4237:
5491:
4715:
3906:
4942:
4414:
192:-test is conservative", and so it is less likely than it should be to find that a variable is significant. However, as either the sample size or the number of cells increases, "the power curves seem to converge to that based on the normal distribution". Tiku (1971) found that "the non-normal theory power of
201:
consequences of such violations are less severe than previously thought. Although these conclusions should not entirely discourage anyone from being concerned about the normality assumption, they have increased the overall popularity of the distribution-dependent statistical tests in all areas of research."
65:). The ANOVA produces an F-statistic, the ratio of the variance calculated among the means to the variance within the samples. If the group means are drawn from populations with the same mean values, the variance between the group means should be lower than the variance of the samples, following the
221:
The normal linear model describes treatment groups with probability distributions which are identically bell-shaped (normal) curves with different means. Thus fitting the models requires only the means of each treatment group and a variance calculation (an average variance within the treatment
4437:
3688:
Each treatment group is summarized by the number of experimental units, two sums, a mean and a variance. The treatment group summaries are combined to provide totals for the number of units and the sums. The grand mean and grand variance are computed from the grand sums. The treatment and grand
200:
The current view is that "Monte-Carlo studies were used extensively with normal distribution-based tests to determine how sensitive they are to violations of the assumption of normal distribution of the analyzed variables in the population. The general conclusion from these studies is that the
4232:{\displaystyle {\begin{aligned}{\overline {Y}}_{1}&={\frac {1}{6}}\sum Y_{1i}={\frac {6+8+4+5+3+4}{6}}=5\\{\overline {Y}}_{2}&={\frac {1}{6}}\sum Y_{2i}={\frac {8+12+9+11+6+8}{6}}=9\\{\overline {Y}}_{3}&={\frac {1}{6}}\sum Y_{3i}={\frac {13+9+11+8+7+12}{6}}=10\end{aligned}}}
5486:{\displaystyle {\begin{aligned}S_{W}=&(1)^{2}+(3)^{2}+(-1)^{2}+(0)^{2}+(-2)^{2}+(-1)^{2}+\\&(-1)^{2}+(3)^{2}+(0)^{2}+(2)^{2}+(-3)^{2}+(-1)^{2}+\\&(3)^{2}+(-1)^{2}+(1)^{2}+(-2)^{2}+(-3)^{2}+(2)^{2}\\=&\ 1+9+1+0+4+1+1+9+0+4+9+1+9+1+1+4+9+4\\=&\ 68\\\end{aligned}}}
3757:
Consider an experiment to study the effect of three different levels of a factor on a response (e.g. three levels of a fertilizer on plant growth). If we had 6 observations for each level, we could write the outcome of the experiment in a table like this, where
187:
However, this is a misconception, based on work done in the 1950s and earlier. The first comprehensive investigation of the issue by Monte Carlo simulation was
Donaldson (1966). He showed that under the usual departures (positive skew, unequal variances) "the
5841:
of the first group differs from the population means of the other groups. However, there is no evidence that the second and third groups have different population means from each other, as their mean difference of one unit is comparable to the standard error.
5781:-test, it is common to carry out some "post-hoc" analysis of the group means. In this case, the first two group means differ by 4 units, the first and third group means differ by 5 units, and the second and third group means differ by only 1 unit. The
2824:
196:
is found to differ from the normal theory power by a correction term which decreases sharply with increasing sample size." The problem of non-normality, especially in large samples, is far less serious than popular articles would suggest.
4251:
3257:
3574:
4710:{\displaystyle {\begin{aligned}S_{B}&=n({\overline {Y}}_{1}-{\overline {Y}})^{2}+n({\overline {Y}}_{2}-{\overline {Y}})^{2}+n({\overline {Y}}_{3}-{\overline {Y}})^{2}\\&=6(5-8)^{2}+6(9-8)^{2}+6(10-8)^{2}=84\end{aligned}}}
5746:
2545:
Given the summary statistics, the calculations of the hypothesis test are shown in tabular form. While two columns of SS are shown for their explanatory value, only one column is required to display results.
61:, which states that samples in all groups are drawn from populations with the same mean values. To do this, two estimates are made of the population variance. These estimates rely on various assumptions (
4442:
369:
3695:
A computer typically determines a p-value from F which determines whether treatments produce significantly different results. If the result is significant, then the model provisionally has validity.
2951:
777:
3729:
In a more complex experiment, where the experimental units (or environmental effects) are not homogeneous, row statistics are also used in the analysis. The model includes terms dependent on
3136:
3039:
2686:
5656:
4947:
3911:
291:
3460:
3360:
2128:
5835:
1992:
728:
5572:
2057:
2692:
810:
681:
528:
1934:
4829:
2535:
5837:. Thus the first group is strongly different from the other groups, as the mean difference is more than 3 times the standard error, so we can be highly confident that the
454:
414:
5770:
at the 5% significance level. One would not accept the null hypothesis, concluding that there is strong evidence that the expected values in the three groups differ. The
1882:
4773:
3674:
3647:
4409:{\displaystyle {\overline {Y}}={\frac {\sum _{i}{\overline {Y}}_{i}}{a}}={\frac {{\overline {Y}}_{1}+{\overline {Y}}_{2}+{\overline {Y}}_{3}}{a}}={\frac {5+9+10}{3}}=8}
2433:
2379:
2323:
643:
592:
2400:
2344:
2233:
2182:
1814:
1765:
1636:
1499:
1470:
1053:
563:
2495:
1667:
1615:
1584:
1553:
1447:
1414:
1386:
1358:
1325:
1292:
1264:
1236:
1203:
1170:
1142:
1114:
932:
3724:
2464:
2261:
2212:
2161:
1842:
1793:
1744:
1716:
1081:
1032:
1004:
976:
483:
614:
3601:
3284:
3142:
2851:
3747:
2285:
1522:
894:
874:
854:
834:
3466:
6024:
Blair, R. C. (1981). "A reaction to 'Consequences of failure to meet assumptions underlying the fixed effects analysis of variance and covariance.'".
836:
over the experimental units can be interpreted several ways. In some experiments, the same experimental unit is subject to a range of treatments;
6091:
143:
6051:
Randolf, E. A.; Barcikowski, R. S. (1989). "Type I error rate when real study values are used as population parameters in a Monte Carlo study".
72:
Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a
6118:
5674:
176:, particularly for small alpha levels and unbalanced layouts. Furthermore, it is also claimed that if the underlying assumption of
127:
3890:-test for this experiment would be that all three levels of the factor produce the same response, on average. To calculate the
303:
2857:
6269:
6255:
734:
3050:
2957:
2598:
161:
The one-way ANOVA can be generalized to the factorial and multivariate layouts, as well as to the analysis of covariance.
5583:
231:
51:
3373:
113:
3290:
6179:
5954:
5901:
2070:
6261:
6208:
5788:
3692:
The three DFs and SSs are calculated from the summaries. Then the MSs are calculated and a ratio determines F.
1947:
6288:
5906:
687:
17:
5502:
3684:
The core ANOVA analysis consists of a series of calculations. The data is collected in tabular form. Then
2537:. The grand mean and grand variance are computed from the grand sums, not from group means and variances.
856:
may point to a particular unit. In others, each treatment group has a distinct set of experimental units;
222:
groups is used). Calculations of the means and the variance are performed as part of the hypothesis test.
69:. A higher ratio therefore implies that the samples were drawn from populations with different mean values.
2819:{\displaystyle \sum _{j}{\frac {(\sum _{i}y_{ij})^{2}}{I_{j}}}-{\frac {(\sum _{j}\sum _{i}y_{ij})^{2}}{I}}}
2009:
97:
100:
that examines the influence of two different categorical independent variables on one dependent variable.
5751:
The critical value is the number that the test statistic must exceed to reject the test. In this case,
782:
108:
The results of a one-way ANOVA can be considered reliable as long as the following assumptions are met:
6293:
6053:
Paper
Presented at the 11th Annual Meeting of the Mid-Western Educational Research Association, Chicago
650:
490:
42:) is a technique to compare whether two or more samples' means are significantly different (using the
1899:
6122:
4784:
2500:
5767:
421:
381:
1851:
4733:
3652:
6233:
5946:
5940:
3610:
204:
For nonparametric alternatives in the factorial layout, see
Sawilowsky. For more discussion see
2406:
2352:
2296:
621:
158:
ANOVA is a relatively robust procedure with respect to violations of the normality assumption.
570:
2385:
2329:
2218:
2167:
1799:
1750:
1621:
1484:
1455:
1038:
535:
66:
3252:{\displaystyle \sum _{j}\sum _{i}y_{ij}^{2}-\sum _{j}{\frac {(\sum _{i}y_{ij})^{2}}{I_{j}}}}
2474:
1642:
1590:
1559:
1528:
1422:
1392:
1364:
1336:
1300:
1270:
1242:
1214:
1178:
1148:
1120:
1092:
907:
6067:
5881:
3702:
2442:
2239:
2190:
2139:
1820:
1771:
1722:
1694:
1059:
1010:
982:
954:
461:
131:
117:
47:
5970:
Welch, B. L. (1951). "On the
Comparison of Several Mean Values: An Alternative Approach".
599:
211:
8:
3580:
3263:
2830:
173:
4837:
Calculate the "within-group" sum of squares. Begin by centering the data in each group
3569:{\displaystyle \sum _{j}\sum _{i}y_{ij}^{2}-{\frac {(\sum _{j}\sum _{i}y_{ij})^{2}}{I}}}
6226:
5987:
3732:
2270:
1507:
879:
859:
839:
819:
6265:
6204:
6175:
6007:
5950:
5916:
4936:
The within-group sum of squares is the sum of squares of all 18 values in this table
169:
147:
6143:
Sawilowsky, S. (1990). "Nonparametric tests of interaction in experimental design".
