1485:
1789:
1251:
1551:
1490:
Notice that the point estimator and the estimated variance are the same as those for a single pairwise comparison. The only difference between the confidence limits for simultaneous comparisons and those for a single comparison is the multiple of the estimated standard deviation.
80:
used to determine the level of significance of the differences considered in Tukey's test has vastly broader application: It is useful for researchers who have searched their collected data for remarkable differences between groups, but then cannot validly determine
998:
1536:
is the standard deviation of the entire design, not just that of the two groups being compared. It is possible to work with unequal sample sizes. In this case, one has to calculate the estimated standard deviation for each pairwise comparison as formalized by
491:
85:
their discovered stand-out difference is using standard statistical distributions used for other conventional statistical tests, for which the data must have been selected at random. Since when stand-out data is compared it was by definition
1480:{\displaystyle {\bar {y}}_{i\bullet }-{\bar {y}}_{j\bullet }\ \pm \ {\frac {\ q_{\ \alpha \ ;\ k\ ;\ N-k}\ }{\ {\sqrt {2\ }}\ }}\ {\widehat {\sigma }}_{\varepsilon }\ {\sqrt {{\frac {2}{n}}\ }}\quad :\quad i,\ j=1,\ldots ,k\quad i\neq j~.}
1784:{\displaystyle {\bar {y}}_{i\bullet }-{\bar {y}}_{j\bullet }\ \pm \ {\frac {\ q_{\ \alpha \ ;\ k\ ;\ N-k}\ }{\ {\sqrt {2\ }}\ }}\ {\widehat {\sigma }}_{\varepsilon }\ {\sqrt {\ {\frac {\ 1\ }{n_{i}}}\ +\ {\frac {\ 1\ }{n_{j}}}\ }}\ }
2055:
Morrison, S.; Sosnoff, J.J.; Heffernan, K.S.; Jae, S.Y.; Fernhall, B. (2013). "Aging, hypertension and physiological tremor: The contribution of the cardioballistic impulse to tremorgenesis in older adults".
1534:
641:
283:
889:
873:
823:
391:
1108:
1926:
183:
1992:
729:
1173:
1042:
318:
230:
90:
selected at random, but rather specifically chosen because it was extreme, it needs a different, stricter interpretation provided by the likely frequency and size of the
1075:
136:
Tukey's test compares the means of every treatment to the means of every other treatment; that is, it applies simultaneously to the set of all pairwise comparisons
2235:
2058:
731:
for all the samples; its calculation is discussed in the following sections. This gives rise to the normality assumption of Tukey's test.
2157:"Logical contradictions in the one-way ANOVA and Tukey–Kramer multiple comparisons tests with more than two groups of observations"
2144:
Linton, L.R.; Harder, L.D. (2007). Lecture notes (Report). Biology 315: Quantitative biology. Calgary, AB: University of
Calgary.
1201:
1497:
744:
568:
115:
91:
77:
2266:
2256:
2027:
993:{\displaystyle q\equiv {\frac {\ {\overline {y}}_{\mathsf {max}}-{\overline {y}}_{\mathsf {min}}\ }{\ S{\sqrt {2/n}}\ }}}
328:
593:
235:
692:
511:
1185:
828:
778:
2261:
20:
486:{\displaystyle q_{\mathsf {s}}={\frac {\ \left|Y_{\mathsf {A}}-Y_{\mathsf {B}}\right|\ }{\ {\mathsf {SE}}\ }}\ ,}
321:
1197:
341:
62:
2007:
1082:
1893:
1872:
It is possible to generate a set of pseudorandom samples of strictly negative measure such that hypothesis
2239:
2002:
379:
193:
142:
1956:
698:
331:
statistical procedure to render the output of this test more transparent to non-statistician audiences.
681:
65:
of the difference between means that have been selected for comparison because of their extreme values.
1145:
1020:
2180:
2098:
73:
354:
There is equal within-subgroup variance across the subgroups associated with each mean in the test (
1232:
1138:
has been tabulated and appears in many textbooks on statistics. In some tables the distribution of
756:
368:
348:
288:
203:
54:
2227:
1832:
Both ANOVA and Tukey–Kramer tests are based on the same assumptions. However, these two tests for
1538:
2170:
2161:
2115:
685:
1053:
2123:
2075:
1209:
589:
obtained from the distribution, the two means are said to be significantly different at level
197:
883:
from these samples. Then the following random variable has a
Studentized range distribution:
2188:
2107:
2067:
355:
58:
880:
647:
1494:
Also note that the sample sizes must be equal when using the studentized range approach.
