Factorial Designs

CENGAGE FACTORIAL DESIGNS STUDY NOTES

INTRODUCTION

Factor: An independent variable (IV) in an experiment, especially those that include two or more IVs.
Factorial Design: A research design that includes two or more factors (IVs).
- A two-factor design has two IVs.
- A single-factor design has one IV.
- Example of a three-factor design: A design represented as $(2 imes 3 imes 2)$ has a total of 12 conditions.

STRUCTURE OF A TWO-FACTOR EXPERIMENT

Example with two IVs: Mode of Presentation (Factor A) and Control of Study Time (Factor B).
- Structure: The levels of one factor determine the columns, and the levels of the second factor determine the rows.
- On Paper:
  - Fixed Time: Exam scores for participants studying text on paper for a fixed time.
  - Self-Regulated Time: Exam scores for participants studying text on paper for self-regulated time.
- On Screen:
  - Fixed Time: Exam scores for participants studying text on screen for a fixed time.
  - Self-Regulated Time: Exam scores for participants studying text on screen for self-regulated time.

COMPLEX DESIGNS

Complex Designs:
- Factorial combination of 2 IVs
- Example: A $(3 imes 3)$ design represents 2 IVs, each with 3 levels, resulting in a factorial combination of 9 conditions.
Application of Complex Designs:
- Research by Dittmar et al. (2006) examined:
- The overall effect of the Version of Picture Book.
- Barbie images caused greater body dissatisfaction compared to Emme and neutral images.
- An overall increase in body dissatisfaction correlated with advancing grade levels (a natural groups variable).
- Notable effects observed for the combined influence of grade and exposure to images.

KNOWLEDGE CHECK 1: INTRODUCTION TO FACTORIAL DESIGNS

Question: How many independent variables are there in a $2 imes 2 imes 2$ factorial design?
- A) 2, B) 3, C) 4, D) 8
- Correct Answer: B) 3

TWO-FACTOR DESIGN

Representation: A two-factor design can be represented by a matrix where each cell corresponds to a separate treatment condition (specific combination of factors).
Data Interpretation: The study yields three separate and distinct sets of information regarding how the two factors independently and jointly affect behavior.

MAIN EFFECTS

Definition: Main effect is the mean differences among levels of one factor.
- In a two-factor study, two main effects exist: one for each factor.
Example Data Representation:
- Overall Means:
- On Paper:
  - Fixed time: $M = 22$ ;
  - Self-regulated time: $M = 18$ ;
  - Overall M = 20.
- On Screen:
  - Fixed time: $M = 18$ ;
  - Self-regulated time: $M = 14$ ;
  - Overall M = 16.

INTERACTION BETWEEN FACTORS

Definition: An interaction occurs when one factor directly influences the effect of another factor.
- Example: Drug interactions where one drug alters the effects of another drug (e.g., exaggerating, minimizing, or blocking effects).
- Independent Factors: No interaction if factors do not affect each other.
Adjusted Data Means for Interaction:
- Example Data Representation:
- On Paper:
  - Fixed time: $M = 20$ ;
  - Self-regulated time: $M = 20$ ;
  - Overall M = 20.
- On Screen:
  - Self-regulated time: $M = 20$ ;
  - Fixed time: $M = 12$ ;
  - Overall M = 16.

ALTERNATIVE VIEWS OF INTERACTION

Interaction exists when the effects of one factor depend on the levels of the second factor.
Graphing Results: Non-parallel lines on a graph indicate an interaction.
Statistical Testing: A statistical test is required to determine if the interaction is significant.

INTERPRETING MAIN EFFECTS AND INTERACTIONS

Statistical Analysis: Significant effects identified by statistical analysis demand caution in interpretation.
Average Distortion: Main effects can give a distorted view of overall outcomes as they are averages which may not depict individual results accurately.

