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Introductory Statisitcs

Proportional Hazards Assumption

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Introductory Statisitcs

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0 Prerequisites
0 Prerequisites
- Chapter Overview
- 0.1 Mathematical Background (Review)
  0.1 Mathematical Background (Review)
- 0.2 Computational Tools
  0.2 Computational Tools
- 0.3 Square Matrices
  0.3 Square Matrices
- 0.4 Linear Algebra and Statistics
  0.4 Linear Algebra and Statistics
- 0.5 Exercises
1 Data Collection
1 Data Collection
- Chapter Overview
- 1.1 Classical Approach—Design Your Data Collection
  1.1 Classical Approach—Design Your Data Collection
- 1.2 Modern Approach—Analyze the Data You Have
  1.2 Modern Approach—Analyze the Data You Have
  - Statistical Models vs Learning Algorithms
  - Prediction vs Inference
- 1.3 Three Learning Paradigms
  1.3 Three Learning Paradigms
- 1.4 Exercises
2 Descriptive Statistics
2 Descriptive Statistics
- Chapter Overview
- 2.1 Exploratory Data Analysis
  2.1 Exploratory Data Analysis
  - Histograms and Density Plots
  - Empirical Distribution and Quantile-Quantile Plots
- 2.2 Shape of Distributions
  2.2 Shape of Distributions
- 2.3 Numerical Summaries
  2.3 Numerical Summaries
- 2.4 Visualization
  2.4 Visualization
- 2.5 Code
  2.5 Code
- 2.6 Exercises
3 Foundations of Probability
3 Foundations of Probability
- Chapter Overview
- 3.1 Probability Theory
  3.1 Probability Theory
- 3.2 Independence
  3.2 Independence
  - Independence of Events
  - Conditional Independence
- 3.3 Random Variables
  3.3 Random Variables
- 3.4 Expectation and Moments
  3.4 Expectation and Moments
- 3.5 Limit Theorems
  3.5 Limit Theorems
- 3.6 Code
  3.6 Code
  - Central Limit Theorem Multi-Distribution Visualization
  - Gambler's Paradox (Law of Large Numbers Failure)
- 3.7 Exercises
4 Distributions
4 Distributions
- Chapter Overview
- 4.1 Discrete Distributions
  4.1 Discrete Distributions
- 4.2 Continuous Distributions
  4.2 Continuous Distributions
- 4.3 Multivariate Structure
  4.3 Multivariate Structure
- 4.4 Code
  4.4 Code
- 4.5 Exercises
5 Sampling Distributions
5 Sampling Distributions
- Chapter Overview
- 5.1 Foundations
  5.1 Foundations
  - Statistics as Random Variables
  - Repeated Sampling Concept
- 5.2 The Four Fundamental Sampling Distributions
  5.2 The Four Fundamental Sampling Distributions
- 5.3 Applications to Common Statistics
  5.3 Applications to Common Statistics
- 5.4 Visualization
  5.4 Visualization
  - Sampling Distribution Visualization
- 5.5 Code
  5.5 Code
- 5.6 Exercises
6 Statistical Estimation
6 Statistical Estimation
- Chapter Overview
- 6.1 Estimator Quality
  6.1 Estimator Quality
- 6.2 Maximum Likelihood Estimation
  6.2 Maximum Likelihood Estimation
- 6.3 Method of Moments
  6.3 Method of Moments
- 6.4 Bayesian Estimation
  6.4 Bayesian Estimation
- 6.5 Code
  6.5 Code
- 6.6 Exercises
7 Estimation of μ and σ²
7 Estimation of μ and σ²
- Chapter Overview
- 7.1 Estimation of the Mean
  7.1 Estimation of the Mean
- 7.2 Estimation of the Variance
  7.2 Estimation of the Variance
- 7.3 Gaussian Maximum Likelihood
  7.3 Gaussian Maximum Likelihood
- 7.4 Estimation Under Non-Normality
  7.4 Estimation Under Non-Normality
  - Heavy-Tailed Distributions
  - Skewed Distributions
- 7.5 Code
  7.5 Code
- 7.6 Exercises
8 Confidence Intervals
8 Confidence Intervals
- Chapter Overview
- 8.1 Foundations
  8.1 Foundations
  - Confidence Level and Coverage
  - Interpretation and Common Pitfalls
- 8.2 One-Sample Intervals
  8.2 One-Sample Intervals
- 8.3 Two-Sample Intervals
  8.3 Two-Sample Intervals
- 8.4 Paired-Sample Intervals
  8.4 Paired-Sample Intervals
