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lect12 - IE 410 Design of Experiments Lecture 12: More on...

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IE 410 Design of Experiments Lecture 12: More on Block Designs Assumption of "Additivity" in Block Designs. Our model is Y ij = μ + τ i + β j + ε ij
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The model implicitly assumes that the factors affect the response Y in an additive way. In the case where both treatments and blocks are fixed effects, this implies that there is no interaction between the factors and blocks.
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I will thus digress temporarily and discuss the topic of INTERACTION effects when factors and blocks are both fixed effects. We will see much more on interaction effects later, but it will be good to see it more than once.
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Consider the following example: We wish to study three recipes by which a chemical can be made. We figure that process temperature may be important, and thus decide to block it. Thus the process will be run at two temperatures.
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Recipe 1 Recipe 2 Recipe 3 Temp 1 X X X Temp 2 X X X
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Suppose that there is an interaction between recipes and temperatures. For example, consider the following INTERACTION PLOT
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Y | 1 | | | 1 1 | | | 2 2 | | 2 |____|_____|_____|_______ R1 R2 R3
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Connecting the 1's with lines, and the 2's with lines completes the plot. Note that in this example the plots for temp 1 and temp 2 are not parallel. Thus there is interaction present (assuming statistical significance). If the response had looked like this:
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Y | | | | 1 1 | | 1 | 2 2 | | 2 |____|_____|_____|_______ R1 R2 R3
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Then there is no interaction. That is, the response can be described by the additive model above. For example note that β 2 - β 1 = 2 (lines) for every recipe in the second plot (just above). Thus the effect of going from temp 1 to temp 2 does not depend on the recipe.
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In the first plot, it is clear that the effect on Y of going from temp 1 to temp 2 DOES depend on the recipe. Thus there is interaction in this case.
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Later we will see models that include interaction terms that look roughly like this: Y ijk = μ + τ i + β j + τβ ij + ε ijk With data sets that look like this:
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Factor A 1 2 3 -------------------------- | X | X | X | | X | X | X | | X | X | X | ------------------------- | X | X | X | | X | X | X | Factor B | X | X | X | ------------------------- | X | X | X | | X | X | X | | X | X | X | -------------------------
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A is at 3 levels, B is at 3 levels, n=3 observations per cell Note that the model with interaction contains 1 + (a-1) + (b-1) + (a-1)(b-1) = ab independent parameters.
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and thus plenty of df left over for SS(error) With each treatment appearing only once in each block, in the RCBD, all the df would be used up in parameter estimates. There would be no df for error. This would make testing impossible.
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lect12 - IE 410 Design of Experiments Lecture 12: More on...

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