Multivariate Analysis

7.

Multiple Linear Regression

7.1.

Regression Model Framework

7.1.1.

Simple vs. Multiple Regression

7.1.2.

Model Specification

7.1.3.

7.1.3.1.

Dependent Variables

7.1.3.2.

Independent Variables

7.1.3.3.

Continuous Variables

7.1.3.4.

Categorical Variables

7.2.

Model Assumptions

7.2.1.

7.2.2.

Independence of Errors

7.2.3.

Homoscedasticity

7.2.4.

Normality of Errors

7.2.5.

No Perfect Multicollinearity

7.2.6.

Fixed X Assumption

7.3.

Parameter Estimation

7.3.1.

Ordinary Least Squares Method

7.3.2.

Matrix Formulation

7.3.3.

Normal Equations

7.3.4.

Geometric Interpretation

7.4.

Model Evaluation

7.4.1.

Coefficient of Determination

7.4.1.1.

7.4.1.2.

Adjusted R-squared

7.4.1.3.

Predicted R-squared

7.4.2.

Standard Error of Estimate

7.4.3.

Analysis of Variance

7.4.3.1.

7.4.3.2.

F-Test for Overall Significance

7.4.3.3.

Sum of Squares Decomposition

7.5.

Coefficient Interpretation

7.5.1.

Unstandardized Coefficients

7.5.2.

Standardized Coefficients (Beta)

7.5.3.

Partial Correlation Coefficients

7.5.4.

Semi-Partial Correlations

7.6.

Hypothesis Testing

7.6.1.

Individual Coefficient Tests

7.6.1.1.

7.6.1.2.

Confidence Intervals

7.6.2.

Joint Hypothesis Tests

7.6.2.1.

F-Tests for Subsets

7.7.

Regression Diagnostics

7.7.1.

Residual Analysis

7.7.1.1.

7.7.1.2.

7.7.1.3.

Normal Probability Plots

7.7.2.

Influence Diagnostics

7.7.2.1.

Leverage Values

7.7.2.2.

Cook's Distance

7.7.2.3.

7.7.2.4.

7.7.3.

Multicollinearity Diagnostics

7.7.3.1.

Variance Inflation Factor

7.7.3.2.

7.7.3.3.

Condition Indices

7.7.3.4.

Eigenvalue Analysis

7.8.

Variable Selection

7.8.1.

Forward Selection

7.8.2.

Backward Elimination

7.8.3.

Stepwise Regression

7.8.4.

All Possible Regressions

7.8.5.

Information Criteria

7.8.5.1.

7.8.5.2.

7.8.5.3.

7.9.

Model Validation

7.9.1.

Cross-Validation

7.9.2.

Split-Sample Validation

7.9.3.

Bootstrap Methods

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8. Multivariate Analysis of Variance