650:
for Tukey's test states that all means being compared are from the same population (i.e.
2184:
530:
189:
2250:
1541:
in 1956, so the procedure for unequal sample sizes is sometimes referred to as the
1045:
188:
and identifies any difference between two means that is greater than the expected
76:(ANOVA), and usually has only been taught in connection with ANOVA. However, the
111:
95:
2035:
2093:
2071:
107:
69:
2127:
2079:
1824:
respectively. The degrees of freedom for the whole design is also applied.
2193:
2156:
1175:
factor. To understand which table it is, we can compute the result for
2119:
82:
1235:
for all pairwise comparisons with confidence coefficient of at least
285:
For unequal sample sizes, the confidence coefficient is greater than
127:
2111:
2175:
2096:(1949). "Comparing individual means in the Analysis of Variance".
1007:
given above is the basis of the critically significant value for
320:
In other words, the Tukey method is conservative when there are
367:
Tukey's test is based on a formula very similar to that of the
680:), the means should be normally distributed (according to the
1048:
rate, or the probability of rejecting a true null hypothesis;
347:
The subgroups associated with each mean in the test are
1529:{\displaystyle \ {\widehat {\sigma }}_{\varepsilon }\ }
122:) (this distribution is similar to the distribution of
1014:
discussed below, and is based on these three factors:
1959:
1896:
1554:
1500:
1254:
1148:
1085:
1056:
1023:
892:
831:
781:
701:
596:
394:
291:
238:
206:
145:
68:
The method was initially developed and introduced by
2050:
Also occasionally described as "honestly", see e.g.
1986:
1920:
1783:
1528:
1479:
1167:
1102:
1069:
1036:
992:
875:is the largest of these sample means, and suppose
867:
817:
723:
635:
485:
312:
277:
224:
177:
636:{\displaystyle \ \alpha ~:~0\leq \alpha \leq 1~.}
278:{\displaystyle \ \alpha ~:~0\leq \alpha \leq 1~.}
510:are the two means being compared, and SE is the
200:, when all sample sizes are equal, is exactly
2236:National Institute of Standards and Technology
1110:the number of degrees of freedom for each mean
868:{\displaystyle \ {\bar {y}}_{\mathsf {max}}\ }
818:{\displaystyle \ {\bar {y}}_{\mathsf {min}}\ }
521:is the sample's test statistic. (The notation
1077:the number of sub-populations being compared;
8:
1859:) may result in logical contradictions when
61:. It can be used to correctly interpret the
2215:(8th ed.). Wiley. Section 3.5.7.
825:is the smallest of these sample means and
2192:
2174:
2139:
2137:
1958:
1895:
1765:
1748:
1731:
1714:
1709:
1700:
1689:
1688:
1668:
1621:
1611:
1593:
1582:
1581:
1568:
1557:
1556:
1553:
1517:
1506:
1505:
1499:
1411:
1409:
1400:
1389:
1388:
1368:
1321:
1311:
1293:
1282:
1281:
1268:
1257:
1256:
1253:
1188:with the same degrees of freedom and the
1152:
1147:
1090:
1089:
1084:
1055:
1022:
974:
969:
945:
944:
934:
917:
916:
906:
899:
891:
849:
848:
837:
836:
830:
799:
798:
787:
786:
780:
706:
705:
700:
595:
559:test statistic can then be compared to a
462:
461:
441:
440:
426:
425:
410:
400:
399:
393:
374:. In fact, Tukey's test is essentially a
290:
237:
205:
163:
150:
144:
16:Statistical test for multiple comparisons
747:. Suppose that we take a sample of size
563:value for the chosen significance level
2018:
1828:Comparing ANOVA and Tukey–Kramer tests
1131:is the total number of observations.)
1103:{\displaystyle \ {\mathsf {df}}\quad }
1094:
1091:
952:
949:
946:
924:
921:
918:
856:
853:
850:
806:
803:
800:
710:
707:
466:
463:
442:
427:
401:
2230:. e-Handbook of Statistical Methods.
