Biostatistics

13.

Multiple Regression Analysis

13.1.

Multiple Linear Regression Model

13.1.1.

Model Specification

13.1.1.1.

Multiple Predictors

13.1.1.2.

Partial Regression Coefficients

13.1.1.3.

13.1.2.

Interpretation of Coefficients

13.1.2.1.

Holding Other Variables Constant

13.1.2.2.

Standardized vs Unstandardized

13.1.3.

13.1.3.1.

Least Squares Estimation

13.1.3.2.

Matrix Formulation

13.1.4.

Model Assessment

13.1.4.1.

Multiple R-Squared

13.1.4.2.

Adjusted R-Squared

13.1.4.3.

F-Test for Overall Model

13.2.

Variable Selection

13.2.1.

Forward Selection

13.2.1.1.

Procedure and Criteria

13.2.1.2.

Advantages and Limitations

13.2.2.

Backward Elimination

13.2.2.1.

Procedure and Criteria

13.2.2.2.

Advantages and Limitations

13.2.3.

Stepwise Selection

13.2.3.1.

Bidirectional Approach

13.2.3.2.

Stopping Criteria

13.2.4.

Best Subsets Regression

13.2.4.1.

All Possible Models

13.2.4.2.

Selection Criteria

13.2.5.

Model Selection Criteria

13.2.5.1.

Akaike Information Criterion (AIC)

13.2.5.2.

Bayesian Information Criterion (BIC)

13.2.5.3.

13.3.

Multicollinearity

13.3.1.

Definition and Detection

13.3.1.1.

Correlation Matrix

13.3.1.2.

Variance Inflation Factor (VIF)

13.3.1.3.

13.3.2.

Consequences of Multicollinearity

13.3.2.1.

Unstable Coefficients

13.3.2.2.

Inflated Standard Errors

13.3.3.

Remedial Measures

13.3.3.1.

Variable Removal

13.3.3.2.

Ridge Regression

13.3.3.3.

Principal Components

13.4.

Model Diagnostics

13.4.1.

Residual Analysis

13.4.1.1.

Standardized Residuals

13.4.1.2.

Studentized Residuals

13.4.1.3.

13.4.2.

Influence Diagnostics

13.4.2.1.

Leverage Values

13.4.2.2.

Cook's Distance

13.4.2.3.

13.4.3.

Assumption Checking

13.4.3.1.

Linearity Assessment

13.4.3.2.

Homoscedasticity Testing

13.4.3.3.

Normality of Residuals

13.4.3.4.

Independence Verification

13.5.

Interaction and Polynomial Terms

13.5.1.

Interaction Effects

13.5.1.1.

Definition and Interpretation

13.5.1.2.

Testing for Interactions

13.5.1.3.

Centering Variables

13.5.2.

Polynomial Regression

13.5.2.1.

Quadratic and Higher-Order Terms

13.5.2.2.

Interpretation Challenges

13.5.2.3.

Overfitting Concerns

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