
509:1
506:N
502:1
499:R
495:i
493:2
490:N
486:2
483:R
479:i
477:1
474:N
469:i
465:t
448:,
445:)
440:1
436:R
430:i
427:2
423:N
414:2
410:R
404:i
401:1
397:N
393:(
388:i
384:t
378:k
373:1
370:=
367:i
356:T
333:N
326:N
322:1
319:C
312:N
305:N
298:N
294:1
291:R
279:3
276:C
271:2
268:C
263:1
260:C
250:2
247:R
239:N
231:N
223:N
213:1
210:R
202:N
194:N
186:N
150:N
142:N
134:N
121:N
113:N
105:N
80:k
73:k
46:k
38:k
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.