225:
The commonly used normal linear models for a completely randomized experiment are:
172:
when there are severe violations of the assumption that each population follows the
6152:
6104:
6100:
6033:
5979:
3749:. Determining the extra terms reduces the number of degrees of freedom available.
177:
6197:
58:
6251:
5858:
5838:
5782:
205:
43:
6156:
6037:
6282:
6068:"Power of the F-Test for Nonnormal Distributions and Unequal Error Variances"
5911:
139:
4727:
The between-group degrees of freedom is one less than the number of groups
181:
146:. If the variances are not known to be equal, a generalization of 2-sample
5896:
2550:
ANOVA table for fixed model, single factor, fully randomized experiment
5991:
212:
The case of fixed effects, fully randomized experiment, unbalanced data
31:
54:
variable "Y" and a single explanatory variable "X", hence "one-way".
5983:
5741:{\displaystyle F={\frac {MS_{B}}{MS_{W}}}\approx 42/4.5\approx 9.3}
142:, a non-parametric alternative to this test should be used such as
5771:
899:
5886:
81:
77:
73:
6232:(2nd ed.). New York: Holt, Rinehart and Winston. p.
485:
is the number of experimental units in the jth treatment group
76:(Gosset, 1908). When there are only two means to compare, the
645:
is the jth treatment effect, a deviation from the grand mean
164:
It is often stated in popular literature that none of these
6009:
Experimental Design: Procedures For The
Behavioral Sciences
594:
is the mean of the observations for the jth treatment group
4431:
Calculate the "between-group" sum of squared differences:
364:{\displaystyle y_{i,j}=\mu +\tau _{j}+\varepsilon _{i,j}}
2946:{\displaystyle {\frac {SS_{Treatment}}{DF_{Treatment}}}}
6072:
Paper
Prepared for United States Air Force Project RAND
153:
6174:(5th ed.). New York: Wiley. p. Section 3–2.
772:{\displaystyle \varepsilon \thicksim N(0,\sigma ^{2})}
5934:
5932:
5791:
5677:
5586:
5505:
4945:
4787:
4736:
4440:
4254:
3909:
3735:
3705:
3655:
3613:
3583:
3469:
3376:
3293:
3266:
3145:
3053:
2960:
2860:
2833:
2695:
2601:
2503:
2477:
2445:
2409:
2388:
2355:
2332:
2299:
2273:
2242:
2221:
2193:
2170:
2142:
2073:
2012:
1950:
1902:
1854:
1823:
1802:
1774:
1753:
1725:
1697:
1645:
1624:
1593:
1562:
1531:
1510:
1487:
1458:
1425:
1395:
1367:
1339:
1303:
1273:
1245:
1217:
1181:
1151:
1123:
1095:
1062:
1041:
1013:
985:
957:
910:
882:
862:
842:
822:
785:
737:
690:
653:
624:
602:
573:
538:
493:
464:
424:
384:
306:
234:
3131:{\displaystyle \sum _{Treatments}(I_{j}-1)s_{j}^{2}}
3034:{\displaystyle {\frac {MS_{Treatment}}{MS_{Error}}}}
2681:{\displaystyle \sum _{Treatments}I_{j}(m_{j}-m)^{2}}
6203:(4th ed.). W H Freeman & Co. p. 764.
5651:{\displaystyle MS_{W}=S_{W}/f_{W}=68/15\approx 4.5}
938:ANOVA data organization, Unbalanced, Single factor
286:{\displaystyle y_{i,j}=\mu _{j}+\varepsilon _{i,j}}
6225:
6196:
6050:
6013:(3 ed.). Pacific Grove, CA, USA: Brooks/Cole.
6006:
5929:
5829:
5740:
5650:
5566:
5485:
4823:
4767:
4709:
4408:
4231:
3779:are the three levels of the factor being studied.
3741:
3718:
3668:
3641:
3595:
3568:
3455:{\displaystyle \sum _{Observations}(y_{ij}-m)^{2}}
3454:
3354:
3278:
3251:
3130:
3033:
2945:
2845:
2818:
2680:
2529:
2489:
2458:
2427:
2394:
2373:
2338:
2317:
2279:
2255:
2227:
2206:
2176:
2155:
2122:
2051:
1986:
1928:
1876:
1836:
1808:
1787:
1759:
1738:
1710:
1661:
1630:
1609:
1578:
1547:
1516:
1493:
1464:
1441:
1408:
1380:
1352:
1319:
1286:
1258:
1230:
1197:
1164:
1136:
1108:
1075:
1047:
1026:
998:
970:
926:
888:
868:
848:
828:
804:
771:
722:
675:
637:
608:
586:
557:
522:
477:
448:
408:
363:
285:
904:One form of organizing experimental observations
812:are normally distributed zero-mean random errors.
6280:
6228:Statistics: Probability, Inference, and Decision
3726:terms are equal so the SS equations simplify.
3355:{\displaystyle {\frac {SS_{Error}}{DF_{Error}}}}
6092:Journal of the American Statistical Association
84:are equivalent; the relation between ANOVA and
6250:
2123:{\displaystyle \sum _{j}\sum _{i}(y_{ij})^{2}}
900:The data and statistical summaries of the data
6224:Winkler, Robert L.; Hays, William L. (1975).
3649:is the estimate of variance corresponding to
6223:
6195:Moore, David S.; McCabe, George P. (2003).
6194:
6085:Tiku, M. L. (1971). "Power Function of the
5577:Thus the within-group mean square value is
144:Kruskal–Wallis one-way analysis of variance
6199:Introduction to the Practice of Statistics
6169:
6142:
6119:"Getting Started with Statistics Concepts"
5902:Multivariate analysis of variance (MANOVA)
5766:=9.3 > 3.68, the results are
4778:so the between-group mean square value is
3699:If the experiment is balanced, all of the
184:properties degenerate much more severely.
6065:
5830:{\displaystyle {\sqrt {4.5/6+4.5/6}}=1.2}
530:is the total number of experimental units
5963:
5868:degrees of freedom in the numerator and
4724:is the number of data values per group.
1987:{\displaystyle \sum _{j}\sum _{i}y_{ij}}
120:(or approximately normally distributed).
5872:degrees of freedom in the denominator.
5496:The within-group degrees of freedom is
723:{\displaystyle \mu _{j}=\mu +\tau _{j}}
128:independent and identically distributed
14:
6281:
5938:
5864:cumulative distribution function with
5567:{\displaystyle f_{W}=a(n-1)=3(6-1)=15}
3900:Calculate the mean within each group:
2540:
6023:
5969:
2052:{\displaystyle \sum _{i}(y_{ij})^{2}}
616:is the grand mean of the observations
6089:-Test Under Non-Normal Situations".
6084:
6004:
154:Departures from population normality
5998:
3679:
416:is an index over experimental units
123:Variances of populations are equal.
96:. An extension of one-way ANOVA is
24:
6244:
6172:Design and Analysis of Experiments
5942:Statistical Methods for Psychology
805:{\displaystyle \varepsilon _{i,j}}
25:
6305:
741:
456:is an index over treatment groups
5891:Includes a one-way ANOVA example
5785:of each of these differences is
2548:
936:
876:may simply be an index into the
676:{\displaystyle \sum \tau _{j}=0}
523:{\displaystyle I=\sum _{j}I_{j}}
126:Responses for a given group are
62:
6217:
6188:
6170:Montgomery, Douglas C. (2001).
6163:
6066:Donaldson, Theodore S. (1966).
1929:{\displaystyle \sum _{i}y_{ij}}
130:normal random variables (not a
6145:Review of Educational Research
6136:
6111:
6105:10.1080/01621459.1971.10482371
6078:
6059:
6044:
6026:Review of Educational Research
6017:
5555:
5543:
5534:
5522:
5338:
5331:
5319:
5309:
5297:
5287:
5275:
5268:
5256:
5246:
5234:
5227:
5208:
5198:
5186:
5176:
5164:
5157:
5145:
5138:
5126:
5119:
5107:
5097:
5078:
5068:
5056:
5046:
5034:
5027:
5015:
5005:
4993:
4986:
4974:
4967:
4824:{\displaystyle MS_{B}=84/2=42}
4688:
4675:
4660:
4647:
4632:
4619:
4597:
4563:
4548:
4514:
4499:
4465:
3882:The null hypothesis, denoted H
3551:
3514:
3443:
3420:
3227:
3200:
3110:
3091:
2801:
2764:
2736:
2709:
2669:
2649:
2530:{\displaystyle \mu _{j}=m_{j}}
2471:Comparing model to summaries:
2111:
2094:
2040:
2023:
766:
747:
103:
13:
1:
449:{\displaystyle j=1,\dotsc ,J}
409:{\displaystyle i=1,\dotsc ,I}
50:technique requires a numeric
4591:
4572:
4542:
4523:
4493:
4474:
4356:
4336:
4316:
4287:
4260:
4245:Calculate the overall mean:
4130:
4025:
3920:
3689:means are used in the model.
