Different Variations of Factorial Designs, Explained

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Different Variations of Factorial Designs

Research Methods in Psychology · Chapter 13: Factorial, Within-Subject, and Small-n DesignsSection 13.1 · lesson 77 of 949 min · free with a free account

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In this lesson

Three architectures for combining factors, and the trade-offs between them.

  • Distinguish between-subjects, within-subjects and mixed factorials
  • Identify the trade-offs each variation makes
  • Match a research question to the right architecture
  • Read notation that signals which factors are which

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Everything said in the video, section by section (1,190 words). Read along, or search it.

Factorial, Within-Subject, and Small-n Designs

Different Variations of Factorial Designs

A factorial design tells you which variables are crossed. It does not tell you where the participants went. This section is about that second question, and it turns out to matter enormously. By the end you will distinguish between-subjects, within-subjects, and mixed factorial designs. You will identify the trade-offs each makes in power, in time, and in threats to validity. You will match a research question to the most appropriate architecture. You will recognize the practical considerations that shape the choice. Sample size, fatigue, and carryover, chiefly. And you will read notation that signals which factors are between and which are within.

The Implicit Question

Where Do the Participants Go

The basic factorial design manipulates two or more independent variables together. That is how you detect main effects and interactions. But the introduction left one question implicit. How are participants distributed across the cells of the design? The answer matters a great deal, more than most students expect. Different distributions produce genuinely different experiments. They have different requirements for sample size. They have different vulnerabilities to specific threats. And they require different statistical analyses to interpret. Three architectures dominate the literature. Choosing among them is one of the most consequential decisions in experimental work.

Architecture One

Between-Subjects Factorials

Start with the between-subjects factorial. Every cell of the design contains a different group of participants. A two by two design needs four separate groups. A two by three needs six. Each participant contributes data to exactly one cell, and never returns. So comparisons across factors are made between groups of different people. This architecture inherits the strengths of any between-subjects design. Clean separation of conditions, with no possibility of practice or carryover effects. And random assignment stays straightforward, because each person is assigned once. The cost is sample size, and the cost is steep. Every cell needs enough participants for adequate power. And the total demand grows multiplicatively as factors are added.

Architecture Two

Within-Subjects Factorials

Now the within-subjects factorial, which inverts almost every one of those properties. Every participant experiences every cell of the design. Consider a two by two design with twenty participants. It generates eighty observations. A between-subjects design with five participants per cell would produce only twenty. Same number of people, four times the data. And because each participant serves as their own control, individual differences drop out. Removing them from the comparison dramatically increases statistical power. The trade-off is exactly what you would guess. Participants who experience multiple conditions may carry effects from one to the next. Through learning, through fatigue, or through expectations about what comes next. The design has to work carefully to manage that risk.

Architecture Three

Mixed Factorials

The third architecture combines the two. In a mixed factorial, at least one factor is manipulated between subjects. And at least one is manipulated within subjects. Here is a typical example from applied research. Two training methods, compared between subjects. Crossed with three levels of task difficulty, tested within subjects. Mixed designs are common because they reflect what is actually practical. The training method cannot be re-administered to the same person. But every participant can attempt all three difficulty levels. They do require analysis techniques that handle both types of factor at once. In exchange they offer a useful middle ground.

Notation

Which Factor Is Which

Sometimes the notation tells you which architecture you are looking at. A two between, by three within, design says it explicitly. The reader learns at a glance how participants are distributed. Often, though, the notation is ambiguous. It gives you numbers and nothing else. When that happens, the methods section should make the architecture clear. If it does not, you cannot evaluate the study's power, or its threats. So reading published research carefully means attending to more than factors and levels. It means checking whether each factor was tested between or within participants.

Head to Head

Sample Size, Power, and Carryover

Put the three architectures side by side on the features that decide most studies. Sample size needed. Between-subjects demands the largest. Within-subjects needs the smallest, and mixed sits in between. Statistical power per participant runs the other way. Between-subjects gives you the lowest, within-subjects the highest. In a mixed design it depends which factor you are asking about. High for the within factor, and lower for the between factor. Now carryover risk, which is the price of that power. Between-subjects has none at all, because nobody sees a second condition. Within-subjects carries high risk, and mixed carries it for the within factor only.

