# What Is a Full Factorial Experiment?

This lesson describes full factorial experiments. Specifically, the lesson answers three questions:

• What is a full factorial experiment?
• What causal effects can we test in a full factorial experiment?
• How should we interpret causal effects?
• What are the advantages and disadvantages of a full factorial experiment?

## What is a Factorial Experiment?

A factorial experiment allows researchers to study the joint effect of two or more factors on a dependent variable. Factorial experiments come in two flavors: full factorials and fractional factorials. In this lesson, we will focus on the full factorial experiment, not the fractional factorial.

### Full Factorial Experiment

A full factorial experiment includes a treatment group for every combination of factor levels. Therefore, the number of treatment groups is the product of factor levels. For example, consider the full factorial design shown below:

A1 A2
B1 B2 B3 B1 B2 B3
C1 Grp 1 Grp 2 Grp 3 Grp 4 Grp 5 Grp 6
C2 Grp 7 Grp 8 Grp 9 Grp 10 Grp 11 Grp 12
C3 Grp 13 Grp 14 Grp 15 Grp 16 Grp 17 Grp 18
C4 Grp 19 Grp 20 Grp 21 Grp 22 Grp 23 Grp 24

Factor A has two levels, factor B has three levels, and factor C has four levels. Therefore, this full factorial design has 2 x 3 x 4 = 24 treatment groups.

Full factorial designs can be characterized by the number of treatment levels associated with each factor, or by the number of factors in the design. Thus, the design above could be described as a 2 x 3 x 4 design (number of treatment levels) or as a three-factor design (number of factors).

### Fractional Factorial Experiments

The other type of factorial experiment is a fractional factorial. Unlike full factorial experiments, which include a treatment group for every combination of factor levels, fractional factorial experiments include only a subset of possible treatment groups.

## Causal Effects

A full factorial experiment allows researchers to examine two types of causal effects: main effects and interaction effects. To facilitate the discussion of these effects, we will examine results (mean scores) from three 2 x 2 factorial experiments:

Experiment I: Mean Scores

A1 A2
B1 5 2
B2 2 5

Experiment II: Mean Scores

C1 C2
D1 5 4
D2 4 1

Experiment III: Mean Scores

E1 E2
F1 5 3
F2 3 1

### Main Effects

In a full factorial experiment, a main effect is the effect of one factor on a dependent variable, averaged over all levels of other factors. A two-factor factorial experiment will have two main effects; a three-factor factorial, three main effects; a four-factor factorial, four main effects; and so on.

#### How to Measure Main Effects

To illustrate what is going on with main effects, let's look more closely at the main effects from Experiment I:

Experiment I: Mean Scores

A1 A2
B1 5 2
B2 2 5

Assuming there were an equal number of observations in each treatment group, we can compute the main effect for Factor A as shown below:

Effect of A at level B1 = A2B1 - A1B1 = 2 - 5 = -3

Effect of A at level B2 = A2B2 - A1B2 = 5 - 2 = +3

Main effect of A = ( -3 + 3 ) / 2 = 0

And we can compute the main effect for Factor B as shown below:

Effect of B at level A1 = A1B2 - A1B1 = 5 - 2 = +3

Effect of B at level A2 = A2B2 - A2B1 = 2 - 5 = -3

Main effect of B = ( 3 - 3 ) / 2 = 0

In a similar fashion, we can compute main effects for Experiment II (see Problem 1) and Experiment III (see Problem 2).

Warning: In a full factorial experiment, you should not attempt to interpret main effects until you have looked at interaction effects. With that in mind, let's look at interaction effects for Experiments I, II, and III.

### Interaction Effects

In a full factorial experiment, an interaction effect exists when the effect of one independent variable depends on the level of another independent variable.

#### When Interactions Are Present

The presence of an interaction can often be discerned when factorial data are plotted. For example, the charts below plot mean scores from Experiment I and from Experiment II:

Experiment I Experiment II In Experiment I, consider how the dependent variable score is affected by level A1 versus level A2. In the presence of B1, the dependent variable score is bigger for A1 than for A2. But in the presense of B2, the reverse is true - the dependent variable score is bigger for A2 than for A1.

In Experiment II, level C1 is associated with a little bit bigger dependent variable score in the presence of D1; but a much bigger dependent variable score in the presence of D2.

In both charts, the way that one factor affects the dependent variable depends on the level of another factor. This is the definition of an interaction effect. In charts like these, the presence of an interaction is indicated by non-parallel plotted lines.

Note: These charts are called interaction plots. For guidance on creating and interpreting interation plots, see Interaction Plots.

#### When Interactions Are Absent

Now, look at the chart below, which plots mean scores from Experiment III:

Experiment III In this chart, E1 has the same effect on the dependent variable, regardless of the level of Factor F. At each level of Factor F, the dependent variable is 2 units bigger with E1 than with E2. So, in this chart, there is no interaction between Factors E and F. And you can tell at a glance that there is no interaction, because the plotted lines are parallel.