1877:{\displaystyle I=\sum I_{j}}
945:Lists of Group Observations
216:
98:two-way analysis of variance
36:one-way analysis of variance
7:
5875:
4768:{\displaystyle f_{b}=3-1=2}
3669:{\displaystyle \sigma ^{2}}
934:is with groups in columns:
10:
6310:
3752:
3642:{\displaystyle MS_{Error}}
6157:10.3102/00346543060001091
6038:10.3102/00346543051004499
2428:{\displaystyle s_{J}^{2}}
2374:{\displaystyle s_{j}^{2}}
2318:{\displaystyle s_{1}^{2}}
1684:Grand Summary Statistics
1683:
1681:Group Summary Statistics
1680:
944:
638:{\displaystyle \tau _{j}}
5922:
5774:for this test is 0.002.
4424:is the number of groups.
587:{\displaystyle \mu _{j}}
5907:Repeated measures ANOVA
2395:{\displaystyle \dotso }
2339:{\displaystyle \dotso }
2228:{\displaystyle \dotso }
2177:{\displaystyle \dotso }
1809:{\displaystyle \dotso }
1760:{\displaystyle \dotso }
1631:{\displaystyle \dotso }
1494:{\displaystyle \vdots }
1465:{\displaystyle \vdots }
1048:{\displaystyle \dotso }
558:{\displaystyle y_{i,j}}
5939:Howell, David (2002).
5831:
5742:
5652:
5568:
5487:
4825:
4769:
4711:
4410:
4233:
3743:
3720:
3670:
3643:
3597:
3570:
3456:
3356:
3280:
3253:
3132:
3035:
2947:
2847:
2820:
2682:
2531:
2491:
2490:{\displaystyle \mu =m}
2460:
2429:
2396:
2375:
2340:
2319:
2281:
2257:
2229:
2208:
2178:
2157:
2124:
2053:
1988:
1930:
1878:
1838:
1810:
1789:
1761:
1740:
1712:
1663:
1662:{\displaystyle y_{ij}}
1632:
1611:
1610:{\displaystyle y_{i3}}
1580:
1579:{\displaystyle y_{i2}}
1549:
1548:{\displaystyle y_{i1}}
1518:
1495:
1466:
1443:
1442:{\displaystyle y_{3j}}
1410:
1409:{\displaystyle y_{33}}
1382:
1381:{\displaystyle y_{32}}
1354:
1353:{\displaystyle y_{31}}
1321:
1320:{\displaystyle y_{2j}}
1288:
1287:{\displaystyle y_{23}}
1260:
1259:{\displaystyle y_{22}}
1232:
1231:{\displaystyle y_{21}}
1199:
1198:{\displaystyle y_{1j}}
1166:
1165:{\displaystyle y_{13}}
1138:
1137:{\displaystyle y_{12}}
1110:
1109:{\displaystyle y_{11}}
1077:
1049:
1028:
1000:
972:
928:
927:{\displaystyle y_{ij}}
890:
870:
850:
830:
806:
773:
724:
677:
639:
610:
588:
559:
524:
479:
450:
410:
365:
287:
5832:
5777:After performing the
5743:
5653:
5569:
5488:
4826:
4770:
4712:
4411:
4234:
3744:
3721:
3719:{\displaystyle I_{j}}
3671:
3644:
3598:
3571:
3457:
3357:
3281:
3254:
3133:
3036:
2948:
2848:
2821:
2683:
2532:
2492:
2461:
2459:{\displaystyle s^{2}}
2430:
2397:
2376:
2341:
2320:
2282:
2258:
2256:{\displaystyle m_{J}}
2230:
2209:
2207:{\displaystyle m_{j}}
2179:
2158:
2156:{\displaystyle m_{1}}
2125:
2054:
1989:
1931:
1879:
1839:
1837:{\displaystyle I_{J}}
1811:
1790:
1788:{\displaystyle I_{j}}
1762:
1741:
1739:{\displaystyle I_{2}}
1713:
1711:{\displaystyle I_{1}}
1664:
1633:
1612:
1581:
1550:
1519:
1496:
1467:
1444:
1411:
1383:
1355:
1322:
1289:
1261:
1233:
1200:
1167:
1139:
1111:
1078:
1076:{\displaystyle I_{j}}
1050:
1029:
1027:{\displaystyle I_{3}}
1001:
999:{\displaystyle I_{2}}
973:
971:{\displaystyle I_{1}}
929:
891:
871:
851:
831:
807:
774:
725:
678:
640:
611:
589:
560:
525:
480:
478:{\displaystyle I_{j}}
451:
411:
366:
288:
67:central limit theorem
6289:Analysis of variance
5945:. Duxbury. pp.
5882:Analysis of variance
5789:
5675:
5584:
5503:
4943:
4785:
4734:
4438:
4252:
3907:
3733:
3703:
3653:
3611:
3581:
3467:
3374:
3291:
3264:
3143:
3051:
2958:
2858:
2831:
2693:
2599:
2555:Source of variation
2501:
2475:
2443:
2407:
2386:
2353:
2330:
2297:
2271:
2240:
2219:
2191:
2168:
2140:
2071:
2010:
1948:
1900:
1852:
1821:
1800:
1772:
1751:
1723:
1695:
1643:
1622:
1591:
1560:
1529:
1508:
1485:
1456:
1423:
1393:
1365:
1337:
1301:
1271:
1243:
1215:
1179:
1149:
1121:
1093:
1060:
1039:
1011:
983:
955:
908:
880:
860:
840:
820:
783:
735:
688:
651:
622:
609:{\displaystyle \mu }
600:
571:
536:
491:
462:
422:
382:
304:
232:
132:simple random sample
118:normally distributed
57:The ANOVA tests the
48:analysis of variance
3596:{\displaystyle I-1}
3507:
3279:{\displaystyle I-J}
3183:
3127:
2846:{\displaystyle J-1}
2564:Degrees of freedom
2551:
2541:The hypothesis test
2424:
2370:
2314:
939:
371:(the effects model)
174:normal distribution
6257:Statistical design
5827:
5738:
5648:
5564:
5483:
5481:
4821:
4765:
4707:
4705:
4406:
4280:
4229:
4227:
3886:, for the overall
3739:
3716:
3666:
3639:
3593:
3566:
3536:
3526:
3490:
3489:
3479:
3452:
3419:
3352:
3276:
3249:
3212:
3196:
3166:
3165:
3155:
3128:
3113:
3090:
3031:
2943:
2843:
2816:
2786:
2776:
2721:
2705:
2678:
2638:
2549:
2527:
2487:
2456:
2425:
2410:
2392:
2371:
2356:
2336:
2315:
2300:
2277:
2253:
2225:
2204:
2174:
2153:
2120:
2093:
2083:
2049:
2022:
1984:
1970:
1960:
1926:
1912:
1874:
1834:
1806:
1785:
1757:
1736:
1708:
1659:
1628:
1607:
1576:
1545:
1514:
1491:
1462:
1439:
1406:
1378:
1350:
1317:
1284:
1256:
1228:
1195:
1162:
1134:
1106:
1073:
1045:
1024:
996:
968:
937:
924:
886:
866:
846:
826:
802:
769:
720:
673:
635:
606:
584:
555:
520:
509:
475:
446:
406:
361:
283:
112:Response variable
6294:Statistical tests
6271:978-0-387-75965-4
6254:(18 April 2008).
6005:Kirk, RE (1995).
5819:
5758:(2,15) = 3.68 at
5716:
5475:
5358:
4934:
4933:
4594:
4575:
4545:
4526:
4496:
4477:
4398:
4371:
4359:
4339:
4319:
4302:
4290:
4271:
4263:
4217:
4156:
4133:
4112:
4051:
4028:
4007:
3946:
3923:
3878:
3877:
3742:{\displaystyle i}
3606:
3605:
3564:
3527:
3517:
3480:
3470:
3377:
3350:
3247:
3203:
3187:
3156:
3146:
3054:
3029:
2941:
2814:
2777:
2767:
2756:
2712:
2696:
2602:
2580:Computational SS
2469:
2468:
2280:{\displaystyle m}
2084:
2074:
2013:
1961:
1951:
1903:
1517:{\displaystyle i}
889:{\displaystyle j}
869:{\displaystyle i}
849:{\displaystyle i}
829:{\displaystyle i}
500:
293:(the means model)
180:is violated, the
16:(Redirected from
6301:
6275:
6238:
6237:
6231:
6221:
6215:
6214:
6202:
6192:
6186:
6185:
6167:
6161:
6160:
6140:
6134:
6133:
6131:
6130:
6121:. Archived from
6115:
6109:
6108:
6099:(336): 913–916.
6082:
6076:
6075:
6063:
6057:
6056:
6048:
6042:
6041:
6021:
6015:
6014:
6012:
6002:
5996:
5995:
5978:(3/4): 330–336.
