ch07 - Chapter 7. Supplemental Text Material S7-1. The...

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Chapter 7. Supplemental Text Material S7-1. The Error Term in a Blocked Design Just as in any randomized complete block design, when we run a replicated factorial experiment in blocks we are assuming that there is no interaction between treatments and blocks. In the RCBD with a single design factor (Chapter 4) the error term is actually the interaction between treatments and blocks. This is also the case in a factorial design. To illustrate, consider the ANOVA in Table 7-2 of the textbook. The design is a 2 2 factorial run in three complete blocks. Each block corresponds to a replicate of the experiment. There are six degrees of freedom for error. Two of those degrees of freedom are the interaction between blocks and factor A , two degrees of freedom are the interaction between blocks and factor B , and two degrees of freedom are the interaction between blocks and the AB interaction. In order for the error term here to truly represent random error, we must assume that blocks and the design factors do not interact. S7-2. The Prediction Equation for a Blocked Design Consider the prediction equation for the 2 4 factorial in two blocks with ABCD confounded from in Example 7-2. Since blocking does not impact the effect estimates from this experiment, the equation would be exactly the same as the one obtained from the unblocked design, Example 6-2. This prediction equation is ± .. . . . . y x x x xx =+ + + + 70 06 108125 4 9375 7 3125 9 0625 8 3125 1341 31 4 This equation would be used to predict future observations where we had no knowledge of the block effect. However, in the experiment just completed we know that there is a strong block effect, in fact the block effect was computed as block effect y y block block = = − 12 18 625 . This means that the difference in average response between the two blocks is –18.625. We should compensate for this in the prediction equation if we want to obtain the correct fitted values for block 1 and block 2. Defining a separate block effect for each block does this, where block effect block effect 9 3125 9 3125 and =− = . . . These block effects would be added to the intercept in the prediction equation for each block. Thus the prediction equations are ± . . .
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This note was uploaded on 03/20/2011 for the course STATISTIC 101 taught by Professor Fandia during the Spring '10 term at UCLA.

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ch07 - Chapter 7. Supplemental Text Material S7-1. The...

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