Model without interaction between the factors
The simplest model for the effect of two factors on a response is an additive one of the form:
(mean response) = (base value) + (effect of factor A) + (effect of factor B )
One implication of this model is that the effect on the response of changing the level of factor A is the same, whatever the level of factor B. In a similar way, the model assumes that the effect of changing factor B is the same whatever the level of factor A.
If this model holds, the effects of each factor can be separately
described by a table of means.
It should be noted that this model does not always hold. For example the effect on the response of changing A may be lower when B is at a low level than when B is at a high level. This is called interaction between the effects of A and B and a different model must be used if interaction is present. Interaction will be discussed later in this section.
Strength of asphaltic concrete
An experiment was conducted by a civil engineer to assess the effect of the compaction method on asphaltic concrete. Two types of aggregate and four compaction types were used in the factorial experiment with three replicates.
The diagram below shows the tensile strengths (psi) of the samples and the best-fitting factor model with no interaction.
Rotate to y-z. The coloured lines show how the tensile strength differs for the two aggregate types for all compaction methods. Observe that :
The fitted lines are all parallel since the no-interaction model assumes that the differences (Basalt - Silicious) are the same for all compaction methods.
Similarly, rotate to y-x. Again the two fitted lines (corresponding to the different aggregate types) are all parallel.
The no-interaction model assumes that the effect of changing the compaction method is the same for both aggregate types.
With the assumption of no interaction, the differences between the effects of the aggregates and compaction methods can be well summarised by the two tables below:
Aggregate type | ||
---|---|---|
Basalt | Silicious | |
Mean strength | 87.25 | 70.25 |
The model therefore estimates that Basalt aggregate is 17psi stronger than Silicious, whatever the compaction method.
Compaction method | ||||
---|---|---|---|---|
Static | Regular kneading |
Low kneading |
Very low kneading |
|
Mean strength | 66.5 |
120.0 |
79.0 |
49.4 |
This table summarises the differences between the compaction methods, whatever the aggregate type.