5967:
5961:
5960:
5936:
5836:
5834:
5833:
5828:
5820:
5815:
5801:
5793:
5747:
5745:
5744:
5739:
5728:
5717:
5715:
5714:
5713:
5700:
5699:
5698:
5685:
5657:
5655:
5654:
5649:
5638:
5627:
5626:
5617:
5612:
5611:
5599:
5598:
5573:
5571:
5570:
5565:
5515:
5514:
5492:
5490:
5489:
5484:
5482:
5473:
5356:
5346:
5345:
5327:
5326:
5305:
5304:
5283:
5282:
5264:
5263:
5242:
5241:
5223:
5216:
5215:
5194:
5193:
5172:
5171:
5153:
5152:
5134:
5133:
5115:
5114:
5093:
5086:
5085:
5064:
5063:
5042:
5041:
5023:
5022:
5001:
5000:
4982:
4981:
4959:
4958:
4840:
4839:
4830:
4828:
4827:
4822:
4811:
4800:
4799:
4774:
4772:
4771:
4766:
4746:
4745:
4716:
4714:
4713:
4708:
4706:
4696:
4695:
4668:
4667:
4640:
4639:
4609:
4605:
4604:
4595:
4587:
4582:
4581:
4576:
4568:
4556:
4555:
4546:
4538:
4533:
4532:
4527:
4519:
4507:
4506:
4497:
4489:
4484:
4483:
4478:
4470:
4454:
4453:
4415:
4413:
4412:
4407:
4399:
4394:
4377:
4372:
4367:
4366:
4365:
4360:
4352:
4346:
4345:
4340:
4332:
4326:
4325:
4320:
4312:
4308:
4303:
4298:
4297:
4296:
4291:
4283:
4279:
4269:
4264:
4256:
4238:
4236:
4235:
4230:
4228:
4218:
4213:
4178:
4173:
4172:
4157:
4149:
4140:
4139:
4134:
4126:
4113:
4108:
4073:
4068:
4067:
4052:
4044:
4035:
4034:
4029:
4021:
4008:
4003:
3968:
3963:
3962:
3947:
3939:
3930:
3929:
3924:
3916:
3784:
3783:
3748:
3746:
3745:
3740:
3725:
3723:
3722:
3717:
3715:
3714:
3680:Analysis summary
3675:
3673:
3672:
3667:
3665:
3664:
3648:
3646:
3645:
3640:
3638:
3637:
3602:
3600:
3599:
3594:
3575:
3573:
3572:
3567:
3565:
3560:
3559:
3558:
3549:
3548:
3535:
3525:
3512:
3506:
3501:
3488:
3478:
3461:
3459:
3458:
3453:
3451:
3450:
3435:
3434:
3418:
3361:
3359:
3358:
3353:
3351:
3349:
3348:
3347:
3322:
3321:
3320:
3295:
3285:
3283:
3282:
3277:
3258:
3256:
3255:
3250:
3248:
3246:
3245:
3236:
3235:
3234:
3225:
3224:
3211:
3198:
3195:
3182:
3177:
3164:
3154:
3137:
3135:
3134:
3129:
3126:
3121:
3103:
3102:
3089:
3040:
3038:
3037:
3032:
3030:
3028:
3027:
3026:
3001:
3000:
2999:
2962:
2952:
2950:
2949:
2944:
2942:
2940:
2939:
2938:
2901:
2900:
2899:
2862:
2852:
2850:
2849:
2844:
2825:
2823:
2822:
2817:
2815:
2810:
2809:
2808:
2799:
2798:
2785:
2775:
2762:
2757:
2755:
2754:
2745:
2744:
2743:
2734:
2733:
2720:
2707:
2704:
2687:
2685:
2684:
2679:
2677:
2676:
2661:
2660:
2648:
2647:
2637:
2561:Sums of squares
2558:Sums of squares
2552:
2536:
2534:
2533:
2528:
2526:
2525:
2513:
2512:
2496:
2494:
2493:
2488:
2465:
2463:
2462:
2457:
2455:
2454:
2434:
2432:
2431:
2426:
2423:
2418:
2401:
2399:
2398:
2393:
2380:
2378:
2377:
2372:
2369:
2364:
2345:
2343:
2342:
2337:
2324:
2322:
2321:
2316:
2313:
2308:
2286:
2284:
2283:
2278:
2262:
2260:
2259:
2254:
2252:
2251:
2234:
2232:
2231:
2226:
2213:
2211:
2210:
2205:
2203:
2202:
2183:
2181:
2180:
2175:
2162:
2160:
2159:
2154:
2152:
2151:
2129:
2127:
2126:
2121:
2119:
2118:
2109:
2108:
2092:
2082:
2058:
2056:
2055:
2050:
2048:
2047:
2038:
2037:
2021:
1993:
1991:
1990:
1985:
1983:
1982:
1969:
1959:
1935:
1933:
1932:
1927:
1925:
1924:
1911:
1883:
1881:
1880:
1875:
1873:
1872:
1843:
1841:
1840:
1835:
1833:
1832:
1815:
1813:
1812:
1807:
1794:
1792:
1791:
1786:
1784:
1783:
1766:
1764:
1763:
1758:
1745:
1743:
1742:
1737:
1735:
1734:
1717:
1715:
1714:
1709:
1707:
1706:
1668:
1666:
1665:
1660:
1658:
1657:
1637:
1635:
1634:
1629:
1616:
1614:
1613:
1608:
1606:
1605:
1585:
1583:
1582:
1577:
1575:
1574:
1554:
1552:
1551:
1546:
1544:
1543:
1523:
1521:
1520:
1515:
1500:
1498:
1497:
1492:
1471:
1469:
1468:
1463:
1448:
1446:
1445:
1440:
1438:
1437:
1415:
1413:
1412:
1407:
1405:
1404:
1387:
1385:
1384:
1379:
1377:
1376:
1359:
1357:
1356:
1351:
1349:
1348:
1326:
1324:
1323:
1318:
1316:
1315:
1293:
1291:
1290:
1285:
1283:
1282:
1265:
1263:
1262:
1257:
1255:
1254:
1237:
1235:
1234:
1229:
1227:
1226:
1204:
1202:
1201:
1196:
1194:
1193:
1171:
1169:
1168:
1163:
1161:
1160:
1143:
1141:
1140:
1135:
1133:
1132:
1115:
1113:
1112:
1107:
1105:
1104:
1082:
1080:
1079:
1074:
1072:
1071:
1054:
1052:
1051:
1046:
1033:
1031:
1030:
1025:
1023:
1022:
1005:
1003:
1002:
997:
995:
994:
977:
975:
974:
969:
967:
966:
940:
933:
931:
930:
925:
923:
922:
895:
893:
892:
887:
875:
873:
872:
867:
855:
853:
852:
847:
835:
833:
832:
827:
811:
809:
808:
803:
801:
800:
778:
776:
775:
770:
765:
764:
729:
727:
726:
721:
719:
718:
700:
699:
682:
680:
679:
674:
666:
665:
644:
642:
641:
636:
634:
633:
615:
613:
612:
607:
593:
591:
590:
585:
583:
582:
565:are observations
564:
562:
561:
556:
554:
553:
529:
527:
526:
521:
519:
518:
508:
484:
482:
481:
476:
474:
473:
455:
453:
452:
447:
415:
413:
412:
407:
370:
368:
367:
362:
360:
359:
341:
340:
322:
321:
292:
290:
289:
284:
282:
281:
263:
262:
250:
249:
178:homoscedasticity
27:Statistical test
21:
6309:
6308:
6304:
6303:
6302:
6300:
6299:
6298:
6279:
6278:
6272:
6247:
6245:Further reading
6242:
6241:
6222:
6218:
6211:
6193:
6189:
6182:
6168:
6164:
6141:
6137:
6128:
6126:
6117:
6116:
6112:
6083:
6079:
6064:
6060:
6049:
6045:
6022:
6018:
6003:
5999:
5984:10.2307/2332579
5968:
5964:
5957:
5937:
5930:
5925:
5878:
5839:population mean
5811:
5797:
5792:
5790:
5787:
5786:
5762:= 0.05. Since
5757:
5724:
5709:
5705:
5701:
5694:
5690:
5686:
5684:
5676:
5673:
5672:
5634:
5622:
5618:
5613:
5607:
5603:
5594:
5590:
5585:
5582:
5581:
5510:
5506:
5504:
5501:
5500:
5480:
5479:
5471:
5465:
5464:
5354:
5348:
5347:
5341:
5337:
5322:
5318:
5300:
5296:
5278:
5274:
5259:
5255:
5237:
5233:
5221:
5220:
5211:
5207:
5189:
5185:
5167:
5163:
5148:
5144:
5129:
5125:
5110:
5106:
5091:
5090:
5081:
5077:
5059:
5055:
5037:
5033:
5018:
5014:
4996:
4992:
4977:
4973:
4963:
4954:
4950:
4946:
4944:
4941:
4940:
4864:
4856:
4848:
4807:
4795:
4791:
4786:
4783:
4782:
4741:
4737:
4735:
4732:
4731:
4704:
4703:
4691:
4687:
4663:
4659:
4635:
4631:
4607:
4606:
4600:
4596:
4586:
4577:
4567:
4566:
4551:
4547:
4537:
4528:
4518:
4517:
4502:
4498:
4488:
4479:
4469:
4468:
4455:
4449:
4445:
4441:
4439:
4436:
4435:
4378:
4376:
4361:
4351:
4350:
4341:
4331:
4330:
4321:
4311:
4310:
4309:
4307:
4292:
4282:
4281:
4275:
4270:
4268:
4255:
4253:
4250:
4249:
4226:
4225:
4179:
4177:
4165:
4161:
4148:
4141:
4135:
4125:
4124:
4121:
4120:
4074:
4072:
4060:
4056:
4043:
4036:
4030:
4020:
4019:
4016:
4015:
3969:
3967:
3955:
3951:
3938:
3931:
3925:
3915:
3914:
3910:
3908:
3905:
3904:
3885:
3808:
3800:
3792:
3778:
3771:
3764:
3755:
3734:
3731:
3730:
3710:
3706:
3704:
3701:
3700:
3682:
3676:of the model.