- 8.5 Sample Size Determination
  8.5 Sample Size Determination
  - Sample Size for Desired Margin of Error
  - Sample Size for Comparing Two Groups
- 8.6 Code
  8.6 Code
- 8.7 Exercises
9 Hypothesis Testing
9 Hypothesis Testing
- Chapter Overview
- 9.1 Foundations
  9.1 Foundations
- 9.2 One-Sample Tests
  9.2 One-Sample Tests
- 9.3 Two-Sample Tests
  9.3 Two-Sample Tests
- 9.4 Paired-Sample Tests
  9.4 Paired-Sample Tests
  - Paired t-test for μ_D
  - When to Use Paired vs Two-Sample
- 9.5 Errors and Power
  9.5 Errors and Power
- 9.6 Multiple Testing
  9.6 Multiple Testing
- 9.7 Code
  9.7 Code
- 9.8 Exercises
10 Chi-Square Tests
10 Chi-Square Tests
- Chapter Overview
- 10.1 Chi-Square Distribution and Asymptotics
  10.1 Chi-Square Distribution and Asymptotics
  - Chi-Square Distribution
  - Degrees of Freedom and Asymptotic Theory
- 10.2 Chi-Square Tests for Categorical Data
  10.2 Chi-Square Tests for Categorical Data
- 10.3 Practical Considerations
  10.3 Practical Considerations
  - Expected Cell Counts and Validity Conditions
  - Effect Size and Cramér's V
- 10.4 Code
  10.4 Code
- 10.5 Exercises
11 Analysis of Variance
11 Analysis of Variance
- Chapter Overview
- 11.1 One-Way Analysis of Variance
  11.1 One-Way Analysis of Variance
  - Model and Assumptions
  - F-Test Procedure
- 11.2 Two-Way Analysis of Variance
  11.2 Two-Way Analysis of Variance
  - Main Effects and Blocking
  - Interaction Effects
- 11.3 Post-Hoc Comparisons
  11.3 Post-Hoc Comparisons
- 11.4 Welch's Analysis of Variance
  11.4 Welch's Analysis of Variance
  - Welch's One-Way Analysis of Variance
  - Welch's Two-Way Analysis of Variance
- 11.5 Assumptions
  11.5 Assumptions
- 11.6 Diagnostics
  11.6 Diagnostics
- 11.7 Practical Applications
  11.7 Practical Applications
- 11.8 Code
  11.8 Code
- 11.9 Exercises
12 Correlation and Causation
12 Correlation and Causation
- Chapter Overview
- 12.1 Correlation
  12.1 Correlation
- 12.2 Ecological Correlation
  12.2 Ecological Correlation
- 12.3 Correlation, Causation, and Confounding
  12.3 Correlation, Causation, and Confounding
- 12.4 Causation
  12.4 Causation
- 12.5 Correlation Tests
  12.5 Correlation Tests
- 12.6 Correlation Matrix and Visualization
  12.6 Correlation Matrix and Visualization
  - Correlation Heatmaps
  - Pair Plots and Scatter Matrices
- 12.7 Code
  12.7 Code
- 12.8 Exercises
13 Linear Regression
13 Linear Regression
- Chapter Overview
- 13.1 Linear Regression
  13.1 Linear Regression
  - Simple Linear Regression
  - Multiple Linear Regression
- 13.2 Estimation and Inference
  13.2 Estimation and Inference
- 13.3 Testing Coefficients
  13.3 Testing Coefficients
- 13.4 Interaction and Polynomial Extensions
  13.4 Interaction and Polynomial Extensions
  - Interaction Terms
  - Polynomial Regression
- 13.5 Assumptions and Diagnostics
  13.5 Assumptions and Diagnostics
- 13.6 Diagnostics
  13.6 Diagnostics
- 13.7 Performance Metrics
  13.7 Performance Metrics
- 13.8 Model Selection Criteria
  13.8 Model Selection Criteria
- 13.9 Splines and Generalized Additive Models
  13.9 Splines and Generalized Additive Models
  - Splines and Generalized Additive Models Overview
  - Generalized Additive Models
- 13.10 Package Usage
  13.10 Package Usage
- 13.10 Code
  13.10 Code
- 13.12 Exercises
14 Normality Tests
14 Normality Tests
- Chapter Overview
- 14.1 Introduction to Normality
  14.1 Introduction to Normality
- 14.2 Graphical Methods
  14.2 Graphical Methods
- 14.3 Descriptive Statistics as Normality Indicators
  14.3 Descriptive Statistics as Normality Indicators
- 14.4 Formal Tests for Normality
  14.4 Formal Tests for Normality
- 14.5 Limitations and Pitfalls
  14.5 Limitations and Pitfalls
- 14.6 Dealing with Non-Normal Data
