![]() ![]() Notice that columns with the same color are just -1 times one another. Notice the contrast defining the main effects (similar colors) - there is an aliasing of these effects. Now, take just the fraction of the full design where ABC = -1 and we place it within its own table: trt ABC is the generator of the 1/2 fraction of the \(2^3\) design. Let's take a look at the first block which is a half fraction of the full design. Now, where ABC is confounded in the fractional factorial we can not say anything about the ABC interaction. ![]() ![]() Just as in the block designs where we had AB confounded with blocks - where we were not able to say anything about AB. So, in this case, either one of these blocks above is a one half fraction of a \(2^3\) design. In an example where we have \(k = 3\) treatments factors with \(2^3 = 8\) runs, we select \(2^p = 2 \text\), which is a fraction of the total number of treatments. In setting up the blocks within the experiment we have been picking the effects we know would be confounded and then using these to determine the layout of the blocks. The treatment combinations in each block of a full factorial can be thought of as a fraction of the full factorial. What we did in the last chapter is consider just one replicate of a full factorial design and run it in blocks. ![]()
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