Lecture 17 - Multifactor Analysis of Variance (Moving on to...

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Multifactor Analysis of Variance (Moving on to Chapter 11) In many situations, there is more than a single factor that is studied through the ANOVA procedure. For example, to study two-factors, A & B without replication (i.e., only one observation per pair of factor levels).
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The total variation in the data is comprised of the variation associated with factor A, factor B and error: SST = SSA + SSB + SSE
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The two-factor ANOVA procedure is testing the hypotheses: H 0A : factor A effects are 0 vs. H aA : at least one factor A effect is 0 and H 0B : factor B effects are 0 vs. H aB : at least one B effect is 0
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The corresponding ANOVA layout is: Source of Var. df SS MS F Factor A I-1 SSA SSA/(I-1) MSA/MSE Factor B J-1 SSB SSB/(J-1) MSB/MSE Error (I-1)(J-1) SSE SSE/(I-1)(J-1) Total IJ-1 SST Reject H 0A if F A > F α , I-1, (I-1)(J-1) Reject H 0B if F B > F α , J-1, (I-1)(J-1)
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Sample Problem: Column Factor (B) R o w F a c t o r ( A ) 1234 1 3 63 93 2 2 1 82 02 22 0 3 3 03 73 33 4
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22 .. 11 .. . 1 . 1 (I rows and J columns) : (1 / ) ) ) (1 / ) ) IJ ij ij I i i J j j ComputingFormulas SST x IJ x SSA J x IJ x SSB I x IJ x SSE SST SSA SSB == = = =− ∑∑
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36 39 36 32 143 18 20 22 20 80 30 37 33 34 134 84 96 91 86 357 75 . 29 12 357 = = x () 2 2 22 2 2 2 2 ~, 357 36 39 ... 34 638 12 357 1 4 143 80 134 581 12 357 1 3 84 96 91 86 29 12 638 581 29 28 ij i j ij xN I D O SST SSA SSB SSE μα β σ =+ + + =++ +− = + = + + = =− =
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Source SS df MS F A 581 2 290 62.1 B2 9 3 9 . 6 2 . 1 < F .05,3,6 =4.76 Error 28 6 4.7 Total 638 11 Reject for A, Fail for B (note how we could have simply rearranged the data layout and done the one-way procedure twice)
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In Experimental Design, a block is a group of objects or people that have been matched. An object or person can be matched with itself, meaning that repeated observations are taken on that object or person that form a block.
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An experiment has a randomized block design if several different levels of one factor are being studied and the objects or people being observed or measured have been matched. Each object or person is randomly assigned to one of the J levels of the factor.
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Usually randomized block designs are analyzed like two factor designs without replication except one of the factors is considered a “noise” factor being blocked out of the results.
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One Way ANOVA with blocking: Problem 11-3 (page 409)
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22 2 2 2 ..
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Lecture 17 - Multifactor Analysis of Variance (Moving on to...

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