Linear Regression

Category: Classical Machine Learning

Overview

Linear regression establishes a mathematical relationship between predictor variables and a continuous outcome variable. The algorithm identifies the optimal linear combination of input features that best predicts the target variable by minimizing the sum of squared residuals between predicted and observed values.

Mathematical Foundation

Linear Model Equation

y = β₀ + β₁x₁ + β₂x₂ + ... + βₚxₚ + ε

Where:

y = dependent variable (outcome)
β₀ = intercept (baseline value when all predictors = 0)
β₁, β₂, ..., βₚ = regression coefficients
x₁, x₂, ..., xₚ = independent variables (predictors)
ε = error term (residual)

Ordinary Least Squares (OLS)

The method minimizes the sum of squared residuals:

minimize: Σ(yᵢ - ŷᵢ)²

This optimization finds coefficient values that provide the best linear fit to the observed data.

Medical Applications

Clinical Research

Dose-response modeling for pharmaceutical studies
Biomarker relationship analysis in clinical trials
Treatment outcome prediction based on patient characteristics
Healthcare cost prediction models

Diagnostic Applications

Predicting laboratory values based on patient demographics
Modeling disease progression rates
Risk factor quantification for preventive medicine

Epidemiological Studies

Population health trend analysis
Environmental factor impact assessment
Disease prevalence modeling

Key Assumptions

1. Linearity

The relationship between independent and dependent variables must be linear. Non-linear relationships require transformation or alternative methods.

2. Independence

Observations must be independent of each other. Violations occur in:

Repeated measures data
Clustered sampling
Time series data

3. Homoscedasticity

Residual variance should be constant across all fitted values. Heteroscedasticity can be detected through:

Residual plots
Breusch-Pagan test
White test

4. Normality of Residuals

Residuals should follow a normal distribution, particularly important for:

Confidence intervals
Hypothesis testing
Prediction intervals

Clinical Considerations

Multicollinearity

Problem: Highly correlated predictors can make coefficient interpretation unreliable.

Solutions:

Variance Inflation Factor (VIF) assessment
Principal component analysis
Ridge or LASSO regression

Outliers and Influential Points

Detection Methods:

Cook's distance
Leverage values
Studentized residuals

Impact: Single outliers can significantly alter regression coefficients and predictions.

Sample Size Requirements

General Rule: Minimum 10-15 observations per predictor variable.

Considerations:

Effect size expectations
Desired statistical power
Expected R² value

Model Validation

Cross-validation for generalizability assessment
Hold-out validation sets
Bootstrap methods for confidence intervals

Performance Metrics

R² (Coefficient of Determination)

Range: 0 to 1
Interpretation: Proportion of variance explained by the model
Medical Context: Higher values indicate better predictive capability

Root Mean Square Error (RMSE)

Units: Same as dependent variable
Interpretation: Average prediction error magnitude
Clinical Relevance: Should be within acceptable clinical tolerance

F-statistic

Purpose: Tests overall model significance
Interpretation: Higher values with low p-values indicate significant predictive ability
Threshold: p < 0.05 for statistical significance

Adjusted R²

Advantage: Penalizes for additional predictors
Use Case: Model comparison with different numbers of variables

Limitations in Medical Context

Extrapolation Risks

Predictions outside the observed data range may be unreliable, particularly concerning for:

Extreme dosages
Unusual patient populations
Novel treatment combinations

Assumption Violations

Many medical datasets violate linear regression assumptions:

Non-linear dose-response relationships
Heteroscedastic error patterns
Non-normal distributions

Interpretation Challenges

Correlation vs. causation distinctions
Confounding variable effects
Clinical vs. statistical significance

Best Practices for Medical Applications

Model Development

Conduct thorough exploratory data analysis
Test all assumptions before model fitting
Consider clinical knowledge in variable selection
Validate findings with independent datasets

Reporting Standards

Report confidence intervals alongside point estimates
Discuss clinical significance of coefficients
Acknowledge model limitations
Provide goodness-of-fit statistics

Clinical Implementation

Validate in target population
Consider integration with clinical workflow
Monitor performance over time
Update models as new data becomes available