Chemistry Reference
In-Depth Information
% Organic
Wave
length
Wave
length
Flow
Flow
pH
pH
FIgure 5.1
Full factorial design experiments for four factors: pH, flow rate, wavelength,
and percent organic in the mobile phase. Runs are indicated by the dots.
5.3.1.1 Full Factorial designs
In a full factorial experiment, all possible combinations of factors are measured.
Each experimental condition is called a “run,” and the results are called “observa-
tions.” The experimental design consists of the entire set of runs. A common full
factorial design is one with all factors set at two levels each, a high and a low value. If
there are k factors, each at two levels, a full factorial design then has 2
k
runs. In other
words, using four factors, there would be 2
4
or 16 design points or runs. To further
illustrate the point, Figure 5.1 illustrates a full factorial design robustness study for
four factors; pH, flow, wavelength, and percent organic in the mobile phase.
5.3.1.2 Fractional Factorial deigns
Full factorial design runs can really start to add up when investigating large numbers of
factors: for nine factors, 512 runs would be needed! (Without even taking into account
replicate injections.) In addition, the design presented in Figure 5.1 assumes linear
responses between factors, but in many cases, curvature is possible, necessitating center
point runs, further increasing the number of runs. For this reason, full factorial designs
are usually not recommended for more than five factors to minimize time and expense.
So, how do analysts investigate more factors, with or without center points? A
carefully chosen fraction or subset of the factor combinations may be all that is
necessary, which is referred to as fractional factorial designs. In general, a “degree
of fractionation (2
−p
),” such as ½, ¼, etc., of the runs called for in the full factorial
design are chosen, as shown in Figure 5.2. In the example above with nine fac-
tors resulting in 512 runs for a full factorial design, fractional factorial designs can
accomplish the same evaluation in as little as 32 runs (using a 1/16 fraction: 512/16,
or 2
k−p
. The latter is arrived at
by taking the full factorial 2
k
* 2
−p
or 2
k−p
.)
Fractional factorial design works mostly due to the “scarcity of effects principle”
that states that while there may be many factors, few may actually be important,
