CH6.SeveralMeans-part2.pdf - Chapter 6 Comparisons of...

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Chapter 6. Comparisons of Several Multivariate Means - Part 2 3. Comparing Several Multivariate Means (One-Way MANOVA) Design Population 1: X 11 , . . . , X 1 n 1 Population 2: X 21 , . . . , X 2 n 2 . . . . . . Population g: X g 1 , . . . , X gn g Assumptions X l 1 , . . . , X ln l is a random sample of size n l from a multivariate normal population with mean µ l and common covariance matrix Σ . The random samples are all independent. (Univariate) One-Way ANOVA Model X li = µ + τ l + ϵ li where g l =1 n l τ l = 0 and ϵ li N (0 , σ 2 ) . The aim is to test H 0 : τ 1 = τ 2 = · · · = τ g = 0 . Decomposition of the sum of square g X l =1 n l X i =1 ( X li ¯ X ·· ) 2 = g X l =1 n l ( ¯ X l · ¯ X ·· ) 2 + g X l =1 n l X i =1 ( X li ¯ X l · ) 2 SS tot = SS tr + SS res . ( SST ) ( SSB ) ( SSW ) 1
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ANOVA Table Source Degrees of of variation Sum of squares (SS) freedom (d.f.) Treatments SSB = g l =1 n l ( ¯ X l · ¯ X ·· ) 2 g 1 Residual (Error) SSW = g l =1 n l i =1 ( X li ¯ X l · ) 2 g l =1 n l g Total (corrected for the mean) SST = g l =1 n l i =1 ( X li ¯ X ·· ) 2 g l =1 n l 1 The usual F -statistics reject H 0 if F = SSB / ( g 1) SSW / ( g l =1 n l g ) > F g 1 , n l g ( α ) . One-Way MANOVA Model X li = µ + τ l + ϵ li where g l =1 n l τ l = 0 and ϵ li N p (0 , Σ ) .
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  • Spring '16
  • 통계학, Xlnl is

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