Half-fraction confounds

On the left side of the equation we now have the 134 product, and from this we conclude that Z134 = h- It is the same conclusion we can reach, though more laboriously, by doing Exercise 4.2. In statistical terminology, we say that the use of the half-fraction confounds the main effect, 2, with the 134 interaction effect. The value of the calculated contrast, I2 (or Z134), is actually an estimate of the sum of the two effects. You can confirm that this is true, by adding the values of the 2 and 134 effects in Table 4.3 and comparing the result with the value of I2 in Table 4.4. 

An alternative design would be to use a half-fraction of the design variables for each run of the chamber. Such a design, before randomization, is shown in Table 2.22. With this design the ABCDE five-factor interaction is confounded with the TxH whole-plot contrast. Under the assumption of negligible three-factor and higher-order interactions all main effects and two-factor interactions can be estimated as well as interactions between the design and the environmental variables. 

In terms of absolute size, main effects tend to be larger than two-factor interactions, which in turn tend to be larger than three-factor interactions, and so on. In the half-fraction factorial design of Table 3.9 the main effects are expected to be significantly larger than the three-factor interactions with which they are confounded. As a consequence it is supposed that the estimate for the main effect and the interaction together is an estimate for the main effect alone. 

Screening designs give information about the main effects in a minimum of experiments. Economy has a price. Some problems can occur. It can happen that the interaction term of two (strongly) significant factors also still is significant. Since this interaction effect can be confounded with a main effect one should be careful to consider the main effect responsible for a calculated effect [44[. This is what would happen if we were to apply a half-fraction design to the data of Table 6.1. Factor C is then confounded... 

If we join the two half-fractions, we will restore the original complete factorial. Taking a combination of the appropriate contrasts, we can recover the effect values without any confounding. For example, I2 and Zg involve the same pair of effects, 2 gmd 134. Their sum results in... 

Example 4.5 Complete Defining Relationship and Confounding Pattern for a Half-Fractional Factorial Example... 

Table 2 is a half-fraction of a full 2 design. Looking carefully at these experiments one can see that the combination of each set of three variables, for example B, C, and D. constitutes a full 2 design. One says then that the full factorial for By C, and D is embedded in the half-fraction factorial design. This also means that, if D is found to be unimportant, one can interpret the experiment as a full factorial design for factors A, B, and C without any confounding between interactions of the remaining factors. 

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