Quarter factorial designs

Confounding can be analysed as above, but now each term will be confounded widi three odier terms for a quarter factorial design (or seven odier terms for an eighdi factorial design). 

As an example, a seven factor quarter fraction factorial design would be written as seen in equation below... 

Now let us try to generate a quarter-fraction factorial design for 6 factors. In this design one fourth of the experiments of a fiill factorial design are... 

Determine the lowest order interaction that must be confounded. For a quarter of a five factorial design, second-order interactions must be confounded. Then, almost... 

The design is called a 2 design. In the same way quarter-fraction factorial designs N — eighth-fraction factorial designs N — 2 ),... can be derived, which... 

The design is a Resolution III design and main effects are confounded with two-variable interaction effects. The design is one quarter of a full, 2 , factorial design. 

We take the full factorial design with the number of variables required and select half the experiments. Each half fraction can in turn be divided into quarter fractions. Whereas for a given number of variables at 2 levels there is only one possible full factorial design, there can be many possible fractional factorial designs and they are not at all equivalent to one another. We saw in paragraph II.C.3 that the 2 design consists of 16 experiments representing all possible combinations of... 

Specific factorial designs can be considered if the resources are insufficient to run all tests prescribed by the complete factorial design (see for example, Feng et al, 2013). This, in particular, is true for the experiments with the number of factors k>3. These designs include fractional factorial designs such as 2 and 2 which are called one-half and one-quarter designs, respectively, because they require respectively a half or a quarter of the total number of tests of the full factor experiment. 

It was decided that four parameters or factors would cover the range of service conditions. Upper dwell time and temperatures were tested at three different values, lower dwell time at two different values and with two different environments. The number of combinations of conditions is therefore 36. A quarter fractional nine trial design necessarily has some confounding and loss of information compared to a full factorial design, but retains the analysis of main effects and some synergies or interactions between factors. 

For a 2 f fractional factorial, first set up the design consisting of 2k f experiments for the first k — f factors, i.e. for a quarter (/ = 2) of a five (=k) factorial experiment, set up a design consisting of eight experiments for the first three factors. 

Four of the runs in Table 4.9 are identical to runs in Table 4.6. The responses for these runs are the same in both tables and represent real values. The other four runs have level combinations for which the experiments had not been performed. Their response values are simulations obtained from the experimental data in Table 4.6. The calculated contrasts are also shown in Table 4.9, where we can observe that their values are in excellent agreement with the estimates of the average and the main effects determined from the 2y design (Table 4.7). Analyzing the results of the 2 quarter-fraction, which would be obtained in the initial stage of the investigation, the research workers could decide if they should perform more runs to arrive at a half-fraction or even the 2 complete factorial, if they should introduce new factors in place of the 1 and 5 variables (which appear to have little influence on the response), or even if they woifld rather change the levels of the variables. ... 

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