2028:"One-way ANOVA – independent samples"
1921:{\displaystyle \ 1-\alpha >0.95\ }
545:, regardless of the original sign of
7:
2059:Journal of the Neurological Sciences
1184:and compare it to the result of the
514:for the sum of the means. The value
178:{\displaystyle \mu _{i}-\mu _{j}\ ,}
110:, it compares all possible pairs of
1987:{\displaystyle \ 1-\alpha =0.975~.}
724:{\displaystyle \ {\mathsf {SE}}\ ,}
378:-test, except that it corrects for
327:This test is often followed by the
2213:Design and Analysis of Experiments
1890:is rejected at significance level
1003:This definition of the statistic
340:The observations being tested are
98:" is an example where it is used.
14:
2155:Gurvich, V.; Naumova, M. (2021).
1869:even if the assumptions do hold.
1168:{\displaystyle \ {\sqrt {2\ }}\ }
1202:cumulative distribution function
1037:{\displaystyle \ \alpha ~\quad }
385:The formula for Tukey's test is
39:Tukey's honest significance test
2211:Montgomery, Douglas C. (2013).
1461:
1430:
1426:
1142:has been tabulated without the
1099:
1066:
1033:
47:honestly significant difference
1587:
1562:
1287:
1262:
842:
792:
745:studentized range distribution
569:studentized range distribution
116:studentized range distribution
78:studentized range distribution
1:
313:{\displaystyle \ 1-\alpha ~.}
225:{\displaystyle \ 1-\alpha \ }
2232:itl.nist.gov/div898/handbook
939:
911:
344:within and among the groups.
329:Compact Letter Display (CLD)
2283:
1070:{\displaystyle \ k~\quad }
755:populations with the same
743:The Tukey method uses the
691:, estimated by the merged
94:; the modern practice of "
21:Tukey mean-difference test
18:
2072:10.1016/j.jns.2013.01.016
1953:is not rejected even at
1816:are the sizes of groups
1186:Student's t-distribution
582:than the critical value
106:The test is named after
63:statistical significance
19:Not to be confused with
735:The studentized range (
356:homogeneity of variance
2240:Department of Commerce
2003:Family-wise error rate
1988:
1922:
1785:
1530:
1481:
1169:
1104:
1071:
1038:
994:
881:pooled sample variance
869:
819:
725:
684:) with the same model
637:
487:
380:family-wise error rate
314:
279:
226:
194:confidence coefficient
179:
1989:
1923:
1786:
1545:which is as follows:
1531:
1482:
1170:
1105:
1072:
1039:
995:
870:
820:
726:
682:central limit theorem
638:
541:with the sign set to
488:
315:
280:
227:
180:
2267:Multiple comparisons
2257:Analysis of variance
1957:
1894:
1552:
1498:
1252:
1146:
1134:The distribution of
1083:
1054:
1021:
890:
829:
779:
699:
594:
567:from a table of the
392:
349:normally distributed
322:unequal sample sizes
289:
236:
204:
143:
114:, and is based on a
74:Analysis of Variance
2194:10.3390/sym13081387
2185:2021Symm...13.1387G
2008:Newman–Keuls method
1543:Tukey–Kramer method
757:normal distribution
537:; the magnitude of
55:multiple comparison
53:, is a single-step
2038:on 17 October 2008
1984:
1918:
1781:
1526:
1477:
1165:
1100:
1067:
1034:
990:
865:
815:
721:
686:standard deviation
633:
483:
363:The test statistic
310:
275:
222:
175:
27:Tukey's range test
2262:Statistical tests
1980:
1962:
1917:
1899:
1780:
1776:
1775:
1771:
1759:
1753:
1747:
1741:
1737:
1725:
1719:
1713:
1708:
1697:
1686:
1682:
1680:
1676:
1675:
1667:
1662:
1648:
1642:
1636:
1630:
1624:
1616:
1610:
1604:
1590:
1565:
1525:
1514:
1503:
1473:
1439:
1424:
1423:
1419:
1408:
1397:
1386:
1382:
1380:
1376:
1375:
1367:
1362:
1348:
1342:
1336:
1330:
1324:
1316:
1310:
1304:
1290:
1265:
1233:confidence limits
1227:Confidence limits
1210:quantile function
1164:
1160:
1159:
1151:
1088:
1065:
1059:
1032:
1026:
988:
986:
982:
965:
960:
942:
914:
904:
864:
845:
834:
814:
795:
784:
775:and suppose that
717:
704:
629:
611:
605:
599:
479:
475:
473:
460:
455:
415:
306:
294:
271:
253:
247:
241:
221:
209:
171:
92:studentized range
2274:
2243:
2228:"Tukey's method"
2216:
2199:
2198:
2196:
2178:
2152:
2146:
2145:
2141:
2132:
2131:
2090:
2084:
2083:
2047:
2045:
2043:
2034:. Archived from
2026:Lowry, Richard.
