In plain words
A way of planning tests that changes several settings at once in a structured pattern, so the effect of each – and how they interact – can be found in far fewer runs than changing one thing at a time.
Going deeper
Why one factor at a time fails
The instinctive way to optimise a growth is to fix everything, vary the temperature, keep the best value, then vary the pressure, and so on. It feels careful and it is efficient-looking, and it answers a question nobody asked: what the best temperature is at that particular pressure. If the two interact – if the right temperature depends on the pressure – the procedure walks to a point that is optimal along each axis separately and is not the optimum at all.
A factorial design varies them together. With three factors at two levels each, the eight corners of a cube cover the space, and the effect of temperature is estimated by comparing the four runs at high temperature with the four at low – using every run, at every combination of the others. The interaction is estimated from the same eight runs, at no extra cost. That is the surprising part the first time: the factorial gets more information from the same number of experiments, not by being clever about any one run but by arranging them.
The families of design
Full factorials grow as two to the power of the number of factors, which is fine for three and unaffordable for eight. Fractional factorials run a carefully chosen subset – a half, a quarter – at the price of aliasing: some effects become indistinguishable from certain interactions. That is an acceptable trade for a screening experiment, where the purpose is to find which two or three of eight factors matter at all, and where high-order interactions are usually negligible.
Once the important factors are known, the question changes from which to how much, and the designs change with it. Adding centre points tests whether the response is curved rather than flat, and response-surface designs – the line of work Box and Wilson opened in 1951 – fit a curved model and locate an optimum rather than a direction. The sequence screen, then optimise, then confirm is the standard one, and skipping the first step is how people end up mapping a response surface in factors that turn out not to matter.
In a growth laboratory
is close to the ideal case: temperature, pressure, carrier gas flow, promoter quantity, growth time and treatment, with interactions between almost all of them, and a body of folklore built from years of one-factor-at-a-time work. A screening design of eight or sixteen runs can establish which factors move coverage, domain size and layer number, in a fraction of the runs that the informal approach consumes.
Two practical points decide whether it works. Run order must be randomised, or blocked deliberately, because furnaces drift and tubes age: without that, a slow drift is silently attributed to whichever factor happened to be varied last. And the response has to be defined before the runs, and measured the same way each time – “it looked better” is not a response variable. Both are the same discipline that control charts and capability indices apply to a running process, brought forward to the stage where the process is still being found.
For specialists
Structured variation of several process parameters at once to find their effects and interactions efficiently.
Where this comes from
- On the experimental attainment of optimum conditions cited by 4,478