9.2.1 Interaction between two factors

We illustrate the idea of an interaction using a thought experiment.

A clinical trial is to be carried out to investigate the effect of the doses of two drugs A and B on a medical condition. Both drugs are available at two dose levels. All four combinations of drug-dose levels will be investigated.

$N$ patients are randomly assigned to each of the possible combinations of drug-dose levels, so that $N/4$ patients receive each combination. The response variable is the increase from pre- to post-treatment red blood cell count. The average increase is calculated for each drug-dose level combination.

In all three outcomes, the level of both drugs affects cell count.

1. 1

   In outcome 1, cell count increases with dose level of both A and B. Since the size and direction of the effect of the dose level of drug A on the cell count is unchanged by changing the dose level of drug B there is no interaction.

   
2. 2

   In outcome 2, there is an interaction; at level 1 of drug B, the cell count is lower for drug A level 2, than for drug A level 1. Conversely, at level 2 of drug B, the cell count is lower for drug A level 1, than for drug A level 2. The direction of the effect of the dose levels of drug A is altered by changing the dose of drug B.

   
3. 3

   In outcome 3, there is also an interaction. In this case increasing dose level of drug A increases cell count, regardless of the level of drug B. But the difference in the response for levels 1 and 2 of drug A is much greater for level 1 of drug B than it is for level 2 of drug B. The size of the effect of the dose levels of drug A is altered by changing the dose of drug B.