2023:
1993:
1991:
1990:
1985:
1978:
1960:
1952:
1951:
1927:
1925:
1924:
1919:
1915:
1897:
1889:
1888:
1868:
1866:
1858:
1835:
1823:
1819:
1815:
1814:
1804:
1803:
1790:
1788:
1787:
1782:
1778:
1777:
1773:
1772:
1770:
1769:
1760:
1757:
1751:
1749:
1745:
1739:
1738:
1736:
1735:
1726:
1723:
1717:
1715:
1711:
1710:
1706:
1705:
1704:
1699:
1698:
1690:
1684:
1683:
1681:
1678:
1677:
1673:
1669:
1665:
1663:
1660:
1659:
1658:
1646:
1640:
1634:
1628:
1622:
1614:
1612:
1608:
1602:
1601:
1600:
1592:
1591:
1583:
1576:
1575:
1567:
1566:
1558:
1535:
1533:
1532:
1527:
1523:
1522:
1521:
1516:
1515:
1507:
1501:
1486:
1484:
1483:
1478:
1471:
1437:
1425:
1421:
1420:
1412:
1410:
1406:
1405:
1404:
1399:
1398:
1390:
1384:
1383:
1381:
1378:
1377:
1373:
1369:
1365:
1363:
1360:
1359:
1358:
1346:
1340:
1334:
1328:
1322:
1314:
1312:
1308:
1302:
1301:
1300:
1292:
1291:
1283:
1276:
1275:
1267:
1266:
1258:
1245:
1243:
1223:
1221:
1215:
1207:
1195:
1193:
1183:
1182:
1174:
1172:
1171:
1166:
1162:
1161:
1157:
1153:
1149:
1141:
1137:
1130:
1126:
1124:
1109:
1107:
1106:
1101:
1098:
1097:
1086:
1076:
1074:
1073:
1068:
1063:
1057:
1043:
1041:
1040:
1035:
1030:
1024:
1013:
1006:
999:
997:
996:
991:
989:
987:
984:
983:
978:
970:
963:
961:
958:
957:
956:
955:
943:
935:
929:
928:
927:
915:
907:
902:
900:
878:
874:
872:
871:
866:
862:
861:
860:
859:
847:
846:
838:
832:
824:
822:
821:
816:
812:
811:
810:
809:
797:
796:
788:
782:
774:
773:
754:
750:
738:
730:
728:
727:
722:
715:
714:
713:
702:
690:
679:
643:
642:
640:
639:
634:
627:
609:
603:
597:
588:
574:
566:
562:
555:
548:
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536:
528:
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517:
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499:
492:
490:
489:
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477:
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469:
458:
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432:
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377:
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319:
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59:statistical test
29:, also known as
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2205:Further reading
2202:
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2112:10.2307/3001913
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648:null hypothesis
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83:how significant
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17:
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5:
2280:
2278:
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2259:
2249:
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2221:External links
2219:
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2200:
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2085:
2066:(1–2): 68–74.
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739:) distribution
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693:standard error
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617:
614:
608:
602:
585:
575:
556:
531:absolute value
518:
512:standard error
507:
500:
494:
493:
482:
468:
465:
451:
444:
439:
435:
429:
424:
419:
409:
403:
398:
364:
361:
360:
359:
352:
345:
336:
333:
309:
303:
300:
297:
274:
268:
265:
262:
259:
256:
250:
244:
218:
215:
212:
190:standard error
186:
185:
174:
166:
162:
158:
153:
149:
133:. See below).
