第三章 變異數分析 - pu.edu.tw

1 F- ,//F 22222121 SS 12 22 S12 S22 v1 v2 F F- (v1 v2 ) F- (ANOVA) F-test F- F- p- 1-way_2010 1 : m ( 1 , .., i , .., m ) Data : m n yij --- i j Group1 i yi2 ..y1n yinymnAvg. (1-way ANOVA) 1-way_2010 2 yij = i + ij , i= 1,.., m, j= 1,.., n ij , N ( 0, 2 ) H0 1= 2=.

2 = m H1 ( i j i j) SST = SSt+ SSE MSt = SSt/(m-1) H0 = 2 MSE = SSE /(N -m) ( ) MSE 2 1-way_2010 3 F = MSt / MSE 1. H0 F F (m-1, N-m) 2. MSt F F p-value = P( F- > F ) p-value< treatment p-value 1 treatment 2 Q : ?

3 ? ? 1-way_2010 4 SourceSSdfMSF- p- TreatmentSStm-1 MStF =MSt/MSEE rrorSSEN-mMSET otalSSTN-1 ANOVA table for 1-way model ANOVA Table 1 F- p- F- df p- df p- p- - p- < 2 p- m 1-way_2010 5 A B C (p42) 1234 A78546B98657C1013111011 : 1-way ANOVA SourceSSdfMSF- p- 8211 ANOVA table p-value = .00569 < 3 (level) 4 1-way_2010 6 A: 67 68 69 70 71 B: 70 71 72 73 74 C: D: 72 E.

4 74 + | | | | | | | 73 + +-----+ | | | | | | | | +-----+ | | | | | | 72 + *--+--* | *--+--* | | | | | | | | | +-----+ +-----+ | | | | | | | 71 + | +-----+ | | +-----+ | | | | | | | | | | *--+--* *--+--* | | | | | | | 70 + +-----+ | | | +-----+ | | |

5 | | | | | | +-----+ | | | | 69 + *--+--* | | | | | | | | | | | | 68 + +-----+ | | | | | | 67 + | ------------+-----------+-----------+--- --------+-----------+----------- G A B C D E 1-way_2010 7 222 )(2 Y-X , tt A vs.

6 B: p-value = , B vs. C: p-value = , D vs. E: p-value = , A B A B D E D E ANOVA F- = ( )/( ). ANOVA F- t- t- 1-way_2010 8 9 A B C 1234 A78546B98657C10131110111234 A731046B410687C101491111 ABCmean SD6 . 1 2 ABCmean SD6 11 . 1-way_2010 10 1 2 6, 7, 11 024681012141601234024681012141601234F- = p-value = .076 3 (mse= ) F- = p-value =.

7 0057 3 (mse= ) 1-way_2010 ANOVA 11 EXCEL: (EXCEL ) EXCEL 1-way_2010 11 ANOVA Treatment quantitative ( ) qualitative ( ) treatment (I) (pairwise comparison) (II) (constrast test) treatment (III) ( ) 1-way_2010 12 1-way_2010 13 I (pairwise comparison) --- 2 ~ 3 jiH allfor :Test 0 a.

8 Fisher s Least significant Difference (LSD) method ( ) LSD yy )( ;2/ji i j 1-way_2010 14 1p- y and y of differenceorder the where D yy mean) (harmonic )n/1(m n , )f,p(;p b. Duncan s Multiple range Test T yy )f,m(;.i c. Tukey s HSD Test i j i j 1-way_2010 15 d. Scheffe s test ( ) f. Dunnett s test ( ) e.

9 The Newman-Keuls Test Duncan 1-way_2010 16 Q: LSD Tukey s HSD Duncan LSD Scheffe (m=4, ni = 4, = ) Bonferron = Tukey HSD = Duncan = Fisher LSD = Scheffe = 1-way_2010 17 Tukey's Studentized Range (HSD) Test for y Alpha Error Degrees of Freedom 16 Error Mean Square Critical Value of Studentized Range Minimum Significant Difference Tukey Grouping Mean N group A 5 C B 5 D C 5 B C 5 A SAS Means with the same letter are not significantly different.

10 1-way_2010 18 Tukey's Studentized Range (HSD) Test for y Comparisons significant at the level are indicated by **. group Comparison Difference Between Means Simultaneous 95% Confidence Limits C - D ** C - B ** C - A ** D - C ** D - B ** D - A ** B - C ** B - D ** B - A SAS 1-way_2010 19 1-way_2010 20 1-way_2010 21 ccimiii0. where ,1 II. (Contrast) --- Contrast (for balanced data) is a contrast of the treatment means, if , cn)Yc(SS ,1/SSMS If p-value < 0 is significant.