### Number of Interactions

The number of interaction effects in a full factorial experiment is determined by the number of factors. A two-factor design (with factors A and B) has one two-way interaction (the AB interaction). A three-factor design (with factors A, B, and C) has one three-way interaction (the ABC interaction) and three two-way interactions (the AB, AC, and BC interactions).

A general formula for finding the number of interaction effects (NIE) in a full factorial experiment is:

NIE =
kΣr=2
kCr =
kΣr=2
k! / [ r!(k - r)! ]

where kCr is the number of combinations of k things taken r at a time, k is the number of factors in the full factorial experiment, and r is the number of factors in the interaction term.

Note: If you are unfamiliar with combinations, see Combinations and Permutations.

## How to Interpret Causal Effects

Recall that the purpose of conducting a full factorial experiment is to understand the joint effects (main effects and interaction effects) of two or more independent variables on a dependent variable. When a researcher looks at actual data from an experiment, small differences in group means are expected, even when independent variables have no causal connection to the dependent variable. These small differences might be attributable to random effects of unmeasured extraneous variables.

So the real question becomes: Are observed effects significantly bigger than would be expected by chance - big enough to be attributable to a main or interaction effect rather than to an extraneous variable? One way to answer this question is with analysis of variance. Analysis of variance will test all main effects and interaction effects for statistical significance. Here is how to interpret the results of that test:

• If no effects (main effects or interaction effects) are statistically significant, conclude that the independent variables do not affect the dependent variable.
• If a main effect is statistically significant, conclude that the main effect does affect the dependent variable.
• If an interaction effect is statistically significant, conclude that the interaction factors act in combination to affect the dependent variable.

Recognize that it is possible for factors to affect the dependent variable, even when the main effects are not statistically significant. We saw an example of that in Experiment I.

Experiment I In Experiment I, both main effects were zero; yet, the interaction effect is dramatic. The moral here is: Do not attempt to interpret main effects until you have looked at interaction effects.

Note: To learn how to implement analysis of variance for a full factorial experiment, see ANOVA With Full Factorial Experiments.

## Advantages and Disadvantages

Analysis of variance with a full factorial experiment has advantages and disadvantages. Advantages include the following:

• The design permits a researcher to examine multiple factors in a single experiment.
• The design permits a researcher to examine all interaction effects.
• The design requires subjects to participate in only one treatment group.

Disadvantages include the following:

• When the experiment includes many factors and levels, sample size requirements may be excessive.
• The need to include all treatment combinations, regardless of importance, may waste resources.

## Test Your Understanding

Problem 1

The table below shows results from a 2 x 2 factorial experiment.

Experiment II: Mean Scores

C1 C2
D1 5 4
D2 4 1

Assuming equal sample size in each treatment group, what is the main effect for both factors?

(A) -2
(B) 3.5
(C) 4
(D) 7
(E) 14

Solution

The correct answer is (A). We can compute the main effect for Factor C as shown below:

Effect of C at level D1 = C2D1 - C1D1 = 4 - 5 = -1

Effect of C at level D2 = C2D2 - C1D2 = 1 - 4 = -3

Main effect of C = ( -1 + -3 ) / 2 = -2

And we can compute the main effect for Factor D as shown below:

Effect of D at level C1 = C1D2 - C1D1 = 4 - 5 = -1

Effect of D at level C2 = C2D2 - C2D1 = 1 - 4 = -3

Main effect of D = ( -1 + -3 ) / 2 = -2

Problem 2

The table below shows results from a 2 x 2 factorial experiment.

Experiment III: Mean Scores

E1 E2
F1 5 3
F2 3 1

Assuming equal sample size in each treatment group, what is the main effect for both factors?

(A) -12
(B) -2
(C) 0
(D) 3
(E) 4

Solution

The correct answer is (B). We can compute the main effect for Factor E as shown below:

Effect of E at level F1 = E2F1 - E1F1 = 3 - 5 = -2

Effect of E at level F2 = E2F2 - E1F2 = 1 - 3 = -2

Main effect of E = ( -2 + -2 ) / 2 = -2

And we can compute the main effect for Factor F as shown below:

Effect of F at level C1 = E1F2 - E1F1 = 3 - 5 = -2

Effect of F at level C2 = E2F2 - E2F1 = 1 - 3 = -2

Main effect of F = ( -2 + -2 ) / 2 = -2

Problem 3

Consider the interaction plot shown below. Which of the following statements are true? (A) There is a non-zero interaction between Factors A and B.
(B) There is zero interaction between Factors A and B.
(C) The plot provides insufficient information to describe the interaction.

Solution

The correct answer is (B). At every level of Factor B, the difference between A1 and A2 is 3 units. Because the effect of Factor A is constant (always 3 units) at every level of Factor B, there is no interaction between Factors A and B.

Note: The parallel pattern of lines in the interaction plot indicates that the AB interaction is zero.