Experiments with two or more factors are encountered frequently. The
best way to carry out such experiments is by using factorial experiments.
[Note]
Factorial experiments are experiments in which all combinations of factors
are investigated in each replicate of the experiment. Factorial experiments
are the only means to completely and systematically study interactions
between factors in addition to identifying significant factors. [Note]
One-factor-at-a-time experiments (where each factor is investigated separately
by keeping all the remaining factors constant) do not reveal the interaction
effects between the factors. Further, in one-factor-at-a-time experiments
full randomization is not possible.
To illustrate factorial experiments consider an experiment where the
response is investigated for two factors, and . Assume that the response is studied
at two levels of factor with representing the lower level of and representing the higher level of . Similarly, let and represent the two levels of factor
that are being investigated in this
experiment. Since there are two factors with two levels, a total of combinations exist (-, -, -, -). Thus, four runs are required for
each replicate if a factorial experiment is to be carried out in this
case. Assume that the response values for each of these four possible
combinations are obtained as shown in Table 6.3.
The effect of factor on the response can be obtained by
taking the difference between the average response when is high and the average response when
is low. The change in the response
due to a change in the level of a factor is called the main effect
of the factor. The main effect of as per the response values in Table
6.3 is:Therefore, when is changed from the lower level to
the higher level, the response increases by 20 units. A plot of the response
for the two levels of at different levels of is shown in Figure 6.8.
The plot shows that change in the level of leads to an increase in the response
by 20 units regardless of the level of . Therefore, no interaction exists
in this case as indicated by the parallel lines on the plot. The main
effect of can be obtained as:
Figure 6.8: Interaction plot for the data in Table 6.3.
Now assume that the response values for each of the four treatment combinations
were obtained as shown in Table 6.4. The main effect of in this case is:
Table 6.4: Two-factor factorial experiment.
It appears that does not have an effect on the response.
However, a plot of the response of at different levels of shows that the response does change
with the levels of but the effect of on the response is dependent on the
level of (see Figure 6.9).
Therefore, an interaction between and exists in this case (as indicated
by the non-parallel lines of the figure). The interaction effect between
and can be calculated as follows:
Figure 6.9: Interaction plot for the data in Table 6.4.
Note that in this case, if a one-factor-at-a-time
experiment were used to investigate the effect of factor on the response, it would lead to
incorrect conclusions. For example, if the response at factor was studied by holding constant at its lower level, then
the main effect of would be obtained as , indicating that the response increases
by 20 units when the level of is changed from low to high. On the
other hand, if the response at factor was studied by holding constant at its higher level then
the main effect of would be obtained as , indicating that the response decreases
by 20 units when the level of is changed from low to high.
and 
. Assume that the response is studied
at two levels of factor 
with 
representing the lower level of 
and 
representing the higher level of 
. Similarly, let 
and 
represent the two levels of factor
that are being investigated in this
experiment. Since there are two factors with two levels, a total of 
combinations exist (
-
, 
-
, 
-
, 
-
). Thus, four runs are required for
each replicate if a factorial experiment is to be carried out in this
case. Assume that the response values for each of these four possible
combinations are obtained as shown in Table 6.3.
