full experimental design

Teach yourself statistics

What is a Full Factorial Experiment?

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

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

Experiment II: Mean Scores

Experiment III: Mean Scores

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:

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 B 1 = A 2 B 1 - A 1 B 1 = 2 - 5 = -3

Effect of A at level B 2 = A 2 B 2 - A 1 B 2 = 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 A 1 = A 1 B 2 - A 1 B 1 = 5 - 2 = +3

Effect of B at level A 2 = A 2 B 2 - A 2 B 1 = 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 interaction 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:

where k C r 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.

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

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

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

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

Effect of C at level D 1 = C 2 D 1 - C 1 D 1 = 4 - 5 = -1

Effect of C at level D 2 = C 2 D 2 - C 1 D 2 = 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 C 1 = C 1 D 2 - C 1 D 1 = 4 - 5 = -1

Effect of D at level C 2 = C 2 D 2 - C 2 D 1 = 1 - 4 = -3

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

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

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

Effect of E at level F 1 = E 2 F 1 - E 1 F 1 = 3 - 5 = -2

Effect of E at level F 2 = E 2 F 2 - E 1 F 2 = 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 C 1 = E 1 F 2 - E 1 F 1 = 3 - 5 = -2

Effect of F at level C 2 = E 2 F 2 - E 2 F 1 = 1 - 3 = -2

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

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.

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.

Experimental Design: Types, Examples & Methods

Saul McLeod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul McLeod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

Experimental design refers to how participants are allocated to different groups in an experiment. Types of design include repeated measures, independent groups, and matched pairs designs.

Probably the most common way to design an experiment in psychology is to divide the participants into two groups, the experimental group and the control group, and then introduce a change to the experimental group, not the control group.

The researcher must decide how he/she will allocate their sample to the different experimental groups.  For example, if there are 10 participants, will all 10 participants participate in both groups (e.g., repeated measures), or will the participants be split in half and take part in only one group each?

Three types of experimental designs are commonly used:

1. Independent Measures

Independent measures design, also known as between-groups , is an experimental design where different participants are used in each condition of the independent variable.  This means that each condition of the experiment includes a different group of participants.

This should be done by random allocation, ensuring that each participant has an equal chance of being assigned to one group.

Independent measures involve using two separate groups of participants, one in each condition. For example:

• Con : More people are needed than with the repeated measures design (i.e., more time-consuming).
• Pro : Avoids order effects (such as practice or fatigue) as people participate in one condition only.  If a person is involved in several conditions, they may become bored, tired, and fed up by the time they come to the second condition or become wise to the requirements of the experiment!
• Con : Differences between participants in the groups may affect results, for example, variations in age, gender, or social background.  These differences are known as participant variables (i.e., a type of extraneous variable ).
• Control : After the participants have been recruited, they should be randomly assigned to their groups. This should ensure the groups are similar, on average (reducing participant variables).

2. Repeated Measures Design

Repeated Measures design is an experimental design where the same participants participate in each independent variable condition.  This means that each experiment condition includes the same group of participants.

Repeated Measures design is also known as within-groups or within-subjects design .

• Pro : As the same participants are used in each condition, participant variables (i.e., individual differences) are reduced.
• Con : There may be order effects. Order effects refer to the order of the conditions affecting the participants’ behavior.  Performance in the second condition may be better because the participants know what to do (i.e., practice effect).  Or their performance might be worse in the second condition because they are tired (i.e., fatigue effect). This limitation can be controlled using counterbalancing.
• Pro : Fewer people are needed as they participate in all conditions (i.e., saves time).
• Control : To combat order effects, the researcher counter-balances the order of the conditions for the participants.  Alternating the order in which participants perform in different conditions of an experiment.

Counterbalancing

Suppose we used a repeated measures design in which all of the participants first learned words in “loud noise” and then learned them in “no noise.”

We expect the participants to learn better in “no noise” because of order effects, such as practice. However, a researcher can control for order effects using counterbalancing.