3660:
3656:
3654:
3651:
3650:
3621:
3617:
3612:
3609:
3608:
3582:
3579:
3578:
3554:
3550:
3541:
3537:
3531:
3521:
3513:
3511:
3502:
3494:
3484:
3474:
3468:
3465:
3464:
3446:
3442:
3427:
3423:
3381:
3375:
3372:
3371:
3331:
3327:
3323:
3304:
3300:
3296:
3294:
3292:
3289:
3288:
3265:
3262:
3261:
3241:
3237:
3230:
3226:
3217:
3213:
3207:
3199:
3197:
3191:
3178:
3170:
3160:
3150:
3144:
3141:
3140:
3122:
3117:
3098:
3094:
3058:
3052:
3049:
3048:
3010:
3006:
3002:
2971:
2967:
2963:
2961:
2959:
2956:
2955:
2910:
2906:
2902:
2871:
2867:
2863:
2861:
2859:
2856:
2855:
2832:
2829:
2828:
2804:
2800:
2791:
2787:
2781:
2771:
2763:
2761:
2750:
2746:
2739:
2735:
2726:
2722:
2716:
2708:
2706:
2700:
2694:
2691:
2690:
2672:
2668:
2656:
2652:
2643:
2639:
2606:
2600:
2597:
2596:
2577:Explanatory SS
2543:
2521:
2517:
2508:
2504:
2502:
2499:
2498:
2476:
2473:
2472:
2450:
2446:
2444:
2441:
2440:
2419:
2414:
2408:
2405:
2404:
2387:
2384:
2383:
2365:
2360:
2354:
2351:
2350:
2331:
2328:
2327:
2309:
2304:
2298:
2295:
2294:
2272:
2269:
2268:
2247:
2243:
2241:
2238:
2237:
2220:
2217:
2216:
2198:
2194:
2192:
2189:
2188:
2169:
2166:
2165:
2147:
2143:
2141:
2138:
2137:
2114:
2110:
2101:
2097:
2088:
2078:
2072:
2069:
2068:
2043:
2039:
2030:
2026:
2017:
2011:
2008:
2007:
1975:
1971:
1965:
1955:
1949:
1946:
1945:
1917:
1913:
1907:
1901:
1898:
1897:
1868:
1864:
1853:
1850:
1849:
1828:
1824:
1822:
1819:
1818:
1801:
1798:
1797:
1779:
1775:
1773:
1770:
1769:
1752:
1749:
1748:
1730:
1726:
1724:
1721:
1720:
1702:
1698:
1696:
1693:
1692:
1650:
1646:
1644:
1641:
1640:
1623:
1620:
1619:
1598:
1594:
1592:
1589:
1588:
1567:
1563:
1561:
1558:
1557:
1536:
1532:
1530:
1527:
1526:
1509:
1506:
1505:
1486:
1483:
1482:
1457:
1454:
1453:
1430:
1426:
1424:
1421:
1420:
1400:
1396:
1394:
1391:
1390:
1372:
1368:
1366:
1363:
1362:
1344:
1340:
1338:
1335:
1334:
1308:
1304:
1302:
1299:
1298:
1278:
1274:
1272:
1269:
1268:
1250:
1246:
1244:
1241:
1240:
1222:
1218:
1216:
1213:
1212:
1186:
1182:
1180:
1177:
1176:
1156:
1152:
1150:
1147:
1146:
1128:
1124:
1122:
1119:
1118:
1100:
1096:
1094:
1091:
1090:
1067:
1063:
1061:
1058:
1057:
1040:
1037:
1036:
1018:
1014:
1012:
1009:
1008:
990:
986:
984:
981:
980:
962:
958:
956:
953:
952:
915:
911:
909:
906:
905:
902:
881:
878:
877:
861:
858:
857:
841:
838:
837:
821:
818:
817:
790:
786:
784:
781:
780:
760:
756:
736:
733:
732:
714:
710:
695:
691:
689:
686:
685:
661:
657:
652:
649:
648:
629:
625:
623:
620:
619:
601:
598:
597:
578:
574:
572:
569:
568:
543:
539:
537:
534:
533:
514:
510:
504:
492:
489:
488:
469:
465:
463:
460:
459:
423:
420:
419:
383:
380:
379:
349:
345:
336:
332:
311:
307:
305:
302:
301:
271:
267:
258:
254:
239:
235:
233:
230:
229:
219:
214:
156:
106:
59:null hypothesis
28:
23:
22:
15:
12:
11:
5:
6307:
6297:
6296:
6291:
6277:
6276:
6270:
6252:George Casella
6246:
6243:
6240:
6239:
6216:
6209:
6187:
6180:
6162:
6135:
6110:
6077:
6058:
6043:
6032:(4): 499–507.
6016:
5997:
5962:
5955:
5927:
5926:
5924:
5921:
5920:
5919:
5917:Welch's t-test
5914:
5909:
5904:
5899:
5894:
5884:
5877:
5874:
5826:
5823:
5818:
5814:
5810:
5807:
5804:
5800:
5796:
5783:standard error
5755:
5749:
5748:
5737:
5734:
5731:
5727:
5723:
5720:
5712:
5708:
5704:
5697:
5693:
5689:
5683:
5680:
5659:
5658:
5647:
5644:
5641:
5637:
5633:
5630:
5625:
5621:
5616:
5610:
5606:
5602:
5597:
5593:
5589:
5575:
5574:
5563:
5560:
5557:
5554:
5551:
5548:
5545:
5542:
5539:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5513:
5509:
5494:
5493:
5478:
5472:
5470:
5467:
5466:
5463:
5460:
5457:
5454:
5451:
5448:
5445:
5442:
5439:
5436:
5433:
5430:
5427:
5424:
5421:
5418:
5415:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5379:
5376:
5373:
5370:
5367:
5364:
5361:
5355:
5353:
5350:
5349:
5344:
5340:
5336:
5333:
5330:
5325:
5321:
5317:
5314:
5311:
5308:
5303:
5299:
5295:
5292:
5289:
5286:
5281:
5277:
5273:
5270:
5267:
5262:
5258:
5254:
5251:
5248:
5245:
5240:
5236:
5232:
5229:
5226:
5224:
5222:
5219:
5214:
5210:
5206:
5203:
5200:
5197:
5192:
5188:
5184:
5181:
5178:
5175:
5170:
5166:
5162:
5159:
5156:
5151:
5147:
5143:
5140:
5137:
5132:
5128:
5124:
5121:
5118:
5113:
5109:
5105:
5102:
5099:
5096:
5094:
5092:
5089:
5084:
5080:
5076:
5073:
5070:
5067:
5062:
5058:
5054:
5051:
5048:
5045:
5040:
5036:
5032:
5029:
5026:
5021:
5017:
5013:
5010:
5007:
5004:
4999:
4995:
4991:
4988:
4985:
4980:
4976:
4972:
4969:
4966:
4964:
4962:
4957:
4953:
4949:
4948:
4932:
4931:
4928:
4925:
4921:
4920:
4917:
4914:
4910:
4909:
4906:
4903:
4899:
4898:
4895:
4892:
4888:
4887:
4884:
4881:
4877:
4876:
4873:
4870:
4866:
4865:
4862:
4857:
4854:
4849:
4846:
4832:
4831:
4820:
4817:
4814:
4810:
4806:
4803:
4798:
4794:
4790:
4776:
4775:
4764:
4761:
4758:
4755:
4752:
4749:
4744:
4740:
4718:
4717:
4702:
4699:
4694:
4690:
4686:
4683:
4680:
4677:
4674:
4671:
4666:
4662:
4658:
4655:
4652:
4649:
4646:
4643:
4638:
4634:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4610:
4608:
4603:
4599:
4593:
4590:
4585:
4580:
4574:
4571:
4565:
4562:
4559:
4554:
4550:
4544:
4541:
4536:
4531:
4525:
4522:
4516:
4513:
4510:
4505:
4501:
4495:
4492:
4487:
4482:
4476:
4473:
4467:
4464:
4461:
4458:
4456:
4452:
4448:
4444:
4443:
4426:
4425:
4417:
4416:
4405:
4402:
4397:
4393:
4390:
4387:
4384:
4381:
4375:
4370:
4364:
4358:
4355:
4349:
4344:
4338:
4335:
4329:
4324:
4318:
4315:
4306:
4301:
4295:
4289:
4286:
4278:
4274:
4267:
4262:
4259:
4240:
4239:
4224:
4221:
4216:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4176:
4171:
4168:
4164:
4160:
4155:
4152:
4147:
4144:
4142:
4138:
4132:
4129:
4123:
4122:
4119:
4116:
4111:
4107:
4104:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4080:
4077:
4071:
4066:
4063:
4059:
4055:
4050:
4047:
4042:
4039:
4037:
4033:
4027:
4024:
4018:
4017:
4014:
4011:
4006:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3966:
3961:
3958:
3954:
3950:
3945:
3942:
3937:
3934:
3932:
3928:
3922:
3919:
3913:
3912:
3883:
3880:
3879:
3876:
3875:
3872:
3869:
3865:
3864:
3861:
3858:
3854:
3853:
3850:
3847:
3843:
3842:
3839:
3836:
3832:
3831:
3828:
3825:
3821:
3820:
3817:
3814:
3810:
3809:
3806:
3801:
3798:
3793:
3790:
3776:
3769:
3762:
3754:
3751:
3738:
3713:
3709:
3697:
3696:
3693:
3690:
3681:
3678:
3663:
3659:
3636:
3633:
3630:
3627:
3624:
3620:
3616:
3604:
3603:
3592:
3589:
3586:
3576:
3563:
3557:
3553:
3547:
3544:
3540:
3534:
3530:
3524:
3520:
3516:
3510:
3505:
3500:
3497:
3493:
3487:
3483:
3477:
