Factorial Experimental Design: Interactions, Main Effects, and Statistical Analysis

Administrative and Grading Policies

Experimental Design Assignments: If the Simple Experimental Design assignment is ready, it should be submitted within the next seven minutes. If a student needs more time, they may take it; however, assignments submitted late will receive a one-point deduction. Grades for this specific assignment are based on accuracy.
Feedback and Accuracy: Feedback provided on the first assignment included notes on what the grade would have been if it were based on accuracy, even though it was actually graded on completion. These notes are intended to guide the current accuracy-based assignments.
Upcoming Homework: The assignment handed out this week will be completion-based to provide a midpoint break in the quarter. Accuracy-based assignments will resume afterward.

Overview of Experimental Designs

Correlational Designs: Researchers observe how variables vary naturally with one another without manipulating any factors.
Simple Experimental Designs: These involve exactly one independent variable (IV) and one dependent variable (DV). The researcher manipulates the IV to observe changes in the DV.
Factorial Designs: Researchers investigate how multiple independent variables interact with one another to affect the same dependent variable. This design allows for a more nuanced understanding of how variables function in real-world scenarios where they rarely act in isolation.

Key Concepts in Factorial Design

Function of Multiple Independent Variables: Factorial designs test the effect of various independent variables on the same dependent variable with the same participants.
Interactions: The technical term for an interaction is the phenomenon where the effects of one independent variable depend on the level of another independent variable.
Main Effects: The effect of one single independent variable on the dependent variable, regardless of the levels or influence of all other independent variables in the study.
Simple Effects: The effect of one independent variable at a specific, constant level of another independent variable. These are used to describe the nature of an interaction.

Scenario: The Condiment Experiment

Design Layout: A grocery store wants to display the condiment people enjoy most to drive sales. While chocolate sauce might win when tasted alone (simple experiment), condiments are usually eaten with food. This necessitates a factorial design.
2x2 Factorial Design:
- Independent Variable 1 (Condiment): Levels are Ketchup and Chocolate Sauce.
- Independent Variable 2 (Food): Levels are Hot Dog and Ice Cream.
- Dependent Variable: Enjoyment rating on a scale of $1$ to $7$ .
Cell Means (Predicted Results):
- Ketchup on Hot Dog: High rating.
- Chocolate Sauce on Hot Dog: Low rating.
- Ketchup on Ice Cream: Low rating.
- Chocolate Sauce on Ice Cream: High rating.
Interaction Interpretation: The preference for a condiment (IV1) depends on the level of food (IV2) it is paired with. One cannot say Ketchup is better than Chocolate Sauce overall; it depends on whether the food is a hot dog or ice cream.

Interpreting Result Patterns from Graphs

Visual Identification of Main Effects:
- IV1 (X-axis variable): Compare the midpoint of the two ends on the left vs. the midpoint of the two ends on the right. If they differ, there is a main effect.
- IV2 (Variable in legend/colors): Compare the midpoint of line 1 vs. the midpoint of line 2. If one line is significantly higher than the other, there is a main effect.
Visual Identification of Interactions:
- If the lines are exactly parallel (slopes are identical), there is no interaction.
- If the lines cross, converge, or diverge (slopes are different), an interaction exists.
Independent Outcomes: In a 2x2 design, there are $2 \times 2 \times 2 = 8$ possible patterns of results. Main effects and interactions are independent; you can have an interaction without main effects, or main effects without an interaction.

The Eight Possible Outcome Patterns in 2x2 Designs

No Effects: Two overlapping, flat, parallel lines.
Main Effect of IV1 Only: Two parallel lines with a slope, but they are either overlapping or very close together.
Main Effect of IV2 Only: Two parallel flat lines that are significantly far apart.
Both Main Effects, No Interaction: Two parallel lines with slopes that are significantly far apart from each other.
Interaction Only: A perfectly equidistant cross. The midpoints for the ends and for the lines are the same, resulting in no main effects.
Main Effect of IV1 and Interaction: Lines are not parallel, and the average of the left endpoints is different from the average of the right endpoints.
Main Effect of IV2 and Interaction: Lines are not parallel, and the midpoint of one line is significantly higher than the midpoint of the other.
Everything Significant: Lines are not parallel (interaction), the horizontal averages differ (IV1 main effect), and the overall vertical averages for the two lines differ (IV2 main effect).