  14.6 Dealing with Non-Normal Data
- 14.7 Applications
  14.7 Applications
- 14.8 Code
  14.8 Code
- 14.9 Exercises
15 Variance Tests
15 Variance Tests
- Chapter Overview
- 15.1 Introduction to Variance Testing
  15.1 Introduction to Variance Testing
- 15.2 Chi-Square Test for Variance
  15.2 Chi-Square Test for Variance
- 15.3 F-Test for Comparing Two Variances
  15.3 F-Test for Comparing Two Variances
- 15.4 Bartlett's Test
  15.4 Bartlett's Test
- 15.5 Robust Tests
  15.5 Robust Tests
- 15.6 Advanced Methods
  15.6 Advanced Methods
- 15.7 Applications
  15.7 Applications
- 15.8 Code
  15.8 Code
- 15.9 Exercises
16 Non-Parametric Tests
16 Non-Parametric Tests
- Chapter Overview
- 16.1 Foundations
  16.1 Foundations
- 16.2 One-Sample Non-Parametric Tests
  16.2 One-Sample Non-Parametric Tests
- 16.3 Paired-Sample Non-Parametric Tests
  16.3 Paired-Sample Non-Parametric Tests
- 16.4 Two-Sample Non-Parametric Tests
  16.4 Two-Sample Non-Parametric Tests
- 16.5 Multi-Group Non-Parametric Tests
  16.5 Multi-Group Non-Parametric Tests
- 16.6 Non-Parametric Correlation
  16.6 Non-Parametric Correlation
  - Spearman's ρ (Revisited)
  - Kendall's τ (Revisited)
- 16.7 Code
  16.7 Code
- 16.8 Exercises
17 Resampling Methods
17 Resampling Methods
- Chapter Overview
- 17.1 Bootstrap Foundations
  17.1 Bootstrap Foundations
- 17.2 Bootstrap Confidence Intervals
  17.2 Bootstrap Confidence Intervals
- 17.3 Bootstrap Hypothesis Testing
  17.3 Bootstrap Hypothesis Testing
- 17.4 Permutation Tests
  17.4 Permutation Tests
- 17.5 Comparison and Practical Guidance
  17.5 Comparison and Practical Guidance
- 17.6 Code
  17.6 Code
- 17.7 Exercises
18 Regularization Techniques
18 Regularization Techniques
- Chapter Overview
- 18.1 Motivation for Regularization
  18.1 Motivation for Regularization
- 18.2 Ridge Regression
  18.2 Ridge Regression
- 18.3 Lasso Regression
  18.3 Lasso Regression
- 18.4 Elastic Net
  18.4 Elastic Net
- 18.5 Hyperparameter Tuning
  18.5 Hyperparameter Tuning
- 18.6 Dimensionality Reduction Methods
  18.6 Dimensionality Reduction Methods
- 18.7 Overview and Comparison
  18.7 Overview and Comparison
- 18.7 Code
  18.7 Code
- 18.9 Exercises
19 Logistic Regression
19 Logistic Regression
- Chapter Overview
- 19.1 Logistic Regression
  19.1 Logistic Regression
- 19.2 Estimation and Inference
  19.2 Estimation and Inference
- 19.3 Model Evaluation
  19.3 Model Evaluation
- 19.4 Regularized Logistic Regression
  19.4 Regularized Logistic Regression
  - L1 and L2 Regularization for Logistic Regression
  - Feature Selection via Penalized Likelihood
- 19.5 Code
  19.5 Code
- 19.6 Exercises
20 Softmax Regression
20 Softmax Regression
- Chapter Overview
- 20.1 Softmax Regression
  20.1 Softmax Regression
- 20.2 Estimation and Optimization
  20.2 Estimation and Optimization
- 20.3 Model Evaluation and Case Studies
  20.3 Model Evaluation and Case Studies
- 20.4 One-vs-Rest and One-vs-One Strategies
  20.4 One-vs-Rest and One-vs-One Strategies
- 20.5 Code
  20.5 Code
- 20.6 Exercises
21 Survival Models
21 Survival Models
- Chapter Overview
- 21.1 Introduction to Survival Analysis
  21.1 Introduction to Survival Analysis
- 21.2 Non-Parametric Methods
  21.2 Non-Parametric Methods
- 21.3 Parametric Survival Models
  21.3 Parametric Survival Models
- 21.4 Cox Proportional Hazards Model
  21.4 Cox Proportional Hazards Model
- 21.5 Model Comparison and Selection
  21.5 Model Comparison and Selection
  - Non-Parametric vs Parametric vs Semi-Parametric
  - Akaike Information Criterion and Concordance Index
- 21.6 Code
  21.6 Code
- 21.7 Exercises
Bonus Chapter
Bonus Chapter
- Code
  Code
  - Kaplan Meier

Proportional Hazards Assumption¶

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