🥖Linear Modeling Theory Unit 11 – Two-Way ANOVA & Interaction Effects

Key Concepts and Definitions

• Two-Way ANOVA analyzes the effect of two independent categorical variables on a continuous dependent variable
• Main effect represents the effect of each independent variable on the dependent variable, ignoring the other independent variable
• Interaction effect occurs when the effect of one independent variable on the dependent variable changes depending on the level of the other independent variable
• Between-subjects design involves different participants in each group or condition
• Within-subjects design (repeated measures) involves the same participants tested under different conditions
  • Requires additional assumptions, such as sphericity
• Factorial design includes all possible combinations of the levels of the independent variables
• Balanced design has an equal number of observations in each cell of the design

Two-Way ANOVA Basics

• Extends One-Way ANOVA by incorporating a second independent variable
• Allows for the examination of both main effects and interaction effects
• Can be used with both between-subjects and within-subjects designs
• Requires a continuous dependent variable and two categorical independent variables
  • Independent variables can have two or more levels
• Partitions the total variance into components attributable to each main effect, the interaction effect, and error
• F-tests are used to assess the significance of main effects and the interaction effect
• Null hypothesis states that there are no differences between group means for each main effect and the interaction effect

Main Effects vs. Interaction Effects

• Main effects represent the overall effect of each independent variable on the dependent variable
  • Calculated by comparing the means of the levels of one independent variable, collapsing across the levels of the other independent variable
• Interaction effects indicate that the effect of one independent variable on the dependent variable depends on the level of the other independent variable
  • Occurs when the differences between the levels of one independent variable are not consistent across the levels of the other independent variable
• Presence of a significant interaction effect can qualify or modify the interpretation of main effects
• Main effects should be interpreted with caution when a significant interaction effect is present
• Simple main effects can be examined to understand the nature of the interaction
  • Involves comparing the levels of one independent variable at each level of the other independent variable

Assumptions and Prerequisites

• Independence of observations assumes that the scores in each cell are independent of each other
• Normality assumes that the dependent variable is normally distributed within each cell of the design
  • Assessed using histograms, Q-Q plots, or statistical tests (Shapiro-Wilk, Kolmogorov-Smirnov)
• Homogeneity of variances assumes that the variance of the dependent variable is equal across all cells
  • Assessed using Levene's test or Brown-Forsythe test
• Absence of outliers assumes that there are no extreme scores that could unduly influence the results
  • Identified using boxplots, z-scores, or Mahalanobis distance
• Adequate sample size ensures sufficient power to detect significant effects
  • Rule of thumb: at least 20 observations per cell
• Measurement of the dependent variable on an interval or ratio scale

Conducting a Two-Way ANOVA

• Specify the research question and hypotheses
• Identify the independent variables and their levels
• Determine the design (between-subjects, within-subjects, or mixed)
• Collect data and check for assumptions
• Calculate descriptive statistics (means, standard deviations) for each cell
• Perform the Two-Way ANOVA using statistical software (SPSS, R, Python)
  • Specify the model, including main effects and the interaction effect
• Examine the ANOVA table for significant effects
  • Main effects and interaction effect are significant if p < alpha (usually .05)
• If significant effects are found, conduct post-hoc tests or simple main effects analyses to determine the nature of the differences
• Report the results, including F-values, degrees of freedom, p-values, and effect sizes

Interpreting Results and Effect Sizes

• A significant main effect indicates that the levels of the independent variable differ in their effect on the dependent variable
  • Interpret the direction of the differences based on the means
• A significant interaction effect indicates that the effect of one independent variable on the dependent variable depends on the level of the other independent variable
  • Examine the pattern of means to understand the nature of the interaction
• Effect sizes quantify the magnitude of the differences or the strength of the relationship
  • Partial eta-squared ($\eta^2_p$) is commonly used for Two-Way ANOVA
    • Interpretation guidelines: small (.01), medium (.06), large (.14)
  • Cohen's f is another effect size measure for ANOVA
    • Interpretation guidelines: small (.10), medium (.25), large (.40)
• Confidence intervals provide a range of plausible values for the population parameters
• Be cautious when interpreting non-significant results, as they may be due to insufficient power or small sample sizes

Visualizing Interactions

• Interaction plots display the means of the dependent variable for each combination of the levels of the independent variables
  • Also known as line plots or profile plots
• Parallel lines indicate no interaction effect, while non-parallel lines suggest an interaction effect
• The degree of non-parallelism reflects the strength of the interaction effect
• Crossing lines indicate a disordinal (crossover) interaction, where the rank order of the levels of one independent variable changes across the levels of the other independent variable
• Diverging or converging lines indicate an ordinal interaction, where the rank order remains the same, but the magnitude of the differences varies
• Interaction plots help in understanding the nature and direction of the interaction effect
• Can be created using statistical software or spreadsheet programs (Excel)

Real-World Applications and Examples

• Education: Examining the effects of teaching method and student background on academic performance
  • Independent variables: teaching method (traditional vs. innovative), student background (low vs. high socioeconomic status)
  • Dependent variable: test scores
• Psychology: Investigating the effects of therapy type and patient characteristics on treatment outcomes
  • Independent variables: therapy type (cognitive-behavioral vs. psychodynamic), patient characteristics (introverted vs. extraverted)
  • Dependent variable: symptom reduction
• Marketing: Assessing the impact of product packaging and price on consumer preferences
  • Independent variables: packaging design (attractive vs. unattractive), price (low vs. high)
  • Dependent variable: purchase intention
• Healthcare: Evaluating the effectiveness of different treatments and patient factors on recovery time
  • Independent variables: treatment (medication vs. physical therapy), patient age (young vs. old)
  • Dependent variable: days to recovery