3473:
3462:
3449:
3445:
3441:
3438:
3433:
3430:
3426:
3422:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3380:
3369:
3365:
3364:
3362:
3346:
3343:
3340:
3337:
3334:
3330:
3326:
3319:
3316:
3313:
3310:
3307:
3303:
3299:
3286:
3275:
3272:
3269:
3259:
3244:
3240:
3233:
3229:
3223:
3220:
3216:
3210:
3206:
3202:
3194:
3190:
3186:
3181:
3176:
3173:
3169:
3163:
3159:
3153:
3149:
3138:
3125:
3120:
3116:
3112:
3109:
3106:
3101:
3097:
3093:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3057:
3046:
3042:
3041:
3025:
3022:
3019:
3016:
3013:
3009:
3005:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2970:
2966:
2953:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2909:
2905:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2870:
2866:
2853:
2842:
2839:
2836:
2826:
2813:
2807:
2803:
2797:
2794:
2790:
2784:
2780:
2774:
2770:
2766:
2760:
2753:
2749:
2742:
2738:
2732:
2729:
2725:
2719:
2715:
2711:
2703:
2699:
2688:
2675:
2671:
2667:
2664:
2659:
2655:
2651:
2646:
2642:
2636:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2605:
2594:
2590:
2589:
2587:
2584:
2581:
2578:
2575:
2572:
2571:
2568:
2565:
2562:
2559:
2556:
2542:
2539:
2524:
2520:
2516:
2511:
2507:
2486:
2483:
2480:
2467:
2466:
2453:
2449:
2438:
2435:
2422:
2417:
2413:
2402:
2391:
2381:
2368:
2363:
2359:
2348:
2346:
2335:
2325:
2312:
2307:
2303:
2292:
2288:
2287:
2276:
2266:
2263:
2250:
2246:
2235:
2224:
2214:
2201:
2197:
2186:
2184:
2173:
2163:
2150:
2146:
2135:
2131:
2130:
2117:
2113:
2107:
2104:
2100:
2096:
2091:
2087:
2081:
2077:
2066:
2063:
2061:
2059:
2046:
2042:
2036:
2033:
2029:
2025:
2020:
2016:
2005:
2003:
2001:
1999:
1995:
1994:
1981:
1978:
1974:
1968:
1964:
1958:
1954:
1943:
1940:
1938:
1936:
1923:
1920:
1916:
1910:
1906:
1895:
1893:
1891:
1889:
1885:
1884:
1871:
1867:
1863:
1860:
1857:
1847:
1844:
1831:
1827:
1816:
1805:
1795:
1782:
1778:
1767:
1756:
1746:
1733:
1729:
1718:
1705:
1701:
1690:
1686:
1685:
1682:
1679:
1676:
1675:
1673:
1670:
1669:
1656:
1653:
1649:
1638:
1627:
1617:
1604:
1601:
1597:
1586:
1573:
1570:
1566:
1555:
1542:
1539:
1535:
1524:
1513:
1502:
1501:
1490:
1480:
1478:
1476:
1474:
1472:
1461:
1450:
1449:
1436:
1433:
1429:
1418:
1416:
1403:
1399:
1388:
1375:
1371:
1360:
1347:
1343:
1332:
1328:
1327:
1314:
1311:
1307:
1296:
1294:
1281:
1277:
1266:
1253:
1249:
1238:
1225:
1221:
1210:
1206:
1205:
1192:
1189:
1185:
1174:
1172:
1159:
1155:
1144:
1131:
1127:
1116:
1103:
1099:
1088:
1084:
1083:
1070:
1066:
1055:
1044:
1034:
1021:
1017:
1006:
993:
989:
978:
965:
961:
950:
947:
946:
943:
921:
918:
914:
901:
898:
885:
865:
845:
825:
814:
813:
799:
796:
793:
789:
768:
763:
759:
755:
752:
749:
746:
743:
740:
730:
717:
713:
709:
706:
703:
698:
694:
683:
672:
669:
664:
660:
656:
646:
632:
628:
617:
605:
595:
581:
577:
566:
552:
549:
546:
542:
531:
517:
513:
507:
503:
499:
496:
486:
472:
468:
457:
445:
442:
439:
436:
433:
430:
427:
417:
405:
402:
399:
396:
393:
390:
387:
373:
372:
358:
355:
352:
348:
344:
339:
335:
331:
328:
325:
320:
317:
314:
310:
295:
294:
280:
277:
274:
270:
266:
261:
257:
253:
248:
245:
242:
238:
218:
215:
213:
210:
206:ANOVA on ranks
155:
152:
148:Welch's t-test
136:
135:
124:
121:
105:
102:
44:F distribution
26:
9:
6:
4:
3:
2:
6306:
6295:
6292:
6290:
6287:
6286:
6284:
6273:
6267:
6263:
6259:
6258:
6253:
6249:
6248:
6235:
6230:
6229:
6220:
6212:
6206:
6201:
6200:
6191:
6183:
6181:9780471316497
6177:
6173:
6166:
6158:
6154:
6151:(1): 91–126.
6150:
6146:
6139:
6125:on 2018-12-04
6124:
6120:
6114:
6106:
6102:
6098:
6094:
6093:
6088:
6081:
6073:
6069:
6062:
6054:
6047:
6039:
6035:
6031:
6027:
6020:
6011:
6010:
6001:
5993:
5989:
5985:
5981:
5977:
5973:
5966:
5958:
5956:0-534-37770-X
5952:
5948:
5944:
5943:
5935:
5933:
5928:
5918:
5915:
5913:
5912:Two-way ANOVA
5910:
5908:
5905:
5903:
5900:
5898:
5895:
5892:
5888:
5885:
5883:
5880:
5879:
5873:
5871:
5867:
5863:
5862:-distribution
5861:
5857:) denotes an
5856:
5852:
5848:
5843:
5840:
5824:
5821:
5816:
5812:
5808:
5805:
5802:
5798:
5794:
5784:
5780:
5775:
5773:
5769:
5765:
5761:
5754:
5735:
5732:
5729:
5725:
5721:
5718:
5710:
5706:
5702:
5695:
5691:
5687:
5681:
5678:
5671:
5670:
5669:
5667:
5663:
5645:
5642:
5639:
5635:
5631:
5628:
5623:
5619:
5614:
5608:
5604:
5600:
5595:
5591:
5587:
5580:
5579:
5578:
5561:
5558:
5552:
5549:
5546:
5540:
5537:
5531:
5528:
5525:
5519:
5516:
5511:
5507:
5499:
5498:
5497:
5476:
5468:
5461:
5458:
5455:
5452:
5449:
5446:
5443:
5440:
5437:
5434:
5431:
5428:
5425:
5422:
5419:
5416:
5413:
5410:
5407:
5404:
5401:
5398:
5395:
5392:
5389:
5386:
5383:
5380:
5377:
5374:
5371:
5368:
5365:
5362:
5359:
5351:
5342:
5334:
5328:
5323:
5315:
5312:
5306:
5301:
5293:
5290:
5284:
5279:
5271:
5265:
5260:
5252:
5249:
5243:
5238:
5230:
5225:
5217:
5212:
5204:
5201:
5195:
5190:
5182:
5179:
5173:
5168:
5160:
5154:
5149:
5141:
5135:
5130:
5122:
5116:
5111:
5103:
5100:
5095:
5087:
5082:
5074:
5071:
5065:
5060:
5052:
5049:
5043:
5038:
5030:
5024:
5019:
5011:
5008:
5002:
4997:
4989:
4983:
4978:
4970:
4965:
4960:
4955:
4951:
4939:
4938:
4937:
4929:
4926:
4923:
4922:
4918:
4915:
4912:
4911:
4907:
4904:
4901:
4900:
4896:
4893:
4890:
4889:
4885:
4882:
4879:
4878:
4874:
4871:
4868:
4867:
4861:
4858:
4853:
4850:
4845:
4842:
4841:
4838:
4836:
4818:
4815:
4812:
4808:
4804:
4801:
4796:
4792:
4788:
4781:
4780:
4779:
4762:
4759:
4756:
4753:
4750:
4747:
4742:
4738:
4730:
4729:
4728:
4725:
4723:
4700:
4697:
4692:
4684:
4681:
4678:
4672:
4669:
4664:
4656:
4653:
4650:
4644:
4641:
4636:
4628:
4625:
4622:
4616:
4613:
4611:
4601:
4588:
4583:
4578:
4569:
4560:
4557:
4552:
4539:
4534:
4529:
4520:
4511:
4508:
4503:
4490:
4485:
4480:
4471:
4462:
4459:
4457:
4450:
4446:
4434:
4433:
4432:
4430:
4423:
4419:
4418:
4403:
4400:
4395:
4391:
4388:
4385:
4382:
4379:
4373:
4368:
4362:
4353:
4347:
4342:
4333:
4327:
4322:
4313:
4304:
4299:
4293:
4284:
4276:
4272:
4265:
4257:
4248:
4247:
4246:
4244:
4222:
4219:
4214:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4174:
4169:
4166:
4162:
4158:
4153:
4150:
4145:
4143:
4136:
4127:
4117:
4114:
4109:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4075:
4069:
4064:
4061:
4057:
4053:
4048:
4045:
4040:
4038:
4031:
4022:
4012:
4009:
4004:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3970:
3964:
3959:
3956:
3952:
3948:
3943:
3940:
3935:
3933:
3926:
3917:
3903:
3902:
3901:
3899:
3895:
3893:
3889:
3873:
3870:
3867:
3866:
3862:
3859:
