Multivariate Data Analysis Chapter 4 Multiple Regression What

Multivariate Data Analysis Chapter 4 – Multiple Regression

What is Multiple Regression Analysis? n n n An Example of Simple and Multiple Regression Setting a Baseline: Prediction Without an Independent Variable Prediction Using A Single Independent Variable – Simple Regression n n The Role of the Correlation Coefficient Specifying the Simple Regression Equation Establishing a Confidence Interval for the Prediction Assessing Prediction Accuracy

What is Multiple Regression Analysis? n Prediction Using Several Independent Variables – Multiple Regression n n The Impact of Multicollinearity The Multiple Regression Equation Adding a Third Independent Variable Summary: simple and straightforward dependence technique

A Decision Process for Multiple Regression Analysis n Stage 1: Objectives of Multiple Regression n Research Problems Appropriate for Multiple Regression n n Prediction with Multiple Regression Explanation with Multiple Regression Specifying a Statistical Relationship Selection of Dependent and Independent Variables

A Decision Process for Multiple Regression Analysis (Cont. ) n Stage 2: Research Design of a Multiple Regression Analysis n Sample Size n n Statistical Power and Sample Size Generalizability and Sample Size Fixed Versus Random Effects Predictors Creating Additional Variables n n n Incorporating Nonmetric Data with Dummy Variables Representing Curvilinear Effects with Polynomials Representing Interaction or Moderator Effects

A Decision Process for Multiple Regression Analysis (Cont. ) n Stage 3: Assumptions in Multiple Regression Analysis n n n Assessing Individual Variables Versus the Variate Linearity of the Phenomenon Constant Variance of the Error Term Independence of the Error Terms Normality of the Error Term Distribution

A Decision Process for Multiple Regression Analysis (Cont. ) n Stage 4: Estimating the Regression Model and Assessing Overall Fit n General Approaches to Variables Selection n n Confirmatory Specification Sequential Search Methods Combinational Approach Overview of the Model Selection Approaches Testing the Regression Variate for Meeting the Regression Assumptions

A Decision Process for Multiple Regression Analysis (Cont. ) n Stage 4: Estimating the Regression Model and Assessing Overall Fit (Cont. ) n Examining the Statistical Significance of Our Model n n n Significance of the Overall Model: The Coefficient of Determination Significance Tests of Regression Coefficients Identifying Influential Observations

A Decision Process for Multiple Regression Analysis (Cont. ) n Stage 5: Interpreting the Regression Variate n n n Using the Regression Coefficients Standardizing the Regression Coefficients: Beta Coefficients Assessing Multicollinearity n n n The Effect of Multicollinearity Identifying Multicollinearity Remedies for Multicollinearity

A Decision Process for Multiple Regression Analysis (Cont. ) n Stage 6: Validation of the Results n n Additional or Split Samples Calculating the PRESS Statistics Comparing Regression Models Predicting with the Model

Chapter 4 A n Assessing Multicollinearity n n A Two-Part Process An Illustration of Assessing Multicollinearity

Chapter 4 a: Identifying Influential Observations n Step 1: Examining Residuals n n n Analysis of Residuals Partial regression plots Step 2: Identifying Leverage Points from the Predictors n n Hat Matrix Mahalanobis distance (D^2)

Identifying Influential Observations (Cont. ) n Step 3: Single-Case Diagnostics Identifying Influential Observations n n n Influences on individual coefficients Overall influence measures Step 4: Selecting and Accommodating Influential Observations