Factorial designs provide the only way to study interactions between treatment A and treatment B. This is because the design has treatment groups with all possible combinations of treatments.
The principles presented earlier about treatment × covariate interactions are relevant to the discussion of treatment A × treatment B interactions in this lesson. These concepts included:
- Detecting interactions may be dependent on the scale of measurement.
- In the presence of interactions, it may not be possible to assess the main effects because the effect of treatment A changes according to the level of treatment B.
- Quantitative interactions refer to the situation in which the direction of the main effects does not change although it could change in magnitude. Qualitative interactions refer to the situation in which the direction of the main effects does change.
The figure above indicates a quantitative interaction. The lines are not parallel but they are not crossing either. The magnitude of the response is dependent on whether treatment A is at a high or low dose. The greatest response is achieved with both Treatment B and Treatment A at high dose. A greater response is observed when Treatment B is at high dose than at mid or low dose, regardless of the dose level of Treatment A, but how much greater is dependent on the level of A. At the lowest dose of A, there is very little difference in the response between the dose levels of B. This is called a quantitative interaction.
The qualitative interaction occurring in the figure above will be difficult to explain. The greatest response is achieved with the low dose of treatment B and the high dose of treatment A. However, if a patient is on low dose of treatment A, the greatest response will be achieved with the high dose of treatment be. Although difficult to sort out, this qualitative interaction is intuitively reasonable for some drug combinations. There may be toxicity or a threshold effect that contribute to making the response greater with only one treatment at the highest dose.