103:
100:
57:procedure and
15:
13:
10:
9:
6:
4:
3:
2:
2279:
2268:
2265:
2263:
2260:
2258:
2255:
2254:
2252:
2241:
2237:
2233:
2229:
2225:
2224:
2220:
2214:
2209:
2208:
2204:
2195:
2190:
2186:
2182:
2177:
2172:
2168:
2164:
2163:
2158:
2151:
2148:
2140:
2138:
2134:
2129:
2125:
2121:
2117:
2113:
2109:
2106:(2): 99–114.
2105:
2101:
2100:
2095:
2089:
2086:
2081:
2077:
2073:
2069:
2065:
2061:
2060:
2049:
2048:
2037:
2033:
2029:
2022:
2019:
2013:
2009:
2006:
2004:
2001:
2000:
1996:
1994:
1981:
1975:
1972:
1969:
1966:
1963:
1947:
1940:
1933:
1912:
1909:
1906:
1903:
1900:
1884:
1877:
1870:
1864:
1857:
1847:
1840:
1836:groups (i.e.
1827:
1825:
1813:
1802:
1766:
1762:
1754:
1742:
1732:
1728:
1720:
1701:
1694:
1691:
1670:
1655:
1652:
1649:
1643:
1637:
1631:
1625:
1618:
1605:
1597:
1594:
1584:
1577:
1572:
1569:
1559:
1548:
1547:
1546:
1544:
1540:
1518:
1511:
1508:
1492:
1474:
1468:
1465:
1462:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1434:
1431:
1427:
1416:
1413:
1401:
1394:
1391:
1370:
1355:
1352:
1349:
1343:
1337:
1331:
1325:
1318:
1305:
1297:
1294:
1284:
1277:
1272:
1269:
1259:
1248:
1247:
1246:
1242:
1234:
1226:
1224:
1211:
1203:
1199:
1196:In addition,
1187:
1180:
1154:
1132:
1123:
1119:
1079:
1060:
1050:
1047:
1027:
1017:
1016:
1015:
979:
975:
971:
966:
936:
930:
908:
896:
893:
886:
885:
884:
882:
839:
789:
771:
767:
763:
758:
751:from each of
746:
734:
732:
718:
694:
687:
683:
678:
668:
661:
654:
649:
644:
630:
624:
621:
618:
615:
612:
606:
600:
581:
570:
550:
532:
513:
480:
449:
437:
433:
422:
417:
407:
396:
388:
387:
386:
383:
381:
373:
362:
357:
353:
350:
346:
343:
339:
338:
334:
332:
330:
325:
323:
307:
301:
298:
295:
272:
266:
263:
260:
257:
254:
248:
242:
216:
213:
210:
199:
195:
191:
172:
164:
160:
156:
151:
147:
139:
138:
137:
134:
132:
117:
113:
109:
101:
99:
97:
93:
89:
84:
79:
75:
71:
66:
64:
60:
56:
52:
48:
44:
40:
36:
32:
28:
22:
2234:. SEMATECH.
2231:
2212:
2166:
2160:
2150:
2103:
2097:
2088:
2063:
2057:
2040:. Retrieved
2036:the original
2031:
2021:
1945:
1938:
1931:
1882:
1875:
1871:
1862:
1852:
1845:
1838:
1831:
1808:
1797:
1793:
1542:
1539:Clyde Kramer
1493:
1489:
1240:
1230:
1178:
1133:
1121:
1117:
1113:
1046:Type I error
1002:
769:
765:
761:
742:
673:
666:
659:
652:
645:
579:
551:
495:
384:
366:
326:
187:
135:
105:
87:
67:
50:
46:
42:
38:
35:Tukey method
34:
31:Tukey's test
30:
26:
25:
2169:(8): 1387.
2094:Tukey, John
1244: are
342:independent
335:Assumptions
102:Development
96:data mining
72:for use in
43:Tukey's HSD
2251:Categories
2176:2104.07552
2099:Biometrics
2042:4 December
2032:Vassar.edu
2014:References
1231:The Tukey
646:Since the
529:means the
108:John Tukey
70:John Tukey
1970:α
1967:−
1907:α
1904:−
1702:ε
1695:^
1692:σ
1653:−
1626:α
1606:±
1598:∙
1588:¯
1578:−
1573:∙
1563:¯
1519:ε
1512:^
1509:σ
1466:≠
1453:…
1402:ε
1395:^
1392:σ
1353:−
1326:α
1306:±
1298:∙
1288:¯
1278:−
1273:∙
1263:¯
1200:offers a
1028:α
940:¯
931:−
912:¯
897:≡
843:¯
793:¯
622:≤
619:α
616:≤
601:α
578:value is
571:. If the
434:−
302:α
299:−
264:≤
261:α
258:≤
243:α
217:α
214:−
161:μ
157:−
148:μ
126:from the
2162:Symmetry
2128:18151955
2080:23385002
1997:See also
1851:= ... =
1811: j
1800: i
1237:
1208:) and a
672:= ... =
232:for any
196:for the
2238:/ U.S.
