03-28-2011 2 Way Anova

# 03-28-2011 2 Way Anova - Lecture 17 Two-Factor ANOVA...

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•3/28/2011 •1 Lecture 17: March 28, 2011 • Two-Factor ANOVA without replication: – You are responsible for pages 456 middle to 464 top third. • Chapter 12 (next class): –Read p. 488-middle p.491 –Skip middle p.491-top p. 493 –Read bottom p.494-middle p.504 • Homework No.11: Two-Factor ANOVA – One problem, perhaps four parts – Ready by tonight or by tomorrow morning. We will get an understanding for what two- factor ANOVA does • by revisiting one-factor ANOVA • and addressing one very important aspect of one-factor ANOVA that we have not yet addressed • and that is • the vertical dimension of the “SS” formulas in the one-factor ANOVA table. Here’s the template for the one-way ANOVA table. Table 11.2 on page 444 of the text • Concentrate on the vertical arrangement of the three “SS” items: SSA, SSE, and SST

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•3/28/2011 •2 Here they are without the scary mathematics. • SST • SSA • SSE Here they are with the scary mathematics. • SST: – “The” total • SSA: – Accounts for explanatory power – Pay attention to the subscript of each treatment mean. It is “j”. • SSE: – No contribution to explaining anything j n c 2 ij j1 i1 SST (Y Y)    c 2 jj j1 SSA n (Y Y) j n c 2 ij j SSE (Y Y ) A Closer Look at the Workings of the Three Sums of Squares: SST= SSA + SSE SST = SSA + SSE Total variation variation explained unexplained or around the mean by the treatments error variation = + Address the components directly above without the squaring and without the summation : Look at the items only within parentheses! = + Switch the two terms on the right hand side: = + Two issues with the equation directly above: On the right-hand side, notice the cancelling effect of . Middle term: error As moves in the direction of the impact is to make SSE smaller. This also means that moves away from . The impact is to make SSA larger. ij YY Y c 2 n(Y Y ) j n c 2 ij j (Y Y ) j n c 2 ij (Y Y) ij j j ij j Y and+Y j Y ij Y ij j j Y
•3/28/2011 •3 Conclusion Thus Far and Where to Go From Here • We have SST=SSA+SSE. • SST is fixed in size. • There is explanatory power through SSA.

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03-28-2011 2 Way Anova - Lecture 17 Two-Factor ANOVA...

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