3856:
3855:
3851:
3848:
3845:
3844:
3840:
3837:
3834:
3833:
3829:
3826:
3823:
3822:
3818:
3815:
3812:
3811:
3805:
3802:
3797:
3794:
3789:
3786:
3785:
3782:
3781:
3780:
3775:
3768:
3761:
3750:
3736:
3727:
3711:
3707:
3694:
3691:
3687:
3686:
3685:
3677:
3661:
3657:
3634:
3631:
3628:
3625:
3622:
3618:
3614:
3590:
3587:
3584:
3577:
3561:
3555:
3545:
3542:
3538:
3532:
3528:
3522:
3518:
3508:
3503:
3498:
3495:
3491:
3485:
3481:
3475:
3471:
3463:
3447:
3439:
3436:
3431:
3428:
3424:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3378:
3370:
3367:
3366:
3363:
3344:
3341:
3338:
3335:
3332:
3328:
3324:
3317:
3314:
3311:
3308:
3305:
3301:
3297:
3287:
3273:
3270:
3267:
3260:
3242:
3238:
3231:
3221:
3218:
3214:
3208:
3204:
3192:
3188:
3184:
3179:
3174:
3171:
3167:
3161:
3157:
3151:
3147:
3139:
3123:
3118:
3114:
3107:
3104:
3099:
3095:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3055:
3047:
3044:
3043:
3023:
3020:
3017:
3014:
3011:
3007:
3003:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2968:
2964:
2954:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2907:
2903:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2868:
2864:
2854:
2840:
2837:
2834:
2827:
2811:
2805:
2795:
2792:
2788:
2782:
2778:
2772:
2768:
2758:
2751:
2747:
2740:
2730:
2727:
2723:
2717:
2713:
2701:
2697:
2689:
2673:
2665:
2662:
2657:
2653:
2644:
2640:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2603:
2595:
2592:
2591:
2588:
2585:
2582:
2579:
2576:
2574:
2573:
2569:
2566:
2563:
2560:
2557:
2554:
2553:
2547:
2538:
2522:
2518:
2514:
2509:
2505:
2484:
2481:
2478:
2451:
2447:
2439:
2436:
2420:
2415:
2411:
2403:
2389:
2382:
2366:
2361:
2357:
2349:
2347:
2333:
2326:
2310:
2305:
2301:
2293:
2290:
2289:
2274:
2267:
2264:
2248:
2244:
2236:
2222:
2215:
2199:
2195:
2187:
2185:
2171:
2164:
2148:
2144:
2136:
2133:
2132:
2115:
2105:
2102:
2098:
2089:
2085:
2079:
2075:
2067:
2064:
2062:
2060:
2044:
2034:
2031:
2027:
2018:
2014:
2006:
2004:
2002:
2000:
1997:
1996:
1979:
1976:
1972:
1966:
1962:
1956:
1952:
1944:
1941:
1939:
1937:
1921:
1918:
1914:
1908:
1904:
1896:
1894:
1892:
1890:
1887:
1886:
1869:
1865:
1861:
1858:
1855:
1848:
1845:
1829:
1825:
1817:
1803:
1796:
1780:
1776:
1768:
1754:
1747:
1731:
1727:
1719:
1703:
1699:
1691:
1688:
1687:
1678:
1677:
1674:
1672:
1671:
1654:
1651:
1647:
1639:
1625:
1618:
1602:
1599:
1595:
1587:
1571:
1568:
1564:
1556:
1540:
1537:
1533:
1525:
1511:
1504:
1503:
1488:
1481:
1479:
1477:
1475:
1473:
1459:
1452:
1451:
1434:
1431:
1427:
1419:
1417:
1401:
1397:
1389:
1373:
1369:
1361:
1345:
1341:
1333:
1330:
1329:
1312:
1309:
1305:
1297:
1295:
1279:
1275:
1267:
1251:
1247:
1239:
1223:
1219:
1211:
1208:
1207:
1190:
1187:
1183:
1175:
1173:
1157:
1153:
1145:
1129:
1125:
1117:
1101:
1097:
1089:
1086:
1085:
1068:
1064:
1056:
1042:
1035:
1019:
1015:
1007:
991:
987:
979:
963:
959:
951:
949:
948:
942:
941:
935:
919:
916:
912:
897:
883:
863:
843:
823:
797:
794:
791:
787:
761:
757:
753:
750:
744:
738:
731:
715:
711:
707:
704:
701:
696:
692:
684:
670:
667:
662:
658:
654:
647:
630:
626:
618:
603:
596:
579:
575:
567:
550:
547:
544:
540:
532:
515:
511:
505:
501:
497:
494:
487:
470:
466:
458:
443:
440:
437:
434:
431:
428:
425:
418:
403:
400:
397:
394:
391:
388:
385:
378:
377:
376:
356:
353:
350:
346:
342:
337:
333:
329:
326:
323:
318:
315:
312:
308:
300:
299:
298:
278:
275:
272:
268:
264:
259:
255:
251:
246:
243:
240:
236:
228:
227:
226:
223:
209:
207:
202:
198:
195:
191:
185:
183:
179:
175:
171:
167:
162:
159:
151:
150:can be used.
149:
145:
141:
133:
129:
125:
122:
119:
115:
111:
110:
109:
101:
99:
95:
92: =
91:
87:
83:
79:
75:
70:
68:
64:
60:
55:
53:
49:
45:
41:
40:one-way ANOVA
37:
33:
19:
18:One way anova
6256:
6227:
6219:
6198:
6190:
6171:
6165:
6148:
6144:
6138:
6127:. Retrieved
6123:the original
6113:
6096:
6090:
6086:
6080:
6071:
6061:
6052:
6046:
6029:
6025:
6019:
6008:
6000:
5975:
5971:
5965:
5941:
5890:
5869:
5865:
5859:
5854:
5850:
5846:
5844:
5778:
5776:
5763:
5759:
5752:
5750:
5665:
5661:
5660:
5576:
5495:
4935:
4859:
4851:
4843:
4834:
4833:
4777:
4726:
4721:
4719:
4428:
4427:
4421:
4242:
4241:
3897:
3896:
3891:
3887:
3881:
3803:
3795:
3787:
3773:
3766:
3759:
3756:
3728:
3698:
3683:
3607:
2567:Mean square
2544:
2470:
903:
815:
374:
296:
224:
220:
203:
199:
193:
189:
186:
182:Type I error
165:
163:
160:
157:
138:If data are
137:
107:
93:
89:
88:is given by
85:
71:
56:
39:
35:
29:
5897:Mixed model
5768:significant
2593:Treatments
168:-tests are
104:Assumptions
6283:Categories
6210:0716796570
6129:2016-09-22
5972:Biometrika
5668:-ratio is
1846:# Observed
1689:# Observed
896:-th list.
816:The index
32:statistics
5733:≈
5719:≈
5643:≈
5550:−
5529:−
5313:−
5291:−
5250:−
5202:−
5180:−
5101:−
5072:−
5050:−
5009:−
4754:−
4682:−
4654:−
4626:−
4592:¯
4584:−
4573:¯
4543:¯
4535:−
4524:¯
4494:¯
4486:−
4475:¯
4357:¯
4337:¯
4317:¯
4288:¯
4273:∑
4261:¯
4159:∑
4131:¯
4054:∑
4026:¯
3949:∑
3921:¯
3658:σ
3588:−
3529:∑
3519:∑
3509:−
3482:∑
3472:∑
3437:−
3379:∑
3271:−
3205:∑
3189:∑
3185:−
3158:∑
3148:∑
3105:−
3056:∑
2838:−
2779:∑
2769:∑
2759:−
2714:∑
2698:∑
2663:−
2604:∑
2506:μ
2479:μ
2390:…
2334:…
2223:…
2172:…
2086:∑
2076:∑
2015:∑
1963:∑
1953:∑
1905:∑
1862:∑
1804:…
1755:…
1626:…
1489:⋮
1460:⋮
1043:…
788:ε
758:σ
742:∼
739:ε
712:τ
705:μ
693:μ
659:τ
655:∑
627:τ
604:μ
576:μ
502:∑
438:…
398:…
347:ε
334:τ
327:μ
269:ε
256:μ
217:The model
114:residuals
63:see below
6262:Springer
5876:See also
4930:12−10=2
4919:7−10=−3
4908:8−10=−2
4897:11−10=1
4886:9−10=−1
4875:13−10=3
3894:-ratio:
2437:Variance
2291:Variance
80:and the
52:response
46:). This
5992:2332579
5947:324–325
5853:,
5772:p-value
5662:Step 5:
4927:8−9=−1
4924:4−5=−1
4916:6−9=−3
4913:3−5=−2
4905:11−9=2
4891:4−5=−1
4883:12−9=3
4872:8−9=−1
4835:Step 4:
4429:Step 3:
4243:Step 2:
3898:Step 1:
3753:Example
140:ordinal
134:(SRS)).
6268:
6207:
6178:
5990:
5953:
5887:F test
5474:
5357:
4902:5−5=0
4894:9−9=0
4880:8−5=3
4869:6−5=1
4720:where
4420:where
3772:, and
3368:Total
3045:Error
2065:Sum Sq
1998:Sum Sq
375:where
170:robust
82:F-test
78:t-test
74:t-test
5988:JSTOR
5923:Notes
5845:Note
6266:ISBN
6205:ISBN
6176:ISBN
5951:ISBN
5756:crit
5664:The
2497:and
2265:Mean
2134:Mean
116:are
38:(or
6234:761
6153:doi
6101:doi
6034:doi
5980:doi
5825:1.2
5809:4.5
5795:4.5
5736:9.3
5730:4.5
5646:4.5
3874:12
3849:11
3841:11
3827:12
3819:13
2586:MS
2583:DF
1942:Sum
1888:Sum
297:or
30:In
6285::
6264:.
