A fractional factorial design that includes half of the runs that a full factorial has would use the notation l raise to the f 1 power. The table shows the 2 4 1 8 run half fraction experiment design and the resulting filtration rate extracted from the table for the full 16 run factorial experiment.

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### This gives us a one half fraction of the 2 4 design.

**Half fractional factorial design**. Lets use the concept of the generator and construct a design for the 2 4 1 fractional factorial. For example if we have 2 levels and 4 factors it would be called a 2 raise to the 4 1 design. A fractional design is a design in which experimenters conduct only a selected subset or fraction of the runs in the full factorial design.

Fractional factorial designs are a good choice when resources are limited or the number of factors in the design is large because they use fewer runs than the full factorial designs. Useful fractional factorial designs for up to 10 factors are summarized here. When we create a fractional factorial design from a full factorial design the first step is to decide on an alias structure.

The results of that example may be used to simulate a fractional factorial experiment using a half fraction of the original 2 4 16 run design. Full factorial experiments can require many runs. Lets look at a fairly simple experiment model with four factors.

The asqc 1983 glossary tables for statistical quality control defines fractional factorial design in the following way. If the number of combinations in a full factorial design is too high to be logistically feasible a fractional factorial design may be done in which some of the possible combinations usually at least half are omitted. This requires less effort and fewer tests but also does not include complete information.

For example with two factors each taking two levels a factorial experiment would have four treatment combinations in total and is usually called a 22 factorial design. Box hunter and hunter 1978 describe a fractional factorial design for studying a chemical reaction to determine what percentage of the chemicals responded in a reactor. Again we want to pick a high order interaction.

A factorial experiment in which only an adequately chosen fraction of the treatment combinations required for the complete factorial experiment is selected to be run. In a typical situation our total number of runs is n 2 k p which is a fraction of the total number of treatments. Hunter new york john wiley sons 1978 and also given in the book.

A fractional factorial design is useful when we cant afford even one full replicate of the full factorial design. We know that to run a full factorial experiment wed need at least 2 x 2 x 2 x 2 or 16 trials. There are very useful summaries of two level fractional factorial designs for up to 11 factors originally published in the book statistics for experimenters by gep.

This defines the generator of the design and from this we can determine which effects are confounded or aliased with which other effects. In this case a fractional factorial design is a reasonable alternative provided that the effects of interest can be estimated.

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