The sample would be split into two groups: experimental (A) and control (B).  For example, group 1 does ‘A’ then ‘B,’ and group 2 does ‘B’ then ‘A.’ This is to eliminate order effects.

Although order effects occur for each participant, they balance each other out in the results because they occur equally in both groups.

3. Matched Pairs Design

A matched pairs design is an experimental design where pairs of participants are matched in terms of key variables, such as age or socioeconomic status. One member of each pair is then placed into the experimental group and the other member into the control group .

One member of each matched pair must be randomly assigned to the experimental group and the other to the control group.

• Con : If one participant drops out, you lose 2 PPs’ data.
• Pro : Reduces participant variables because the researcher has tried to pair up the participants so that each condition has people with similar abilities and characteristics.
• Con : Very time-consuming trying to find closely matched pairs.
• Pro : It avoids order effects, so counterbalancing is not necessary.
• Con : Impossible to match people exactly unless they are identical twins!
• Control : Members of each pair should be randomly assigned to conditions. However, this does not solve all these problems.

Experimental design refers to how participants are allocated to an experiment’s different conditions (or IV levels). There are three types:

1. Independent measures / between-groups : Different participants are used in each condition of the independent variable.

2. Repeated measures /within groups : The same participants take part in each condition of the independent variable.

3. Matched pairs : Each condition uses different participants, but they are matched in terms of important characteristics, e.g., gender, age, intelligence, etc.

Learning Check

Read about each of the experiments below. For each experiment, identify (1) which experimental design was used; and (2) why the researcher might have used that design.

1 . To compare the effectiveness of two different types of therapy for depression, depressed patients were assigned to receive either cognitive therapy or behavior therapy for a 12-week period.

The researchers attempted to ensure that the patients in the two groups had similar severity of depressed symptoms by administering a standardized test of depression to each participant, then pairing them according to the severity of their symptoms.

2 . To assess the difference in reading comprehension between 7 and 9-year-olds, a researcher recruited each group from a local primary school. They were given the same passage of text to read and then asked a series of questions to assess their understanding.

3 . To assess the effectiveness of two different ways of teaching reading, a group of 5-year-olds was recruited from a primary school. Their level of reading ability was assessed, and then they were taught using scheme one for 20 weeks.

At the end of this period, their reading was reassessed, and a reading improvement score was calculated. They were then taught using scheme two for a further 20 weeks, and another reading improvement score for this period was calculated. The reading improvement scores for each child were then compared.

4 . To assess the effect of the organization on recall, a researcher randomly assigned student volunteers to two conditions.

Condition one attempted to recall a list of words that were organized into meaningful categories; condition two attempted to recall the same words, randomly grouped on the page.

Experiment Terminology

Ecological validity.

The degree to which an investigation represents real-life experiences.

Experimenter effects

These are the ways that the experimenter can accidentally influence the participant through their appearance or behavior.

Demand characteristics

The clues in an experiment lead the participants to think they know what the researcher is looking for (e.g., the experimenter’s body language).

Independent variable (IV)

The variable the experimenter manipulates (i.e., changes) is assumed to have a direct effect on the dependent variable.

Dependent variable (DV)

Variable the experimenter measures. This is the outcome (i.e., the result) of a study.

Extraneous variables (EV)

All variables which are not independent variables but could affect the results (DV) of the experiment. Extraneous variables should be controlled where possible.

Confounding variables

Variable(s) that have affected the results (DV), apart from the IV. A confounding variable could be an extraneous variable that has not been controlled.

Random Allocation

Randomly allocating participants to independent variable conditions means that all participants should have an equal chance of taking part in each condition.

The principle of random allocation is to avoid bias in how the experiment is carried out and limit the effects of participant variables.

Order effects

Changes in participants’ performance due to their repeating the same or similar test more than once. Examples of order effects include:

(i) practice effect: an improvement in performance on a task due to repetition, for example, because of familiarity with the task;

(ii) fatigue effect: a decrease in performance of a task due to repetition, for example, because of boredom or tiredness.