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= (interaction) 4 - A - B (main effect) - A B (interaction)
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- ひでたつ よしなが
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1 1 (two-way ANOVA) - - A B 1
2 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= (interaction) 4 - A - B (main effect) - A B (interaction)
3 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: N ij, N ia N jb N N ij = N A i N B j / N 3
4 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
5 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
6 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
7 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 = ( ) 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
8 15 ( SSA = (T A 1 ) / N A 1 + (T A ) / N A - T / N = 71 / / / 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 / / / / 30 = MSB = SSB / (K B - 1) = / (3-1) = 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 = = MSAB = SSAB / (K A - 1) (K B - 1) = / ( - 1) (3-1) =
9 A 17 5% 1, 4 F, F > 4.6 F = MSA / MSW = 0.14 / 1.08 = B 18 5%, 4 F F > 3.40 F = MSB / MSW = / 1.08 =
10 19 5%, 4 F F > 3.40 F = MSAB / MSW = / 1.08 = Source SS df MS F p A > 0.7 B < 0.001*** A B < 0.001*** Within:
11 Male Female Score T1 T T3 Treatment Score T1 T T3 0 Male Female Treatment Sex 11
12 3 Score T1 T T Male Female Male Female Treatment Sex Treatment Sex 4 Within-Subject design - A - B two-way ANOVA 1 1
13 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
14 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
15 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 = MSW MSB - MSW ( ε ij ) - MSB ( ε ij ) + ( Σ N i a i / (K - 1) ) MSB MSW - MSB
16 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
17 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
18 35 MSA, MSAB MSA ( ε ij ) + + ( ) + ( MSAB: ( ε ij ) + ( ) Within-Subject 18
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MINITAB アシスタントホワイトペーパー本書は Minitab 統計ソフトウェアのアシスタントで使用される方法およびデータチェックを開発するため Minitab の統計専門家によって行われた調査に関する一連の文書群を構成する文書の 1 つです ゲージ R&R 分析 ( 交差 ) 概要 測定システ
MINITAB アシスタントホワイトペーパー本書は Minitab 統計ソフトウェアのアシスタントで使用される方法およびデータチェックを開発するため Minitab の統計専門家によって行われた調査に関する一連の文書群を構成する文書の 1 つです ゲージ R&R 分析 ( 交差 ) 概要 測定システムの分析は 生産工程を適切に監視および改善するために 事実上あらゆる種類の製造業で行われています 一般的な測定システムの分析では
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- k k k = y. k = ky. y du dx = ε ux ( ) ux ( ) = ax+ b x u() = ; u( ) = AE u() = b= u () = a= ; a= d x du ε x = = = dx dx N = σ da = E ε da = EA ε A x A x x - σ x σ x = Eε x N = EAε x = EA = N = EA k =
日本呼吸器学会雑誌第48巻第6号
Fig.1 Cutaneousfindingsandpathologicalfindingsofatransbronchiallungbiopsy(TBLB)andlichenifiedeczemaoftherightforearm.Thepatienthadlichenifiedeczemaonhisextremitiesandbodytrunk (1a:rightforearm,1b:leftthigh).Therewasinfiltrationoftheinflammatorylymphocytes,edemaandabrasioninthetype2alveolarepithelium
n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz
1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x