# Factorial Design Variations

Here, we’ll look at a number of different factorial designs. We’ll begin with a two-factor design where one of the factors has more than two levels. Then we’ll introduce the three-factor design. Finally, we’ll present the idea of the incomplete factorial design.

## A 2x3 Example

For these examples, let’s construct an example where we wish to study of the effect of different treatment combinations for cocaine abuse. Here, the dependent measure is severity of illness rating done by the treatment staff. The outcome ranges from 1 to 10 where higher scores indicate more severe illness: in this case, more severe cocaine addiction. Furthermore, assume that the levels of treatment are:

**Factor 1: Treatment**

- psychotherapy
- behavior modification

**Factor 2: Setting**
* inpatient
* day treatment
* outpatient

Note that the setting factor in this example has three levels.

The first figure shows what an effect for setting outcome might look like. You have to be very careful in interpreting these results because higher scores mean the patient is doing *worse*. It’s clear that inpatient treatment works best, day treatment is next best, and outpatient treatment is worst of the three. It’s also clear that there is no difference between the two treatment levels (psychotherapy and behavior modification). Even though both graphs in the figure depict the exact same data, I think it’s easier to see the main effect for setting in the graph on the lower left where setting is depicted with different lines on the graph rather than at different points along the horizontal axis.

The second figure shows a main effect for treatment with psychotherapy performing better (remember the direction of the outcome variable) in all settings than behavior modification. The effect is clearer in the graph on the lower right where treatment levels are used for the lines. Note that in both this and the previous figure the lines in all graphs are parallel indicating that there are no interaction effects.

Now, let’s look at a few of the possible interaction effects. In the first case, we see that day treatment is never the best condition. Furthermore, we see that psychotherapy works best with inpatient care and behavior modification works best with outpatient care.

The other interaction effect example is a bit more complicated. Although there may be some main effects mixed in with the interaction, what’s important here is that there is a unique combination of levels of factors that stands out as superior: psychotherapy done in the inpatient setting. Once we identify a “best” combination like this, it is almost irrelevant what is going on with main effects.

## A Three-Factor Example

Now let’s examine what a three-factor study might look like. We’ll use the same factors as above for the first two factors. But here we’ll include a new factor for dosage that has two levels. The factor structure in this 2 x 2 x 3 factorial experiment is:

**Factor 1: Dosage**

- 100 mg.
- 300 mg.

**Factor 2: Treatment**

- psychotherapy
- behavior modification

**Factor 3: Setting**

- inpatient
- day treatment
- outpatient

Notice that in this design we have 2x2x3=12 groups! Although it’s tempting in factorial studies to add more factors, the number of groups always increases multiplicatively (is that a real word?). Notice also that in order to even show the tables of means we have to have to tables that each show a two factor relationship. It’s also difficult to graph the results in a study like this because there will be a large number of different possible graphs. In the statistical analysis you can look at the main effects for each of your three factors, can look at the three two-way interactions (e.g., treatment vs. dosage, treatment vs. setting, and setting vs. dosage) and you can look at the one three-way interaction. Whatever else may be happening, it is clear that one combination of three levels works best: 300 mg. and psychotherapy in an inpatient setting. Thus, we have a three-way interaction in this study. If you were an administrator having to make a choice among the different treatment combinations you would be best advised to select that one (assuming your patients and setting are comparable to the ones in this study).

## Incomplete Factorial Design

It’s clear that factorial designs can become cumbersome and have too many groups even with only a few factors. In much research, you won’t be interested in a **fully-crossed factorial design** like the ones we’ve been showing that pair every combination of levels of factors. Some of the combinations may not make sense from a policy or administrative perspective, or you simply may not have enough funds to implement all combinations. In this case, you may decide to implement an incomplete factorial design. In this variation, some of the cells are intentionally left empty – you don’t assign people to get those combinations of factors.

One of the most common uses of incomplete factorial design is to allow for a control or placebo group that receives no treatment. In this case, it is actually impossible to implement a group that simultaneously has several levels of treatment factors and receives no treatment at all. So, we consider the control group to be its own cell in an incomplete factorial rubric (as shown in the figure). This allows us to conduct both relative and absolute treatment comparisons within a single study and to get a fairly precise look at different treatment combinations.