10.2 Multiple linear regression

Multiple linear regression expands on simple linear regression by including multiple predictors. It's like juggling several balls instead of just one. The model aims to find the best relationship between a response variable and multiple explanatory variables, estimating how each predictor impacts the outcome.

Interpreting the results involves examining coefficients, significance tests, and goodness of fit measures. It's like decoding a puzzle, where each piece reveals something about the relationships between variables. Understanding these elements helps assess the model's reliability and predictive power.

Multiple Linear Regression

Extension of Simple Linear Regression

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• Multiple linear regression incorporates multiple explanatory variables (predictors) into the model, extending the concepts of simple linear regression
• The general form of a multiple linear regression model is $Y = β₀ + β₁X₁ + β₂X₂ + ... + βₚXₚ + ε$, where:
  • $Y$ is the response variable
  • $X₁, X₂, ..., Xₚ$ are the explanatory variables
  • $β₀, β₁, β₂, ..., βₚ$ are the regression coefficients
  • $ε$ is the random error term
• The goal is to find the best-fitting linear relationship between the response variable and the explanatory variables by minimizing the sum of squared residuals
• The least squares method estimates the regression coefficients, similar to simple linear regression

Interpretation of Regression Coefficients

• Each regression coefficient represents the change in the response variable for a one-unit change in the corresponding explanatory variable, holding all other explanatory variables constant
• The intercept ($β₀$) represents the expected value of the response variable when all explanatory variables are equal to zero
• Example: In a multiple linear regression model predicting house prices based on square footage and number of bedrooms, the coefficient for square footage represents the change in house price for a one-unit increase in square footage, keeping the number of bedrooms constant

Interpreting Coefficients

Significance Testing

• The significance of regression coefficients is assessed using hypothesis tests (t-tests) and p-values
• A low p-value (typically < 0.05) indicates that the corresponding explanatory variable has a significant impact on the response variable
• A high p-value suggests that the variable may not be important in the model
• Example: If the p-value for the coefficient of the number of bedrooms is 0.02, it suggests that the number of bedrooms has a significant impact on house prices

Confidence Intervals

• Confidence intervals provide a range of plausible values for the true regression coefficients
• They indicate the uncertainty associated with the estimated coefficients
• Example: A 95% confidence interval for the coefficient of square footage might be (50, 100), suggesting that the true change in house price for a one-unit increase in square footage is likely between $50 and$100, with 95% confidence

Model Fit and Prediction

Goodness of Fit Measures

• The coefficient of determination ($[R](https://www.fiveableKeyTerm:r)²$) measures the proportion of variance in the response variable explained by the explanatory variables
  • A higher $R²$ indicates a better fit of the model to the data
• The adjusted $R²$ penalizes the addition of irrelevant variables, providing a more conservative measure of the model's goodness of fit
• The F-test assesses the overall significance of the multiple linear regression model
  • It tests the null hypothesis that all regression coefficients (except the intercept) are equal to zero
  • A low p-value for the F-test indicates that at least one of the explanatory variables has a significant impact on the response variable

Predictive Power Assessment

• Residual analysis assesses the assumptions of multiple linear regression (linearity, homoscedasticity, normality of residuals, independence of errors)
  • Diagnostic plots, such as residual plots and Q-Q plots, help identify violations of these assumptions
• Cross-validation techniques (k-fold cross-validation, leave-one-out cross-validation) assess the predictive power of the model on unseen data and detect overfitting
• Example: Using 5-fold cross-validation, the model's performance is evaluated on five different subsets of the data, providing an estimate of its predictive accuracy on new data

Issues in Multiple Regression

Multicollinearity

• Multicollinearity occurs when there is a high correlation among the explanatory variables, leading to unstable and unreliable estimates of the regression coefficients
• Symptoms of multicollinearity:
  • Large standard errors for the regression coefficients
  • Coefficients with unexpected signs or magnitudes
  • High pairwise correlations among the explanatory variables
• Variance Inflation Factors (VIFs) quantify the severity of multicollinearity for each explanatory variable
  • A VIF greater than 5 or 10 is often considered indicative of problematic multicollinearity
• Addressing multicollinearity:
  • Remove one or more of the correlated explanatory variables
  • Combine the correlated variables into a single variable
  • Use regularization techniques (ridge regression, lasso regression)

Model Selection

• Model selection involves choosing the best subset of explanatory variables to include in the multiple linear regression model
• Criteria for model selection:
  • Goodness of fit
  • Predictive power
  • Model complexity
• Stepwise selection methods (forward selection, backward elimination, stepwise regression) iteratively add or remove variables based on their statistical significance or contribution to the model's fit
• Information criteria (Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC)) compare and select among different models while balancing goodness of fit and model complexity
• Example: Using forward selection, variables are added one at a time to the model based on their contribution to the model's fit, until no further improvement is observed