1 (two-way ANOVA) - - A B 1
3 M=8.4 M=3 M=.8 M=4.7 M=5.6 M=3 M=5. M=4.6 M=7 M=3 M=4 - - 1 (interaction) 4 - A - B (main effect) - A B (interaction)
two-way ANOVA 5 1 A - H0: µ A 0 = µ A 1 = = µ A n - H1: not H0 B - H0: µ B 0 = µ B 1 = = µ B k - H1: not H0 3 - H0: A B - H1: 6 - - - N ij, N ia N jb N N ij = N A i N B j / N 3
7 - X ijk (i, j) k - N ij (i, j) - N ia, N jb A (B) i ( j) - N Nij - T ij, M ij (i, j) - T ia, T jb A (B) i ( j) - M ia, M jb A (B) i ( j) - T, M - K A, K B A (B) A 8 1 A MSA MSA = SSA / (K A - 1) SSA = Σ i N i A ( M i A - M ) = Σ i ( T i A / N i A ) - T / N MSW MSW = SSW / (N - K A K B ) SSW = Σ ijk ( X ijk - M ij ) = Σ ijk X ijk - Σ ij (T ij / N ij ) 4
A 9 MSA/MSW (K A - 1), (N - K A K B ) F - A B one-way ANOVA two-way ANOVA B 10 B MSB MSB = SSB / (K B - 1) SSB = Σ j N B j ( M B j - M ) = Σ j ( T B j / N B j ) - T / N MSB/MSW (K B - 1), (N - K A K B ) F - MSW 5
A B 11 A, B MSAB MSAB = SSAB / (K A - 1) (K B - 1) SSAB = Σ ij N ij ( M ij - M i A - M j B + M ) = Σ ij ( T ij / N ij ) - SSA - SSB - T / N (Mij - M) (Mi A - M) + (Mj B - M) MSAB/MSW (K A - 1)(K B - 1) (N - K A K B ) F -MSW 1 Source SS df MS F p ( ) ( ) ( ) A SSA K A - 1 MSA MSA / MSW B SSB K B - 1 MSB MSB / MSW A B SSAB (K A - 1)(K B - 1) MSAB MSAB / MSW Within: SSW N - K A K B MSW 6
13 5% 10, 9, 8, 7, 7 5, 4, 3,, 1 4, 3, 3,, 6, 6, 6, 5, 5 4, 4, 3,, 6, 6, 5, 5, 4 A B N 11 = N 1 = N 13 = N 1 = N = N 3 = 5 N A 1 = N A = 15, N B 1 = N B = N B 3 = 10 N = 30 14 ( ) T 11 = 4, T 1 = 15, T 13 = 14, T 1 = 8, T = 15, T 3 = 6, T 1 A = 71, T A = 69, T 1 B = 70, T B = 30, T 3 B = 40 M 11 = 8.4, M 1 = 3, M 13 =.8, M 1 = 5.6, M = 3, M 3 = 5., M A 1 = 4.7, M A = 4.6, M B 1 = 7, M B = 3, M B 3 = 4 T = 140, M = 4.33, K A =, K B = 3, Σ X ijk = 800 7
15 ( SSA = (T A 1 ) / N A 1 + (T A ) / N A - T / N = 71 / 15 + 69 / 15-140 / 30 = 0.14 MSA = SSA / (K A - 1) = 0.14 / ( - 1) = 0.14 SSB = (T B 1 ) / N A 1 + (T B ) / N B + (T 3B ) / N B 3 - T / N = 70 / 10 + 30 / 10 + 40 / 10-140 / 30 = 86.67 MSB = SSB / (K B - 1) = 86.67 / (3-1) = 43.33 SSW = Σ X ijk - [T 11 / N 11 + T 1 / N 1 + T 13 / N 13 + T 1 / N 1 + T / N + T 3 / N 3 ] = 6 16 MSW = SSW / (N - K A K B ) = 6 / (30 - *3) = 1.08 SSAB = [T 11 / N 11 + T 1 / N 1 + T 13 / N 13 + T 1 / N 1 + T / N + T 3 / N 3 ] - SSA - SSB - T / N = 774-0.14-86.67-653.33 = 33.86 MSAB = SSAB / (K A - 1) (K B - 1) = 33.86 / ( - 1) (3-1) = 16.93-8
