The simplest factorial design is the 2 × 2 factorial with two levels of factor A crossed with two levels of factor B to yield four treatment combinations. A special case of the 2 × 2 factorial with a placebo and an active formulation of factor A crossed with a placebo and an active formulation of factor B. This yields the four treatment regimens:

For example, here you could have a placebo for each treatment. In one case you might have a placebo injection for A and a placebo pill for B. Such a design allows the comparison of the levels of factor A (A main effects), the comparison of the levels of factor B (B main effects), and the investigation of A × B interactions.

There are some issues to consider prior to conducting of a factorial clinical trial.

First, the treatments must be amenable to being administered in combination without changing dosage in the presence of each other treatment.

Second, it must be acceptable to not administer the individual treatments, (i.e., a placebo is ethical) or administer them at lower doses if that will be required for the combination.

Third, we must be genuinely interested in learning about treatment combinations required for the factorial design. Otherwise, some of the treatment combinations are unnecessary, yet without them, the advantages of the factorial design are diminished.

Fourth, the therapeutic questions must be chosen appropriately, e.g., treatments that use different mechanisms of action are more suitable candidates for a factorial clinical trial.

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:

1. Detecting interactions may be dependent on the scale of measurement.
2. 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.
3. 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.

A special case of a partial factorial design that occasionally is used in clinical research is the incomplete 2 × 2 factorial design with three treatment groups consisting of drug A, drug B, and drug A in combination with drug B:

Notice that the Placebo A + Placebo B group is not included in the design, hence the incompleteness. The incomplete factorial design has become popular.

 Why?

Combination therapies can be marketable and profitable. If a company can combine the active ingredients for treatment A and treatment B into one pill/tablet/capsule, more symptoms are relieved with one dose of medicine. For example, combining an antihistamine with a decongestant for cold symptoms produces a new cold remedy that will alleviate two major symptoms with one capsule. Additionally once the company has created the new combination product, the company applies for a new patent, extending the years of profitable returns from the research dollars expended to develop the intial products. Approval of a combination therapy however, requires evidence demonstrating the superiority of the AB combination therapy to the A monotherapy and the B monotherapy. A logical experimental design to demonstrate these results would be the incomplete factorial.

Suppose that the response is continuous and that we want to compare the means \(\mu_{A}, \mu_{B}\), and \(\mu_{AB}\), which represent the population means for the A monotherapy, the B monotherapy, and the AB combination therapy, respectively. The research objective is to show the superiority of the combination therapy over the individual therapies.

Assuming that the higher response is more beneficial, a one-sided hypothesis testing format can be constructed as \(H_{0}: {\mu_{A} ≥ \mu_{AB} \text{ or } \mu_{B} ≥ \mu_{AB}}\) versus \(H_{1}: {\mu_{A} < \mu_{AB} \text{ and }\mu_{B} < \mu_{AB}}\)

Notice that the null hypothesis indicates that the AB combination therapy is not better than at least one of the monotherapies, whereas the alternative indicates that the AB combination is better than the A monotherapy and the B monotherapy.

How do we do this?

The appropriate test statistic to use for this situation is called the “min” test. If the data are normally distributed, construct two two-sample t statistics, one comparing the AB combination therapy to the A monotherapy (call it \(t_{A}\)) and the other comparing the AB combination therapy to the B monotherapy (call it \(t_{B}\)).

\(t_A=(\bar{Y}_{AB}-\bar{Y}_A)/s \sqrt{\dfrac{1}{n_{AB}}+\dfrac{1}{n_{A}}}\) , \(t_B=(\bar{Y}_{AB}-\bar{Y}_B)/s \sqrt{\dfrac{1}{n_{AB}}+\dfrac{1}{n_{B}}}\)

where

\(\bar{Y}_{A}=\dfrac{1}{n_A}\sum_{i=1}^{A}Y_{A, i} , \bar{Y}_{B}=\dfrac{1}{n_B}\sum_{i=1}^{B}Y_{B, i} , \bar{Y}_{AB}=\dfrac{1}{n_AB}\sum_{i=1}^{AB}Y_{AB, i} \)

and

\(s^2= \dfrac{1}{n_{A}+n_{B}+n_{AB}-3} \left( \sum_{i=1}^{n_A}\left( Y_{A,i}-\bar{Y}_A \right)^2 + \sum_{i=1}^{n_B}\left( Y_{B,i}-\bar{Y}_B \right)^2 + \sum_{i=1}^{n_{AB}}\left( Y_{AB,i}-\bar{Y}_{AB} \right)^2 \right)\)

The null hypothesis is rejected at the \(\alpha\) significance level in favor if the alternative hypothesis when each of \(t_{A}\) and \(t_{B}\) is statistically significant at the \(\alpha\) significance level.

It is called the min test because in this situation it is comparable to rejecting the null hypothesis if

\(\text{minimum}(t_A, t_b) > t_{n_A+n_B+n_{AB}-3, 1-\alpha}\)

As a simple example, suppose that:

\(\bar{Y}_A=20, \bar{Y}_B=21, \bar{Y}_{AB}=24, n_A=n_B=n_{AB}=50, s=10\)

Then \(t_{A} = 2, t_{B} = 1.5\), and \(\text{minimum}\left(t_{A}, t_{B}\right) = 1.5\), which is not greater than \(t_{147, 0.95} = 1.66\). Thus, the null hypothesis cannot be rejected at the 0.05 significance level, i.e., the AB combination is not significantly better than the A monotherapy and the B monotherapy. It is close, but there clearly is not enough statistical evidence to show significant difference.

In this lesson, among other things, we learned to:

• identify the conditions that would allow a factorial design to be useful
• recognize the difference between qualitative and quantitative interactions
• recognize the situation for which a ‘min’ test is the appropriate analysis