9.4: Factorial Designs (Summary)
- Page ID
- 309670
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Key Takeaways
Key Terms and Concepts
FACTORIAL DESIGNS
Experimental designs with two or more independent variables.
BETWEEN-SUBJECTS FACTORIAL DESIGN
Factorial design with different participants in each combination of conditions.
WITHIN-SUBJECTS FACTORIAL DESIGN
Factorial design with each participant tested in all conditions.
MIXED FACTORIAL DESIGN
Factorial design with some between-subjects and some within-subjects factors.
NON-MANIPULATED INDEPENDENT VARIABLE
A participant characteristic (like age or gender) treated as an independent variable.
MAIN EFFECT
The overall effect of one independent variable, averaged across levels of other variables.
INTERACTION
When the effect of one independent variable depends on the level of another.
SPREADING INTERACTIONS
Interactions where lines on a graph diverge but do not cross.
CROSS-OVER INTERACTION
Interaction where lines cross on a graph, showing a reversal of effects.
SIMPLE EFFECTS
A way of breaking down an interaction to better understand it.
Test Your Knowledge (answers at end of section)
1. In a 2 × 3 factorial design, how many total treatment conditions (cells) are created?
A) 5 conditions
B) 6 conditions
C) 8 conditions
D) 9 conditions
2. A researcher designs a study with three independent variables: teaching method (lecture vs. discussion), class size (small vs. large), and time of day (morning vs. afternoon vs. evening). How many total conditions are in this factorial design, and what advantage does this design have over conducting three separate studies?
A) 8 conditions; it saves time by testing all variables at once
B) 7 conditions; it reduces the number of participants needed
C) 12 conditions; it allows examination of interactions among the three variables that separate studies could not detect
D) 6 conditions; it simplifies the statistical analysis
3. What does an interaction effect indicate in a factorial experiment?
A) Both independent variables have strong main effects
B) The effect of one independent variable depends on the level of the other independent variable
C) Neither independent variable affects the dependent variable
D) The independent variables are perfectly correlated
4. A researcher studies the effects of caffeine (0 mg vs. 200 mg) and task difficulty (easy vs. hard) on performance. Results show: For easy tasks, performance is high regardless of caffeine (no caffeine = 90%, caffeine = 92%). For hard tasks, performance is low without caffeine (45%) but high with caffeine (85%). What pattern of effects is present?
A) Main effect of caffeine only
B) Main effects of both caffeine and task difficulty, but no interaction
C) Interaction effect between caffeine and task difficulty; caffeine improves performance only on hard tasks
D) No significant effects detected
Answer Key
1. B - 6 conditions
In a factorial design, the number of conditions equals the product of the levels of each factor. A 2 × 3 design has two factors: one with 2 levels and one with 3 levels. Multiplying these gives 2 × 3 = 6 total conditions. Each condition represents a unique combination of one level from each factor. For example, if Factor A has levels A1 and A2, and Factor B has levels B1, B2, and B3, the six conditions are: A1B1, A1B2, A1B3, A2B1, A2B2, and A2B3.
2. C - 12 conditions; it allows examination of interactions among the three variables that separate studies could not detect
This is a 2 × 2 × 3 factorial design with 12 total conditions (2 × 2 × 3 = 12). The major advantage of factorial designs over separate single-variable studies is the ability to examine interactions—how variables combine and influence each other. For example, discussion method might work better than lecture in small classes but not in large classes, or morning classes might be effective for lecture but afternoon for discussion. These interaction effects cannot be detected when variables are studied separately. Factorial designs provide a more complete picture of how variables work together in realistic combinations, which is often more informative than studying each variable in isolation.
3. B - The effect of one independent variable depends on the level of the other independent variable
An interaction effect occurs when the effect of one independent variable on the dependent variable differs depending on the level of the other independent variable. In other words, the two factors combine in ways that cannot be predicted from their individual (main) effects alone. For example, a teaching method might be effective for younger students but not for older students, or a drug might work well with therapy but not alone. When graphing an interaction, the lines representing different levels of one factor are typically non-parallel, indicating that the effect of one variable changes across levels of the other.
4. C - Interaction effect between caffeine and task difficulty; caffeine improves performance only on hard tasks
This is a classic interaction effect. The effect of caffeine depends on task difficulty: caffeine has virtually no effect on easy tasks (2% difference) but a large effect on hard tasks (40% difference). When graphed, the lines for easy and hard tasks would be non-parallel—the easy task line would be flat across caffeine conditions while the hard task line would show a steep increase. There is also likely a main effect of task difficulty (easy tasks produce higher performance overall) and possibly a main effect of caffeine (averaging across both task types), but the key finding is the interaction: you cannot accurately describe caffeine's effect without specifying the task difficulty. This demonstrates why interactions are often the most interesting and informative results in factorial experiments.