Main effects in resolution III designs
In resolution III designs, main effects are confounded with 2-factor interactions. In order to interpret them, it is therefore necessary to make the assumption that the 2-factor (and higher-order) interactions are either zero or negligible compared to the main effects.
Main effects in resolution IV designs
In resolution IV designs, no main effects are confounded with 2-factor interactions. As a result, the main effects can be interpreted even in the presence of 2-factor interactions.
However if there is an AB interaction, care must be taken in how the main effect for A is described. The AB interaction means that the effect of changing A is different for the high level of B and the low level of B. The estimated main effect for A is the average of these two separate effects.
Ranking factor main effects
Fractional factorial experiments are often conducted as screening designs to help decide on which factors are most important for further study.
The most important factors are those whose main effects are furthest from zero.
Equivalently, they are the factors whose explained sum of squares are greatest.
Graphical display
If there are several factors, a graphical display of the factor effects (or the sums of squares explained by the factors) can help to highlight which factors stand out from the rest. Several alternative displays are possible:
Yield in chemical experiment
A 25-2 fractional factorial experiment was conducted to estimate the effects of the following factors on the yield of a chemical reaction:
The levels of D and E were defined by:
D = ABC
E = AC
The experiment uses 8 runs and is of resolution III since all main effects are orthogonal but are confounded with 2-factor interactions.
Factors | Response | |||||
---|---|---|---|---|---|---|
Run | A | B | C | D = ABC | E = AC | Yield |
1 | -1 | -1 | -1 | -1 | +1 | 23.2 |
2 | -1 | -1 | +1 | +1 | -1 | 23.8 |
3 | -1 | +1 | -1 | +1 | +1 | 16.8 |
4 | -1 | +1 | +1 | -1 | -1 | 16.2 |
5 | +1 | -1 | -1 | +1 | -1 | 16.9 |
6 | +1 | -1 | +1 | -1 | +1 | 23.4 |
7 | +1 | +1 | -1 | -1 | -1 | 15.5 |
8 | +1 | +1 | +1 | +1 | +1 | 18.1 |
The main effects are the differences between the mean yield at high and low levels of the factors. These are a quarter of the sum of the cross-products of the ±1 columns and the yield column. For example,
main effect for D = (-23.2 + 23.8 + 16.8 - 16.2 + 16.9 - 23.4 - 15.5 + 18.1) / 4 = -0.675
The estimated effect of changing the condensation time, D, from its low to high level is therefore a decrease in yield of 0.675. The table below shows all main effects
Factor | A | B | C | D | E |
---|---|---|---|---|---|
Estimate of effect | -1.525 | -5.175 | 2.275 | -0.675 | 2.275 |
From this table, we conclude that the factor that affects yield most is the B (amount of material 1) and it should be kept low to maximise yield. Factors C (solvent volume) and E (amount of material 2) have the next most important effects and should both be high.
A subsequent experiment might focus on factors B, C and E with enough runs to estimate their interaction.