Lecture 7 - Factorial design & ANOVA

Page 1: Introduction

• Factorial Design & ANOVA

• PSYC 71, Week 7 Lecture

• Date: February 19th, Winter 2025

Page 2: Reminders

• Week 7:

  • Due: Stats HW #2 by Sunday at 11:59pm (available on Canvas)

  • Week 8:

    • Exam 2: First half of lecture (12:00 - 12:50pm)

    • Due: Stats HW #3 by Sunday at 11:59pm (released on Canvas one week prior)

  • Week 9:

    • Due: Poster (PDF version submitted through Canvas by Sunday at 11:59pm)

Page 3: Lecture Outline

• Topics:

  • Factorial design & notation

  • Between, within, and mixed designs

  • Main effects & interactions

  • Eyeballing effects

  • Factorial ANOVA model

  • Decomposing cell means into main effects and interactions

  • Tests of effects in the two-factor model

  • ANOVA output table for the two-factor model

Page 4: Factorial Design

• Focus on analyzing multiple independent variables.

Page 5: Experiment with 2 Independent Variables

• Example Questions:

  • Which is more delicious: hot food or cold food?

  • Which is more delicious: hamburgers or ice cream?

• Answer: It depends on various combinations of food types.

Page 6: Manipulating Multiple Factors

• Investigates the interaction effect of one independent variable with another.

• Factorial Design: Each level of one IV combined with each level of the others for all combinations.

Page 7: Factorial Design Definition

• Investigates the effect of two or more independent variables (factors) on the dependent variable.

• Factors: Price and style affecting wine rating.

• Example of levels: Cheap/Expensive and Sweet/Dry can result in different ratings.

Page 8: Factorial Notation

• Written like a multiplication expression: # x # x #

  • # of IVs = number of factors

  • # of levels = specific values of each factor

  • # of conditions = product of the factors

• Examples include:

  • 2x2 = 2 IVs, each with 2 levels = 4 conditions

  • 2x2x2 = 3 IVs, each with 2 levels = 8 conditions

  • 3x3 = 2 IVs, each with 3 levels = 9 conditions

Page 9: Peer Instruction #1

• Scenario: Researchers study the impact of stereotype threat on intelligence tests with 4 conditions.

• Notation for Design:

  • a) 2x2 factorial

  • b) 1x4 factorial

  • c) 3x3 factorial

  • d) None of the above

Page 10: Between, Within, & Mixed Factorials

• Definitions and examples of these designs will follow.

Page 11: Between- vs. Within-Subjects Factorial Design

• Between-Subjects Design: Each subject in just one condition.

• Within-Subjects Design: Each subject in ALL conditions.

• Mixed Design: Some factors between subjects, some within; subjects in more than one but not all conditions.

Page 12: Between-Subjects Example

• Different conditions assessed by different subjects:

  • Sweet, Cheap / Sweet, Expensive / Dry, Cheap / Dry, Expensive

Page 13: Within-Subjects Example

• Subjects participate in all conditions;

  • Order does not have to be counterbalanced, could be randomized.

Page 14: Mixed Design Example

• Subjects experience more than one condition but not all; example shown with price and style combinations.

Page 15: Additional Mixed Design Example

• More variety in how participants experience combinations of factors.

Page 16: Peer Instruction #2

• Scenario: Robert's study on age and emotion's effect on memory with conditional manipulation:

• Design type:

  • a) 2x2 between-subjects factorial

  • b) 2x2 within-subjects factorial

  • c) 2x2 mixed design factorial

  • d) None of the above

Page 17: Main Effects and Interactions

• Main effect - on average, are the levels of one factor different from each other?

• Simple effect - at a specific level of one factor, are the levels of the other factor different from each other?

• interaction - is the pattern/effect at one level of a factor different from the pattern/effect at another level of that factor?

Page 18: Kinds of Statistical Effects

• Main Effect: Average differences between levels of a factor.

• Simple Effect: Differences at a specific level of one factor.

• Interaction: Differences in patterns across levels of factors.

Page 19: Main Effects Calculation

• Example calculations shown using price and style, with marginal means detailed.

Page 20: Main Effects Continued

• Reiteration of the main effects of style across different types of wine ratings.

Page 21: Simple Effects Calculation

• Sample simple effects calculated for each type of wine helping illustrate interactions.

Page 22: Interactions Explained

• The dependence of effect of one factor on the level of another demonstrated through wine rating differences.

Page 23: Interactions & Effects

Page 24: No Interaction Visual

• Explanation of parallel lines indicating no interaction between factors.

Page 25: Peer Instruction #3

• Evaluation of potential statistical effects from provided data table.

Page 26: Peer Instruction #4

• Evaluation of effects shown in a figure based on quality ratings.

Page 27: Peer Instruction #5

• Reiteration of evaluating effects from another figure representation.

Page 28: More Practice

• Eyeballing Effects: Worksheet available on Canvas to aid practice.

Page 29: Factorial ANOVA Model

• Introduction to models related to factorial ANOVA.

Page 30: One Factor General Linear Model

• Equations demonstrating the general model structure for one factor.

Page 31: Two Factor Linear Model

• Description of how to formulate a two-factor linear model with examples.

Page 32: Decomposing Group Means

• Steps to analyze variance of component scores relative to residual variance after group mean decomposition.

Page 33: Testing Main Effect Offsets

• Concept of treatment offsets within routes and time when examining deviations.

Page 34: Estimating Interaction Offsets

• Detailed calculations illustrating how offsets derived from observed cell means reflect interaction laws.

Page 35: Checking the Math on Cell Means

• Confirming calculations around the grand mean of given data.

Page 36: Tests of Effects for Two Factors

• Overview of tests regarding main effects and interactions in factorial ANOVA.

Page 37: ANOVA Table With Two Factors

• Structure and specifics of how to populate an ANOVA table for two-factor designs.

Page 38: Peer Instruction #6

• Task to complete Annotated ANOVA table for the Times to Campus data, prompting calculation and estimation.