Independently adjustable factors
The two-factor analyses that we have described in this section require an equal number of replicates at all combinations of levels of the two factors. This means that we must be able to independently control the levels of both factors.
In some experiments, it is possible to identify two factors that we believe will affect the response (e.g. temperature and humidity in a warehouse affect deterioration of a product). However it may be impossible in practice to independently adjust both factors in the warehouse — increasing temperature causes a change in humidity but humidity cannot be changed directly.
Despite our identification of two factors, only one can be directly adjusted so we could only conduct a 1-factor experiment in the warehouse. This would not allow separation of the effects of the two underlying factors.
In a laboratory however, it may be possible to independently alter both factors, allowing separate assessment of their effects.
Mixtures
An extreme example of this practical restriction occurs when a product contains a mixture of two constituents, A and B. The quality of the product depends on both constituents, but the proportion of B is one minus the proportion of A, so they are directly related.
For a mixture of two constituents, only a one-factor experiment is possible.
Imprecisely controlled factor levels
A further practical problem with conducting factorial experiments arises when the control of one or more factors is separated from the actual numerical value of the factor. For example, we can attempt to control the temperature in a warehouse with a thermostat, but outside temperature may result in actual temperatures in parts of the warehouse that differ from the thermostat setting.
In the experiment, we may decide to use three different temperatures (levels of this factor), but the actual warehouse temperatures may differ between runs of the experiment — close but not exactly equal to the three target temperatures.
If the actual temperatures are recorded, they should be used as explanatory variables instead of the target factor levels. However the analysis is more complicated than the methods that have been described in this section.
We do not offer solutions to this problem here, but simply alert the reader to the need for a different analysis.