chapter9 - Chapter 9 More Complicated Experimental Designs...

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Chapter 9 More Complicated Experimental Designs
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Randomized Block Design (RBD) t > 2 Treatments (groups) to be compared b Blocks of homogeneous units are sampled. Blocks can be individual subjects. Blocks are made up of t subunits Subunits within a block receive one treatment. When subjects are blocks, receive treatments in random order. Outcome when Treatment i is assigned to Block j is labeled Y ij • Effect of Trt i is labeled α i • Effect of Block j is labeled β j • Random error term is labeled ε ij Efficiency gain from removing block-to-block variability from experimental error
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Randomized Complete Block Designs Model : 2 1 ) ( 0 ) ( 0 σ ε α β μ = = = + + = + + + = = ij ij t i i ij j i ij j i ij V E Y Test for differences among treatment effects: H 0 : 1 = ... = t = 0 ( 1 = ... = t ) H A : Not all i = 0 (Not all i are equal) Typically not interested in measuring block effects (although sometimes wish to estimate their variance in the population of blocks). Using Block designs increases efficiency in making inferences on treatment effects
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RBD - ANOVA F -Test (Normal Data) Data Structure: ( t Treatments, b Subjects) Mean for Treatment i : Mean for Subject (Block) j : Overall Mean: Overall sample size: N = bt ANOVA: T reatment, B lock, and E rror Sums of Squares . i y j y . .. y ( 29 ( 29 ( 29 ( 29 ) 1 )( 1 ( 1 1 1 2 .. . . 1 2 .. . 1 2 .. . 1 1 2 .. - - = - - = + - - = - = - = - = - = - = - = ∑∑ ∑ ∑ = = = = t b df SSB SST TSS y y y y SSE b df y y t SSB t df y y b SST bt df y y TSS E j i ij B b j j T t i i t i b j Total ij
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RBD - ANOVA F -Test (Normal Data) ANOVA Table: Source SS df MS F Treatments SST t -1 MST = SST /( t -1) F = MST / MSE Blocks SSB b -1 MSB = SSB /( b -1) Error SSE ( b -1)( t -1) MSE = SSE /[( b -1)( t -1)] Total TSS bt -1 H 0 : α 1 = ... = t = 0 ( μ 1 = ... = t ) H A : Not all i = 0 (Not all i are equal) ) ( : : . . : . . ) 1 )( 1 ( , 1 , obs t b t obs obs F F P val P F F R R MSE MST F S T - = - - -
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Pairwise Comparison of Treatment Means Tukey’s Method- q in Table 11, p. 701 with ν = ( b -1)( t - 1) ( 29 ij j i ij j i j i ij W y y W y y b MSE v t q W ± - - = . . . . : Interval Confidence s Tukey' if Conclude ) , ( μ α Bonferroni’s Method - t -values from table on class website with = ( b -1)( t -1) and C = t ( t -1)/2 ( 29 ij j i ij j i j i v C ij B y y B y y b MSE t B ± - - = . . . . , , 2 / : Interval Confidence s ' Bonferroni if Conclude 2
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Expected Mean Squares / Relative Efficiency Expected Mean Squares: As with CRD, the Expected Mean Squares for Treatment and Error are functions of the sample sizes ( b , the number of blocks), the true treatment effects ( α 1 ,…, t ) and the variance of the random error terms ( σ 2 ) By assigning all treatments to units within blocks, error variance is (much) smaller for RBD than CRD (which
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This note was uploaded on 06/04/2011 for the course STA 6166 taught by Professor Staff during the Fall '08 term at University of Florida.

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chapter9 - Chapter 9 More Complicated Experimental Designs...

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