A 17 5% 1, 4 F, F > 4.6 F = MSA / MSW = 0.14 / 1.08 = 0.13 - B 18 5%, 4 F F > 3.40 F = MSB / MSW = 43.34 / 1.08 = 40.13-9
19 5%, 4 F F > 3.40 F = MSAB / MSW = 16.93 / 1.08 = 15.68-0 Source SS df MS F p A 0.14 1 0.14 0.13 > 0.7 B 86.67 43.33 40.13 < 0.001*** A B 33.86 16.93 15.68 < 0.001*** Within: 6.00 4 1.08 10
1 10 8 Male Female Score 6 4 0 T1 T T3 Treatment Score 10 8 6 4 T1 T T3 0 Male Female Treatment Sex 11
3 Score 10 8 6 4 T1 T T3 10 8 6 4 0 0 Male Female Male Female Treatment Sex Treatment Sex 4 Within-Subject design - A - B two-way ANOVA 1 1
5 -X ij (i, j) -K A, K B A (B) -N K A K B -T ia, T jb A (B) i (j) -M ia, M jb A (B) i (j) -T, M 6 A MSA MSA = SSA / (K A - 1) SSA = K B Σ i ( M A i - M ) = ( Σ i T A i ) / K B - T / N B MSB MSB = SSB / (K B - 1) SSB = K A Σ j ( M B j - M ) = ( Σ j T B j ) / K A - T / N MSAB MSAB = SSAB / (K A - 1) (K B - 1) SSAB = Σ ij ( X ij - M i A - M j B + M ) = Σ ij X ij - SSA - SSB - T / N 13
7 F = MSA / MSAB -MSA -MSAB (K A -1) (K A - 1)(K B - 1) 8 Source SS df MS F p ( ) ( ) ( ) A SSA K A - 1 MSA MSA/MSAB B SSB K B - 1 MSB A B SSAB (K A - 1)(K B - 1) MSAB MSW A 14
9 1 Y ij = µ + a i + ε ij µ a i A ( a i = µ i - µ ) ε ij ( ε ij = Y ij - µ i ) - ε ij - H0 all a i = 0 - H1 not all a i = 0 1 30 MSW MSB - MSW ( ε ij ) - MSB ( ε ij ) + ( Σ N i a i / (K - 1) ) MSB MSW - MSB 0 1 15
31 Y ijk = µ + a i + b j + (ab) ij + ε ijk µ a i A ( a i = µ i - µ ) b j B ( b j = µ j - µ) (ab) ij ( (ab) ij = µ ij - µ i - µ j + µ ) ε ijk - 3 A - H0 all a i = 0 - H1 not all a i = 0 B - H0 all b j = 0 - H1 not all b j = 0 - H0 all (ab) ij = 0 - H1 not all (ab) ij = 0 16
33 MSW, MSA, MSB, MSAB - MSW ( ε ijk ) - MSA ( ε ijk ) + ( Σ N A i a i / (K A - 1) ) - MSB ( ε ijk ) + ( Σ N B j b j / (K B - 1) ) - MSAB: ( ε ijk ) + ( Σ N ij (ab) ij / (K A - 1) (K B - 1) ) 1 Within-Subject design 34 Y ij = µ + a i + b j + (ab) ij + ε ij µ a i A b j B (ab) ij ε ij - H0 all a i = 0 - H1 not all a i = 0 17
35 MSA, MSAB MSA ( ε ij ) + + ( ) + ( MSAB: ( ε ij ) + ( ) 1 36 - - - - Within-Subject 18