Head to Head

Time, Assignment, and Fit

Three more features complete the comparison. Time per participant is shortest for between-subjects designs. It is longest for within-subjects, where each person does everything. Mixed designs are intermediate again. Random assignment requirements differ too. Between-subjects assignment is straightforward, a single allocation per person. Within-subjects designs need order randomization instead. And mixed designs need both kinds at once. Finally, when is each one best? Between-subjects when conditions cannot be re-administered to the same person. Within-subjects when conditions are repeatable and brief. And mixed when one factor naturally fits each architecture.

Step One

Start With the Manipulation

So how do you actually choose? Start with the manipulation itself. Some manipulations simply cannot be repeated within the same person. Once a participant has learned a piece of information, they cannot unlearn it. There is no second condition available for that person. Manipulations of that kind force a between-subjects approach. The decision is made for you before you start. Other manipulations are inherently repeatable. A stimulus can be presented many times. A task can be performed under several different instructions. A sequence of trials can be randomized across conditions. Manipulations of that second kind permit a within-subjects approach.

Step Two

Weigh Power Against Carryover

When both architectures are technically possible, the real weighing begins. Statistical power on one side, carryover risk on the other. A within-subjects design extracts much more information from each participant. But it requires careful counterbalancing. Without that, order effects contaminate the manipulation you care about. A between-subjects design is simpler to run, and simpler to analyze. Its cost is that it demands a larger sample to detect the same effect. Neither of those is a small consideration. And mixed designs offer a compromise when the factors fall on different sides of the divide.

Not a Default

Architecture Is a Methodological Decision

One warning before leaving this topic. In some research traditions, between-subjects designs are the unspoken default. In others, within-subjects designs are. Neither default is correct, and inheriting one is not a decision. Three things should drive the choice. The research question you are asking. The nature of the manipulation, which may settle it outright. And the specific threats to validity that matter most for your topic. Get it wrong and the consequences scale with how wrong you were. A mismatched architecture wastes participants in the best case. In the worst case it produces results nobody can interpret.

Vocabulary

The Terms This Chapter Runs On

Five terms carry the rest of this chapter. A between-subjects factorial assigns different participants to each cell. A within-subjects factorial has every participant experience every cell. A mixed factorial combines at least one factor of each type. A carryover effect is the influence of one condition on later conditions. That covers learning, fatigue, and expectations, and it only threatens within-subjects designs. And counterbalancing is the procedure that varies the order of conditions across participants. Its job is to neutralize order-related artifacts. The next section takes counterbalancing apart in detail.

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FAQ

Questions students ask us

About Different Variations of Factorial Designs

What are the three types of factorial designs based on how participants are assigned?
The three types are between-subjects, within-subjects, and mixed factorial designs. In a between-subjects factorial, each cell contains a different group of participants. In a within-subjects factorial, every participant experiences every cell. A mixed factorial manipulates at least one factor between subjects and at least one within subjects. The choice affects sample size, statistical power, time per participant, carryover risk, and the analysis required.
Which type of factorial design needs the fewest participants?
A within-subjects factorial needs the fewest participants, because every person contributes data to every cell. In a 2 x 2 design, 20 participants yield 80 observations, whereas a between-subjects version with 5 people per cell yields only 20. Each person also serves as their own control, which removes individual differences and raises power. The cost is carryover risk from learning, fatigue, or expectations, so condition order must be counterbalanced.
What is an example of a mixed factorial design?
A typical mixed factorial compares two training methods between subjects and crosses them with three levels of task difficulty tested within subjects. The training method cannot be re-administered to the same person, but everyone can attempt all three difficulty levels. Mixed designs are common because they match what is practical, and they need analyses, such as mixed ANOVA, that handle between-subjects and within-subjects factors together.
How can you tell whether a factor is between-subjects or within-subjects in a published study?
Check the design notation and the methods section. Some notation states it directly, as in a 2 (between) x 3 (within) design. Often the notation gives only numbers, such as 2 x 3, so the methods section has to say how participants were distributed across cells. Without that information, a reader cannot judge the study's statistical power or its exposure to carryover effects.

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