Moderated Multiple Regression

Announcements and Course Progress

• Tutorials and Practical Testing:   - Tutorials are continuing this week with a focus on Practical Test 6.   - Practical Test 6 involves Jamovi interpretation for Hierarchical Multiple Regression (HMR) and conducting moderated multiple regression in Jamovi (Part I).   - Practical Test 5 marks were released to Learn.UQ on Monday evening, including general feedback.   - Practical Test 7 will be administered in Week 11 tutorials. This is a 15-minute paper test requiring a pen/pencil/eraser. It covers moderated multiple regression, calculating mean-centered scores, interaction terms (bring a calculator), and describing relationships from Jamovi output and figures. Mock tests will be released by the end of the week.

• Assignments:   - Assignment 1 (Factorial ANOVA) marks were released Monday evening.   - Assignment 2 (Multiple Regression) will be released Friday evening. Students are encouraged to map the assignment onto current class learning.

• Class Study:   - PSYC3010 Class Study II is available at https://tinyurl.com/PSYC3010S2. Students should complete this for the correlation/regression block.

• Featured Pets:   - The week's featured pets include: Skipper, Coco, Tato, and Poochie.

Review of Hierarchical Multiple Regression (HMR)

• Definition: Multiple regression where predictors are entered into the model sequentially in a pre-specified order based on logic or theoretical grounds.
• Use in Interactions: HMR is used to test for interactions between predictors by:   1. Adding predictors at Step 1.   2. Adding the interaction term at Step 2.

Introduction to Interactions in Multiple Regression

• Interaction Defined: A situation where a particular effect or relationship between variables changes as a function of another variable.
• Purpose: Interactions add nuance to established effects by indicating boundary conditions—the conditions under which an effect is likely to be stronger, weaker, non-existent, or reversed.
• Moderated Multiple Regression (MMR): This is the specific method used to test for interactions in the context of regression. Moderation is said to occur when there is a significant interaction between two predictors.

Comparison: Two-Way Factorial ANOVA vs. Moderated Multiple Regression

• Conceptual Similarity: Both examine if the effect of one variable on a dependent variable/criterion changes as a function of a second variable.
• Factorial ANOVA:   - Examines whether the effect of Factor $A$ on the DV changes based on Factor $B$.   - Uses categorical variables (groups/conditions).   - ANOVA does not function with continuous variables.
• Moderated Multiple Regression:   - Examines whether the relationship between a focal predictor ($X$) and a criterion ($Y$) changes as a function of a moderator ($W$).   - Predictors ($X$ and $W$) can be continuous or categorical.

The Moderated Multiple Regression Framework

• Predictor Designations: In MMR, we typically designate one predictor as the Focal Predictor ($X$) and the other as the Moderator ($W$).
• Statistical Components:   - The model includes the additive relationships (direct effects) and the interaction term ($XW$) to see if the latter contributes to predicting the criterion ($Y$).
• Three Key Tests in MMR:   1. Direct effect of $X$: Does $X$ significantly predict $Y$? (Relationship between $X$ and $Y$).   2. Direct effect of $W$: Does $W$ significantly predict $Y$? (Relationship between $W$ and $Y$).   3. Interaction between $X$ and $W$: Does the $XW$ interaction significantly predict $Y$? Does the $X$-$Y$ relationship change at different levels of $W$?
• Independence of Tests: These three tests are conceptually distinct. A significant interaction cannot be inferred from significant direct effects, or vice versa.

The MMR Linear Model

• Additive Effects (No Interaction):$\hat{Y} = b_X X + b_W W + a$
• Interactive Effects (Additive + Interaction):$\hat{Y} = b_X X + b_W W + b_{XW} XW + a$
• Standardized Model:$\hat{z}_{\hat{Y}} = \beta_X z_X + \beta_W z_W + \beta_{XW} z_{XW}$

Analysis Plan for Moderated Multiple Regression

• Standard Procedure:   1. Use Hierarchical Multiple Regression.   2. Step 1: Enter only the predictors ($X$ and $W$) to test additive "direct effects."   3. Step 2: Enter the interaction term ($XW$) to see if it significantly increases the variance accounted for in $Y$. This is tested using the significance of $R^2$ change ($F_{\text{change}}$).   4. Post-hoc: If the interaction is significant, interpret it using simple slopes analysis.

Visualization: Plots and Slopes

• 3D Visualization: While interactions can be shown in 3D (where the relationship between $X$ and $Y$ varies across the plane of $W$), these are difficult to report.
• 2D Customary Plots:   - The y-axis is the criterion ($Y$).   - The x-axis is the focal predictor ($X$).   - The moderator ($W$) is represented by separate lines (typically representing "low" and "high" levels).
• Rules for Interpretation:   - If the lines are parallel, there is no interaction.   - If the lines have different slopes, it indicates an interaction (e.g., $X$ is positively associated with $Y$ when $W$ is low, but negatively associated when $W$ is high).