Ethics and Rules for Graphing for Assignments

Categorical Variables: If an independent variable is categorical (e.g., Ketchup vs. Chocolate), it MUST be plotted using a bar graph. Points are deducted (four points) for using a line graph for categorical data.
Trend Lines: Trend lines may be added over bar graphs to help visualize the interaction/slopes, but the bars must remain.
Axes: The Y-axis always represents the dependent variable (e.g., Enjoyment Rating). The X-axis represents IV1, and different colors/bars represent IV2.

Hypothesis Writing for Factorial Designs

Structure: Hypotheses must cover the main effect of IV1, the main effect of IV2, and the interaction between them.
Main Effect Hypothesis Format: "If the theory that [Rationale/Theory] is true, then we should obtain a significant main effect of [IV] such that [DV] will be significantly higher when [Level A] than [Level B], regardless of the level of [Other IV]."
Interaction Hypothesis Paragraph: A complete interaction hypothesis must state that a significant interaction is expected AND describe all four simple effects (comparing relevant rows and columns).
- Example: "I hypothesize a significant interaction between condiments and food. Specifically, when given Ketchup, we predict enjoyment will be significantly higher for hot dogs than ice cream. When given Chocolate Sauce, we predict enjoyment will be significantly higher for ice cream than hot dogs. Given hot dogs, Ketchup will be higher than Chocolate Sauce. Given ice cream, Chocolate Sauce will be higher than Ketchup."

Numerical Calculation of Effects: The Forensic Study Scenario

Scenario Background: Forensic psychologist Dr. Webb studied recall ability based on two variables: Race of the Perpetrator (Same vs. Different) and Type of Crime (Violent vs. Nonviolent).
Data Set:
- Nonviolent Crime + Same Race: $50$
- Nonviolent Crime + Different Race: $40$
- Violent Crime + Same Race: $30$
- Violent Crime + Different Race: $20$
Significance Threshold: Differences $\ge 10$ points are considered significant.
Marginal Means (for Main Effects):
- Main Effect of Race: Average of Same Race $(50 + 30) / 2 = 40$ vs. Average of Different Race $(40 + 20) / 2 = 30$ . Difference is $|40 - 30| = 10$ . (Significant).
- Main Effect of Crime: Average of Nonviolent $(50 + 40) / 2 = 45$ vs. Average of Violent $(30 + 20) / 2 = 25$ . Difference is $|45 - 25| = 20$ . (Significant).
Interaction Calculation (Difference of Differences):
- Difference 1 (Col 1: Nonviolent - Violent for Same Race): $50 - 30 = 20$
- Difference 2 (Col 2: Nonviolent - Violent for Different Race): $40 - 20 = 20$
- Final Calculation: $|20 - 20| = 0$ . (Not significant; lines are parallel).

Simple Effect Analysis (Forensic Scenario)

Simple Effect 1: Effect of crime type at Same Race: $50$ (Nonviolent) vs. $30$ (Violent). Difference = $20$ (Significant).
Simple Effect 2: Effect of crime type at Different Race: $40$ (Nonviolent) vs. $20$ (Violent). Difference = $20$ (Significant).
Simple Effect 3: Effect of race at Nonviolent crime: $50$ (Same) vs. $40$ (Different). Difference = $10$ (Significant).
Simple Effect 4: Effect of race at Violent crime: $30$ (Same) vs. $20$ (Different). Difference = $10$ (Significant).

Questions & Discussion

Question: What are the numbers 50, 30, 40, and 20 measuring in the Dr. Webb study?
Answer: These are cell means representing the dependent variable, which is the recall score (number of questions answered correctly on a scale related to the crime viewing).
Question: Does the direction of the interaction matter?
Answer: Interaction is calculated as the difference between differences. It does not matter if you subtract rows first or columns first; the absolute resulting value will be the same.
Question: How should the hypothesis be formatted?
Answer: It should use an "If [Theory], then [Prediction]" format and must include the term "significant" to distinguish findings from background noise.