When an experiment is conduced with fewer experiment units than the number required for a single complete replicate of a factorial design, it should come as no surprise that something is lost.

Confounded effects

When only a subset of the runs in a complete factorial design are used, some interactions get 'mixed up' with main effects.

When main effects and interactions are mixed up in this way, they are said to be confounded or aliased with each other.

It is necessary to assume that high-order interactions between the factors are either not present or negligible in relation to the main effects of the factors. If the number of experimental units is much smaller than the number required for a complete experiment, it may be necessary to assume that there are no interactions between the variables at all.

Hardness of biscuits

This experiment is an extreme example of a small subset of the 215 = 32,768 runs needed for a complete factorial design. With 16 runs, it is only possible to estimate the overall mean and 15 other quantities, so only the 15 main effects of the 15 factors can be estimated — there is no independent information to estimate interactions.

Moreover, the main effects of all factors are confounded with various interactions between other factors. For example, factor L had the largest main effect — we estimate that adding this additive will increase the hardness of the biscuits by 6.0.

  Additive Hardness
Run   ...   D E   ...   L   ...   DE   ...   Y
1 ... +1 +1 ... +1 ... +1 ... 53.3
2 ... -1 +1 ... -1 ... -1 ... 46.6
3 ... +1 +1 ... +1 ... +1 ... 53.8
4 ... -1 +1 ... -1 ... -1 ... 44.6
5 ... +1 -1 ... -1 ... -1 ... 44.8
6 ... -1 -1 ... +1 ... +1 ... 58.9
7 ... +1 -1 ... -1 ... -1 ... 56.5
8 ... -1 -1 ... +1 ... +1 ... 60.5
9 ... +1 -1 ... -1 ... -1 ... 48.2
10 ... -1 -1 ... +1 ... +1 ... 47.4
11 ... +1 -1 ... -1 ... -1 ... 54.8
12 ... -1 -1 ... +1 ... +1 ... 45.9
13 ... +1 +1 ... +1 ... +1 ... 56.3
14 ... -1 +1 ... -1 ... -1 ... 50.2
15 ... +1 +1 ... +1 ... +1 ... 62.3
16 ... -1 +1 ... -1 ... -1 ... 44.7

However the term for an interaction between factors D and E, found by multiplying the +1 and -1 values for D and E, is identical to the levels of L, so the main effect of 6 percent for L could alternatively have been caused by a strong interaction between the effects of D and E.

The main effect for factor L is confounded with the interaction between D and E — it is impossible to say whether the increase of 6 percent is caused by L, a DE interaction or some combination.

Use of the estimated main effects in the previous page to rank the importance of the factors therefore requires the assumption that the interactions are much smaller than the main effects of the factors.