Combining parameters is a very efficient way of reducing the number of experiments, but caution is required. When two or more
parameters are combined, they will be studied at the same level in the design; for example, whenever parameter A is set at
“level +1”, parameter B will also be set to “level +1”, and whenever parameter A has “level -1”, parameter B will also be
at “level -1”. Scientific judgement, therefore, must be used when allocating combinations to ensure that the effects of the
parameters do not cancel each other out. For instance, if increasing the pH increases the resolution and the effect of the
% organic on the resolution is unknown, it is unwise to group pH with % organic in case increasing the % organic decreases
the resolution. As a rule, parameters can be combined if they have the same direction of effect (e.g., if increasing the pH
and the % organic will increase the resolution) or if they affect different method performance characteristics. Categorical
parameters (e.g., column batch and column age) can also be combined, though often the effect of these parameters is the hardest
to predict. Studies using these concepts have been termed RMR because they do not investigate all method parameters separately.
The parameters with the highest risk are included as single factors in the design.
Reducing the statistical resolution of the design
Reducing the statistical resolution is another way of decreasing the number of experiments. Reducing the resolution means
that not all the terms (single parameters or interactions) are estimated independently. The consequence of this is that some
of the terms are statistically ‘aliased’ (a statistically combined), where the estimated effect is the combination of the
true effect of each of the aliased terms. Depending on the number of factors, several resolutions will be available (Figure 2).
Figure 2: Statistical resolutions available for a 7 factor design and scoping experiment.
In DX7 (Design‑Expert 7.1 software; Stat‑Ease Inc., MN, USA), the statistical resolution is indicated by both the colours
and notation. In the notation 2k-p, 2 indicates that the design is a 2-level fractional factorial design; k is the number
of factors; and p is the fraction exponent. Suppose there are 7 factors (k=7):
- When p=0, 2k is a full design and corresponds to 27= 128 experiments (runs)
- When p = 1, this corresponds to a fractional design. The fraction is 2-1 = 0.5; the design is cut in half. This equates to 27-1 = 64 experiments (runs)
- When p = 2, this also corresponds to a fractional design. The fraction is 2-2 = 0.25; the design is a quarter of the full design. This equates to 27-2 = 32 experiments (runs)
And so on.
The most limited form of RMR is a scoping study.17,18 Typically, four experiments make up a scoping study; two centre points and two extreme sets of conditions. At one extreme,
the parameters are set to their chosen levels (either “level +1 or -1”) which, for all parameters together, form the condition
most likely to produce that extreme response. At the other extreme, the parameters are set to their alternative level. The
extremes depend on prior knowledge of the parameters and are not necessarily set at all the low and all the high parameters’
settings - similar to that described earlier with respect to the effect of pH and % organic on the resolution. The centre
points represent the nominal method conditions and give an indication of repeatability; the extremes are thought to lead to
the lowest and highest expected results. Scoping experiments are often used to check the repeatability and the experimental
design space (assessment of whether the factor range settings are appropriate) before committing resources to a more detailed
study. A scoping study is extremely unlikely to be suitable for determining robustness of most analytical methods.
For RMR, a resolution III design is typically used. Although this design does not allow direct estimation of interactions,
it still tests the whole design space. In addition, should an important interaction exist, this will generally be identified
even though more experiments (e.g., by ‘folding over’ the design) may be required to clarify whether it is the interaction
or an aliased main effect. As robustness is being tested, few (if any) important effects are expected, which reduces the likelihood
of needing to clarify aliases. Scientific judgement can often be used to assign the likely main effect/interaction. In addition
to this, if an interaction exists, usually one, if not both, of the main effects involved will also show (hierarchical evidence).
Within each category of resolution, the extent of the aliasing increases as the number of experiments is reduced. In addition
to accepting a higher degree of aliasing, careful allocation of parameters, or groups of parameters, to each main factor (denoted
by a letter in DX7) can assist with retaining separation of important effects.