Argonauta 8: 27-37 (2003) 1 ANOVA, analysis of variance t 1980 90 ANOVA ANOVA Underwood, 1997 ANOVA ANOVA ANOVA 1975 Sokal & Rohlf 1981 1990 Underwood 1997 1997 Zar 1999 1-way ANOVA 2-way, 3-way, ANOVA, multiway, factorial A, B, C 1 27
A B C A B C --- --- --- X --- --- --- --- --- --- --- --- --- --- --- --- Y --- --- --- --- --- --- --- --- --- --- --- --- Z --- --- --- --- --- --- 1 A, B, C 1 A, B, C, X, Y, Z, Model I Model II A, B, C A, B, C ANOVA parametrics nonparametrics parametrics 1998 nonparametrics parametrics Sokal & Rohlf 1981 t U 2000 1-way ANOVA parametrics 28
A B A, B, C l, m, n A B C --------------------------------- a 1 b 1 c 1 a 2 b 2 c 2 A, B : : : C a l b m c n --------------------------------- a b c a, b, c l m n parametrics 1-way ANOVA Bartlett-test B C B/C k 1 Sokal & Rohlf 1981, Underwood 1997 Hartley, Scheffe, Leven, Cochlan t 2000 M S T (M a i ) 2 + (M b i ) 2 + (M c i ) 2 S C l(m a )2 +m(m b) 2 +n(m c) 2 S R (a a i ) 2 + (b b i ) 2 + (c c i ) 2 29
S summation, T, C, R Total, Column, Row S C l, m, n S T, S C, S R S T = S C + S R S R ANOVA S T = S C + S R S T S C S R = S T S C S R A, B, C A, B, C A, B, C S C, S R k 1, (l 1 ) + (m 1) + (n 1) V T, V C V R Fcal = V C / V R Fcal calculated F F l + m + n 3, k 1 F Fcal > F 80 Fcal SS df MS F S T l + m + n 1 S C k 1 V C V C / V R 0.002 S R l + m + n 3 V R 1-way ANOVA P 30
SS Summation of Squares MS Mean of Squares ij = + j + ij i, j i j ij j j ij j + + parametrics ij Fcal = V C /V R F F Fcal F cal t t 1-way ANOVA 2 Kruskal-Wallis 1-way ANOVA nonparametrics Kruskal-Wallis 2 nonparametrics U 3 H Hcal = {12 / N ( N + 1 ) } { Ri 2 / ni 3 ( N +1)} N k Ri i ni i 2 2 2 2 Hcal = {12 / ( l + m + n ) ( l + m + n + 1) } ( R A + R B + R C ) H k 1 2 2 Hcal k=3 N<18 2 Kruskal-Wallis H 12 / N ( N + 1 ) 3 ( N +1) 2 Ri 2 / ni 3 31
2 H Model I 2 t P=0.05 20 1 1 = 2 > 3 1, 2, 3 a priori a posteriori 2 Duncan Multiple Range Test Tukey Test, SNK Test Student Newman Keuls Test Newman Keuls Test Tukey Test 2, 3 A B SE= (V R / n A B l = m = n 1 / n 1/2 (1 / l +1 / m) l, m A, B q = (a b) / SE a, b A, B l+m+n 3 q qcal > q A, B q t t t q q q A, B, C B C A C A B AB AB C 32
A B C a b c SNK Test Tukey Test q qcal q Tukey Test Nonparametric Multiple Comparison nonparametrics Kruskal Wallis Tukey Test nonparametrics r A, r B, r C A B SE= N(N+1)/12 (1/l+1/m) SE Q AB = ( r A r B ) / SE Q AB Q Q Q AB > Q AB AC, BC SNK Test nonparametrics Nested ANOVA nested nest mixed model pure model mixed model Nested Anova 33
X Y Z A B C D E F ----------------------------------- a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 a 3 b 3 c 3 d 3 e 3 f 3 a 4 b 4 c 4 d 4 e 4 f 4 Nested Anova Nested Anova X Y Z A C E a 1 a 2 c 1 e 1 a 3 a 4 B D b 1 d 1 F f 1 XYZ ABC abc XYZ SS T SS H SS L SS E T, total; H, high level; L, low level; E, error SS T = SS H + SS L + SS E 34
SS df MS Fcal SS H 3 1 SS H / 2 =V H V H / V L SS L 3 (2 1) SS L / 3 =V L V L / V E SS E 3 2 (4 1) SS E /18 =V R 5 Nested ANOVA F F Fcal > F 5 S 2 E= V E S 2 L= (V L V E ) / 4 4 S 2 H= (V H V L ) / (4 2) 2 S 2 S 2 E / (S 2 E + S 2 L + S 2 H) Nested ANOVA ijk = + i + ij + ijk ijk i j k i i ij i j i ijk,, + + + 2-way ANOVA 35
1 replicate replicate F replicate A B C D P AP1 AP2 BP1 BP2 CP1 CP2 DP1 DP2 Q AQ1 AQ2 BQ1 BQ2 CQ1 CQ2 DQ1 DQ2 R AR1 AR2 BR1 BR2 CR1 CR2 DR1 DR2 A, B, C, D P, Q, R replicate 2-way Anova S T S R S C S RC S E S R C = S RC S R S C 36
S T = S R + S C + S R C + S E SS df MS Fcal S T 2 3 4 1 S R 3 1 V R =S R /2 V R /V E S C 4 1 V C =S C /3 V C /V E S RC 3 4 1 S R C (3 1) (4 1) V R C =S R C /6 V R C /V E S E 3 4 (2 1) V E =S E /12 2-way ANOVA F ijk = + i + j + ij + i j k + + + + Time Height Treatment 2-way ANOVA nonparametrics Friedman 37