2181:Bibcode
2120:3001913
1865:> 2
1115:( df =
879:is the
527:|
2126:
2118:
2078:
1979:
1961:
1928:while
1916:
1898:
1794:where
1779:
1774:
1758:
1752:
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1740:
1724:
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1712:
1707:
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1679:
1674:
1666:
1661:
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1635:
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1623:
1615:
1609:
1603:
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1438:
1422:
1407:
1385:
1379:
1374:
1366:
1361:
1347:
1341:
1335:
1329:
1323:
1315:
1309:
1303:
1214:qtukey
1206:ptukey
1163:
1158:
1150:
1127:where
1087:
1064:
1058:
1031:
1025:
985:
964:
959:
903:
863:
833:
813:
783:
716:
703:
689:σ
628:
610:
604:
598:
586:α
580:larger
565:α
523:|
496:where
478:
472:
459:
454:
414:
305:
293:
270:
252:
246:
240:
220:
208:
192:. The
170:
2171:arXiv
2116:JSTOR
1976:0.975
1190:same
552:This
372:-test
131:-test
112:means
41:, or
2124:PMID
2076:PMID
2044:2008
1913:0.95
1910:>
1820:and
1805:and
1239:1 −
1218:for
1181:= 2
1044:the
503:and
51:test
2189:doi
2108:doi
2068:doi
2064:326
549:.)
533:of
198:set
88:not
2253::
2187:.
2179:.
2167:13
2165:.
2159:.
2136:^
2122:.
2114:.
2102:.
2074:.
2062:.
2030:.
1944:=
1937:=
1881:=
1844:=
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1120:−
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382:.
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324:.
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2242:.
2197:.
2191::
2183::
2173::
2130:.
2110::
2104:5
2082:.
2070::
2046:.
1982:.
1973:=
1964:1
1949:3
1946:μ
1942:2
1939:μ
1935:1
1932:μ
1901:1
1886:2
1883:μ
1879:1
1876:μ
1867:,
1863:k
1855:k
1853:μ
1849:2
1846:μ
1842:1
1839:μ
1834:k
1822:j
1818:i
1809:n
1798:n
1767:j
1763:n
1755:1
1743:+
1733:i
1729:n
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1650:N
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1638:k
1632:;
1619:q
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1475:.
1469:j
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1456:,
1450:,
1447:1
1444:=
1441:j
1435:,
1432:i
1428::
1417:n
1414:2
1371:2
1356:k
1350:N
1344:;
1338:k
1332:;
1319:q
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1285:y
1270:i
1260:y
1241:α
1222:.
1220:q
1212:(
1204:(
1198:R
1194:.
1192:α
1179:k
1155:2
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1136:q
1129:N
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1122:k
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1095:f
1092:d
1061:k
1011:α
1009:q
1005:q
980:n
976:/
972:2
967:S
953:n
950:i
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937:y
925:x
922:a
919:m
909:y
894:q
877:S
857:x
854:a
851:m
840:y
807:n
804:i
801:m
790:y
772:)
770:σ
766:μ
764:(
762:N
753:k
749:n
737:q
719:,
711:E
708:S
676:k
674:μ
670:3
667:μ
663:2
660:μ
656:1
653:μ
631:.
625:1
613:0
607::
584:q
576:s
573:q
561:q
557:s
554:q
547:x
543:+
539:x
535:x
525:x
519:s
516:q
508:B
505:Y
501:A
498:Y
481:,
467:E
464:S
450:|
443:B
438:Y
428:A
423:Y
418:|
408:=
402:s
397:q
376:t
370:t
351:.
308:.
296:1
273:.
267:1
255:0
249::
211:1
173:,
165:j
152:i
129:t
124:t
120:q
118:(
45:(
23:.
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