Week 10B Multiple Regression - Tagged

Course Overview

• Course: PY2501 Research Methods & Data Analysis

• University: Aston University, Birmingham, UK

• Lecturer: Dr. Ryan DiLinn

• Week 10 Lecture: Focus on Multiple Regression

Lecture Breakdown

• Part 1: Recap of Simple Regression

• Part 2: Introduction to Multiple Regression

  • Definition and Purpose

  • Reporting & Interpreting Results

• Part 3: Assumptions of Multiple Regression

  • Linearity, Normality, Homoscedasticity, No Outliers, No Multicollinearity

• Part 4: Regression Formula Review

  • Mathematical Representation of Regression

Recap of Simple Regression

• Definition: Explores the relationship between two continuous variables.

  • Examples:

    • Exam scores vs. Exam anxiety

    • Running distance vs. Sweat produced

    • Album sales vs. Advertising budget

• Goal: Establish a cause and effect relationship where X predicts Y.

• Method: Fitting a line to data to predict Y from X.

  • Example Prediction: If Exam Anxiety = 39, then Y (Exam Performance) is predicted to be 80%.

Key Terms in Regression

• Beta Estimate: Indicates the slope of the regression line.

• t-Statistic: Measures the reliability of the predictor.

• p-value: Tests if the predictor is significant (often p<0.05).

• F-Statistic: Tests the overall significance of the regression model.

• R2 Value: Coefficient of determination, indicating the proportion of variance explained by the model.

  • Adjusted R2: Adjusted for the number of predictors in the model.

Quick Quiz Highlights

• Difference Between Correlation & Regression: (A) Regression predicts Y from X.

• Beta Value Interpretation: (G) A change in X influences Y’s changes.

• Interpretation of Regression Results: Use practical examples such as advertising budget predicting Spotify subscriptions.

Multiple Regression Explained

• Expansion of Simple Regression: Used when more than one independent variable (IV) predicts a dependent variable (DV).

  • General Formula: Y = b0 + b1X1 + b2X2 + ... + bnXn

  • Example: Y = Album sales = b0 + (b1 * Advertising Budget) + (b2 * Airplay).

• How it Works: Each IV receives its regression line indicating its influence on Y.

Assumptions of Multiple Regression

1. Linearity: Relationship must be linear.

2. Normality of Residuals: Deviations from the regression line should be normally distributed.

3. Homoscedasticity: Residuals should have constant variance across levels of IV.

4. No Outliers: No data points should significantly deviate from the main data trend.

5. No Multicollinearity: IVs should not be perfectly correlated; VIF score helps assess this.

Addressing Violated Assumptions

• Pre-Experiment: Design experiments with adequate sample sizes (N > 50).

• Post-Experiment: Potentially remove outliers or transform data.

Advanced Regression Formula

• Formula: Yi = b0 + b1Xi

  • Interpretations:

    • b0: Y-intercept

    • b1: Slope indicating change in Y for a unit change in X.

• Example Application: Plugging values into the regression formula to predict album sales.

Approaches to Multiple Regression

• Forced Entry: All predictors added at once.

• Hierarchical Entry: Predefined IVs added first, followed by new predictors.

• Stepwise Entry: Only significant IVs are retained based on correlation with the DV.

Conclusion & Report Requirements

• Reporting Guidelines:

  • Report b-values, t-statistics (& p-values) for individual predictors.

  • Report the F-statistic (& its p-value) for collective prediction significance.

  • Report R-squared to convey variance explained by the model.

Additional Resources

• Contact: Dr. Ryan DiLinn via email r.blything@aston.ac.uk

• Assignment Guidance: Check the Blackboard for specifics on submissions.

• Quiz Updates: Practicals on questionnaires and multiple regression will be assessed in upcoming quizzes.