INDEPENDENCE OF MAIN EFFECTS AND INTERACTIONS

A two-factor study enables evaluation of:
- Mean differences from the main effect of factor A.
- Mean differences from the main effect of factor B.
- Mean differences from the interaction of factors A and B.

KNOWLEDGE CHECK 2: INTERPRETING MAIN EFFECTS AND INTERACTIONS

Question: Possible outcomes from a $2 imes 2$ factorial design?
- A) Two main effects and an interaction.
- B) Three main effects and no interaction.
- C) Four main effects and an interaction.

BETWEEN-SUBJECTS AND WITHIN-SUBJECTS DESIGN

Between-Subjects Design:
- Requires a large number of participants; individual differences may confound the results.
- Avoids order effects.
Within-Subjects Design:
- Each participant undergoes numerous treatments; often time-consuming and risks attrition.
- Increases the risk of testing effects but only requires one group of participants, thus eliminating individual differences.

MIXED DESIGNS

Definition: Mixed designs incorporate both within- and between-subjects variables.
Application: Common in factorial studies where one factor is between-subjects and the other is within-subjects, often used when a factor threatens validity.

EXPERIMENTAL AND NONEXPERIMENTAL OR QUASI-EXPERIMENTAL RESEARCH STRATEGIES

Experimental Design: All factors are true independent variables manipulated by the researcher.
Non-Experimental Design: All factors are quasi-independent variables that are not manipulated, but are still referred to as factors.

COMBINED STRATEGIES: EXPERIMENTAL AND QUASI-EXPERIMENTAL

Involves two different research strategies within the same factorial design.
- One factor acts as a true IV (experimental strategy) and the second factor is a quasi-independent variable (nonexperimental or quasi-experimental strategy).
- Types: May include preexisting participant characteristics or time.

PRETEST–POSTTEST CONTROL GROUP DESIGNS

Example: A two-factor mixed design where:
- One factor (treatment/control) is a between-subjects factor.
- Other factor (pretest–posttest) is a within-subjects factor.
Data Representation:
- Treatment Group: Pretest and posttest scores for participants receiving treatment.
- Control Group: Pretest and posttest scores for those not receiving treatment.

HIGHER-ORDER FACTORIAL DESIGNS

Concept: Extends the basic two-factor design to more complex designs involving three or more factors (higher-order factorial designs).
Three-Factor Design: Evaluates main effects for all three factors, and interprets higher-order interactions similarly to two-factor interactions.
Note: Designs with more than three factors can yield complex and difficult-to-analyze results.

STATISTICAL ANALYSIS OF FACTORIAL DESIGNS

Statistical analysis depends on whether:
- Factors are between-subjects, within-subjects, or mixed.
Standard Practices:
- Compute mean for each treatment condition (cell)
- Utilize ANOVA to assess statistical significance of mean differences.

EXPANDING AND REPLICATING A PREVIOUS STUDY

Replication: Repeating a prior study using the same IVs exactly.
Expansion: Introducing a second factor through new conditions or participant characteristics to assess generalizability of previously reported effects to new populations or situations.

REDUCING VARIANCE IN BETWEEN-SUBJECTS DESIGNS

Purpose: Use a participant variable as a second factor to reduce within-group variance.
Outcome: Decreases individual differences within each group without sacrificing external validity.

WHY WORRY ABOUT ALL THESE INTERACTIONS?

Most outcomes in psychological studies emerge from interactions rather than main effects, indicating a need for careful analysis of interdependencies.

IDENTIFYING FACTORIAL DESIGNS IN EMPIRICAL JOURNAL ARTICLES

Method Section: Details study design.
Factorial Notation: Presented as ___ x ____ x ____.
Results Section: Evaluates significance of main effects and interactions, references significance or p values, and discusses MANOVA and F-tests.

IDENTIFYING FACTORIAL DESIGNS IN POPULAR PRESS ARTICLES

Look for phrases indicating interactions, such as "it depends" or "only when."
Be alert for mentions of participant variables (e.g., age, gender, ethnicity).