Procedural Example: Relationship Satisfaction

• Scenario: A study of $N=30$ people examining relationship satisfaction.
• Variables:   - Focal Predictor ($X$): External Stressors ($1 = \text{none}$, $10 = \text{many}$).   - Moderator ($W$): Communication Quality ($1 = \text{very ineffective}$, $10 = \text{very effective}$).   - Criterion ($Y$): Relationship Satisfaction ($1 = \text{very low}$, $10 = \text{very high}$).
• Hypotheses:   1. External stressors will uniquely negatively predict satisfaction (controlling for communication).   2. Communication quality will uniquely positively predict satisfaction (controlling for stressors).   3. Communication quality will moderate the stressors-satisfaction relationship, attenuating (weakening) it at higher levels of communication quality.

Step-by-Step implementation in Jamovi

Step 1: Mean-Centering and Calculating Interaction

• The Procedure:   1. Mean-center $X$ and $W$ by subtracting the mean of each variable from individual scores.   2. Multiply the mean-centered scores to create the interaction term ($XW$).
• Why Mean-Center?   - Reduces Multicollinearity: Predictors and their interaction term are often highly intercorrelated; centering reduces this.   - Interpretability: A score of $0$ on a centered variable represents the average. Positive scores are above average, negative scores are below average.
• Jamovi Commands:   - c_Stressors = External Stressors - VMEAN(External Stressors)   - c_Communication = Communication Quality - VMEAN(Communication Quality)   - c_Interaction = c_Stressors * c_Communication

Step 2: Testing Significance

• Overall Model Statistics (Step 1):   - External stressors and communication explain $82\%$ of variance.   - $F(2, 27) = 62.91, p < .001$.
• Change Statistics (Step 2):   - Adding the interaction explains an additional $8\%$ of variance.   - $F_{\text{change}}(1, 26) = 21.15, p < .001$. This confirms a significant interaction.
• Step 2 Overall Statistics:   - Full model explains $90\%$ of variance ($R^2 = .90$).   - $F(3, 26) = 80.28, p < .001$.
• Individual Predictor Statistics (Step 1):   - Stressors: $\beta = -0.46, 95\%\,CI\,[-.62, -.29], t(27) = -5.64, p < .001$. Uniquely accounts for $21\%$ of variance ($sr^2 = .21$).   - Communication: $\beta = 0.74, 95\%\,CI\,[.58, .91], t(27) = 9.17, p < .001$. Uniquely accounts for $55\%$ of variance ($sr^2 = .55$).
• Individual Predictor Statistics (Step 2):   - Focus on the interaction term: $\beta = 0.31, 95\%\,CI\,[.17, .45], t(26) = 4.60, p < .001$. Uniquely accounts for $8\%$ of variance ($sr^2 = .08$).
• Regression Equations:   - Unstandardized: $\hat{Y} = -0.26X_{\text{centered}} + 0.48W_{\text{centered}} + 0.08XW + 6.06$   - Standardized: $\hat{z}_{\hat{Y}} = -0.33 z_X + 0.72 z_W + 0.31 z_{XW}$

Step 3: Simple Slopes Analysis

• Definition: Testing the relationship between $X$ and $Y$ at specific "low" ($-1\,SD$) and "high" ($+1\,SD$) levels of the moderator ($W$).
• Procedural Steps:   1. Create two new moderator variables: $W_{\text{low}} = W_{\text{centered}} - (-1\,SD)$ and $W_{\text{high}} = W_{\text{centered}} - (1\,SD)$.   2. Create two new interaction terms representing the interaction at these low and high levels ($XW_{\text{low}}$ and $XW_{\text{high}}$).   3. Run SMR twice (once per level).
• Jamovi Simple Slopes Results:   - Low Communication Quality: External stressors is a significant negative predictor ($\beta = -0.66, t(26) = -8.74, p < .001$).   - High Communication Quality: External stressors is not a significant predictor ($\beta = 0.00, t(26) = 0.01, p = .996$).
• Conclusion: Hypothesis 3 is supported; communication quality moderates the relationship by attenuating the negative effect of stressors at high levels.

Step 4: Plotting the Interaction

• Graphing via Simple Regression Equations:   - Low Comm ($W$): $\hat{Y} = -0.52X_{\text{centered}} + 4.57$   - High Comm ($W$): $\hat{Y} = 0.00X_{\text{centered}} + 7.54$
• Calculating Coordinates: Substitute $-1\,SD$ and $+1\,SD$ of $X$ ($X = \pm 2.61$) into equations:   - Low Comm Line: Low Stressor ($\hat{Y} = 5.93$), High Stressor ($\hat{Y} = 3.21$).   - High Comm Line: Low Stressor ($\hat{Y} = 7.54$), High Stressor ($\hat{Y} = 7.54$).
• Visual Result: The line for high communication quality is horizontal (no relationship), while the line for low communication quality has a steep negative slope.

Summary of Findings

• Hypothesis 1: Supported (Direct effect of stressors).
• Hypothesis 2: Supported (Direct effect of communication).
• Hypothesis 3: Supported (Moderation and attenuation).
• Next Topic: Mediation and Indirect Effects.