6260:.
6149:60
6147:.
6097:66
6095:.
6070:.
6030:51
6028:.
5986:.
5976:38
5974:.
5949:.
5931:^
5722:42
5640:15
5632:68
5562:15
5477:68
4819:42
4805:84
4701:84
4679:10
4392:10
4223:10
4211:12
4193:11
4181:13
4094:11
4082:12
3871:8
3868:4
3863:7
3860:6
3857:3
3852:8
3846:5
3838:9
3835:4
3830:9
3824:8
3816:8
3813:6
3765:,
2570:F
1402:33
1374:32
1346:31
1280:23
1252:22
1224:21
1158:13
1130:12
1102:11
779:,
208:.
34:,
6274:.
6236:.
6213:.
6184:.
6159:.
6155::
6132:.
6107:.
6103::
6087:F
6074:.
6055:.
6040:.
6036::
5994:.
5982::
5959:.
5893:)
5889:(
5870:y
5866:x
5860:F
5855:y
5851:x
5849:(
5847:F
5822:=
5817:6
5813:/
5806:+
5803:6
5799:/
5779:F
5764:F
5760:α
5753:F
5726:/
5711:W
5707:S
5703:M
5696:B
5692:S
5688:M
5682:=
5679:F
5666:F
5636:/
5629:=
5624:W
5620:f
5615:/
5609:W
5605:S
5601:=
5596:W
5592:S
5588:M
5559:=
5556:)
5553:1
5547:6
5544:(
5541:3
5538:=
5535:)
5532:1
5526:n
5523:(
5520:a
5517:=
5512:W
5508:f
5469:=
5462:4
5459:+
5456:9
5453:+
5450:4
5447:+
5444:1
5441:+
5438:1
5435:+
5432:9
5429:+
5426:1
5423:+
5420:9
5417:+
5414:4
5411:+
5408:0
5405:+
5402:9
5399:+
5396:1
5393:+
5390:1
5387:+
5384:4
5381:+
5378:0
5375:+
5372:1
5369:+
5366:9
5363:+
5360:1
5352:=
5343:2
5339:)
5335:2
5332:(
5329:+
5324:2
5320:)
5316:3
5310:(
5307:+
5302:2
5298:)
5294:2
5288:(
5285:+
5280:2
5276:)
5272:1
5269:(
5266:+
5261:2
5257:)
5253:1
5247:(
5244:+
5239:2
5235:)
5231:3
5228:(
5218:+
5213:2
5209:)
5205:1
5199:(
5196:+
5191:2
5187:)
5183:3
5177:(
5174:+
5169:2
5165:)
5161:2
5158:(
5155:+
5150:2
5146:)
5142:0
5139:(
5136:+
5131:2
5127:)
5123:3
5120:(
5117:+
5112:2
5108:)
5104:1
5098:(
5088:+
5083:2
5079:)
5075:1
5069:(
5066:+
5061:2
5057:)
5053:2
5047:(
5044:+
5039:2
5035:)
5031:0
5028:(
5025:+
5020:2
5016:)
5012:1
5006:(
5003:+
4998:2
4994:)
4990:3
4987:(
4984:+
4979:2
4975:)
4971:1
4968:(
4961:=
4956:W
4952:S
4863:3
4860:a
4855:2
4852:a
4847:1
4844:a
4816:=
4813:2
4809:/
4802:=
4797:B
4793:S
4789:M
4763:2
4760:=
4757:1
4751:3
4748:=
4743:b
4739:f
4722:n
4698:=
4693:2
4689:)
4685:8
4676:(
4673:6
4670:+
4665:2
4661:)
4657:8
4651:9
4648:(
4645:6
4642:+
4637:2
4633:)
4629:8
4623:5
4620:(
4617:6
4614:=
4602:2
4598:)
4589:Y
4579:3
4570:Y
4564:(
4561:n
4558:+
4553:2
4549:)
4540:Y
4530:2
4521:Y
4515:(
4512:n
4509:+
4504:2
4500:)
4491:Y
4481:1
4472:Y
4466:(
4463:n
4460:=
4451:B
4447:S
4422:a
4404:8
4401:=
4396:3
4389:+
4386:9
4383:+
4380:5
4374:=
4369:a
4363:3
4354:Y
4348:+
4343:2
4334:Y
4328:+
4323:1
4314:Y
4305:=
4300:a
4294:i
4285:Y
4277:i
4266:=
4258:Y
4220:=
4215:6
4208:+
4205:7
4202:+
4199:8
4196:+
4190:+
4187:9
4184:+
4175:=
4170:i
4167:3
4163:Y
4154:6
4151:1
4146:=
4137:3
4128:Y
4118:9
4115:=
4110:6
4106:8
4103:+
4100:6
4097:+
4091:+
4088:9
4085:+
4079:+
4076:8
4070:=
4065:i
4062:2
4058:Y
4049:6
4046:1
4041:=
4032:2
4023:Y
4013:5
4010:=
4005:6
4001:4
3998:+
3995:3
3992:+
3989:5
3986:+
3983:4
3980:+
3977:8
3974:+
3971:6
3965:=
3960:i
3957:1
3953:Y
3944:6
3941:1
3936:=
3927:1
3918:Y
3892:F
3888:F
3884:0
3807:3
3804:a
3799:2
3796:a
3791:1
3788:a
3777:3
3774:a
3770:2
3767:a
3763:1
3760:a
3737:i
3712:j
3708:I
3662:2
3635:r
3632:o
3629:r
3626:r
3623:E
3619:S
3615:M
3591:1
3585:I
3562:I
3556:2
3552:)
3546:j
3543:i
3539:y
3533:i
3523:j
3515:(
3504:2
3499:j
3496:i
3492:y
3486:i
3476:j
3448:2
3444:)
3440:m
3432:j
3429:i
3425:y
3421:(
3416:s
3413:n
3410:o
3407:i
3404:t
3401:a
3398:v
3395:r
3392:e
3389:s
3386:b
3383:O
3345:r
3342:o
3339:r
3336:r
3333:E
3329:F
3325:D
3318:r
3315:o
3312:r
3309:r
3306:E
3302:S
3298:S
3274:J
3268:I
3243:j
3239:I
3232:2
3228:)
3222:j
3219:i
3215:y
3209:i
3201:(
3193:j
3180:2
3175:j
3172:i
3168:y
3162:i
3152:j
3124:2
3119:j
3115:s
3111:)
3108:1
3100:j
3096:I
3092:(
3087:s
3084:t
3081:n
3078:e
3075:m
3072:t
3069:a
3066:e
3063:r
3060:T
3024:r
3021:o
3018:r
3015:r
3012:E
3008:S
3004:M
2997:t
2994:n
2991:e
2988:m
2985:t
2982:a
2979:e
2976:r
2973:T
2969:S
2965:M
2936:t
2933:n
2930:e
2927:m
2924:t
2921:a
2918:e
2915:r
2912:T
2908:F
2904:D
2897:t
2894:n
2891:e
2888:m
2885:t
2882:a
2879:e
2876:r
2873:T
2869:S
2865:S
2841:1
2835:J
2812:I
2806:2
2802:)
2796:j
2793:i
2789:y
2783:i
2773:j
2765:(
2752:j
2748:I
2741:2
2737:)
2731:j
2728:i
2724:y
2718:i
2710:(
2702:j
2674:2
2670:)
2666:m
2658:j
2654:m
2650:(
2645:j
2641:I
2635:s
2632:t
2629:n
2626:e
2623:m
2620:t
2617:a
2614:e
2611:r
2608:T
2523:j
2519:m
2515:=
2510:j
2485:m
2482:=
2452:2
2448:s
2421:2
2416:J
2412:s
2367:2
2362:j
2358:s
2311:2
2306:1
2302:s
2275:m
2249:J
2245:m
2200:j
2196:m
2149:1
2145:m
2116:2
2112:)
2106:j
2103:i
2099:y
2095:(
2090:i
2080:j
2045:2
2041:)
2035:j
2032:i
2028:y
2024:(
2019:i
1980:j
1977:i
1973:y
1967:i
1957:j
1922:j
1919:i
1915:y
1909:i
1870:j
1866:I
1859:=
1856:I
1830:J
1826:I
1781:j
1777:I
1732:2
1728:I
1704:1
1700:I
1655:j
1652:i
1648:y
1603:3
1600:i
1596:y
1572:2
1569:i
1565:y
1541:1
1538:i
1534:y
1512:i
1435:j
1432:3
1428:y
1398:y
1370:y
1342:y
1331:3
1313:j
1310:2
1306:y
1276:y
1248:y
1220:y
1209:2
1191:j
1188:1
1184:y
1154:y
1126:y
1098:y
1087:1
1069:j
1065:I
1020:3
1016:I
992:2
988:I
964:1
960:I
920:j
917:i
913:y
884:j
864:i
844:i
824:i
798:j
795:,
792:i
767:)
762:2
754:,
751:0
748:(
745:N
716:j
708:+
702:=
697:j
671:0
668:=
663:j
631:j
580:j
551:j
548:,
545:i
541:y
516:j
512:I
506:j
498:=
495:I
471:j
467:I
444:J
441:,
435:,
432:1
429:=
426:j
404:I
401:,
395:,
392:1
389:=
386:i
357:j
354:,
351:i
343:+
338:j
330:+
324:=
319:j
316:,
313:i
309:y
279:j
276:,
273:i
265:+
260:j
252:=
247:j
244:,
241:i
237:y
194:F
190:F
166:F
94:t
90:F
86:t
20:)
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