Unit 3967: Advanced Statistical Methods and Applications

1. Multivariate Analysis

1.1 Introduction to Multivariate Data

• Definition and examples of multivariate data
• Importance in various disciplines

1.2 Factor Analysis

• Concept and objectives
• Exploratory vs. confirmatory factor analysis
• Extraction methods (principal axis factoring, maximum likelihood)
• Rotation techniques (varimax, oblimin)

1.3 Principal Component Analysis (PCA)

• Purpose and applications
• Covariance and correlation matrices
• Eigenvalues and eigenvectors interpretation
• Dimensionality reduction

1.4 Canonical Correlation Analysis

• Concept and use cases
• Deriving canonical variables
• Interpretation of canonical correlations

2. Time Series Analysis

2.1 Overview of Time Series Data

• Components: trend, seasonality, cyclicity, and noise
• Stationarity concepts

2.2 Autoregressive Integrated Moving Average (ARIMA) Models

• Model components: AR, I, MA
• Identification using ACF and PACF
• Model estimation and diagnostics

2.3 Seasonal Decomposition of Time Series

• Additive and multiplicative models
• Decomposition techniques (classical and STL)

2.4 Forecasting Methods

• Model validation and accuracy
• Applications of forecasting in practice

3. Bayesian Statistics

3.1 Foundations of Bayesian Inference

• Bayes’ theorem and probability interpretation
• Prior, likelihood, and posterior distributions

3.2 Common Prior Distributions

• Conjugate priors
• Informative vs. non-informative priors

3.3 Markov Chain Monte Carlo (MCMC) Algorithms

• Purpose and overview
• Gibbs sampling and Metropolis-Hastings algorithms

3.4 Bayesian Hierarchical Models

• Structure and applications
• Model fitting and interpretation

4. Survival Analysis

4.1 Introduction to Time-to-Event Data

• Censoring mechanisms
• Survival and hazard functions

4.2 Kaplan-Meier Survival Curves

• Estimation and interpretation
• Comparing survival curves

4.3 Cox Proportional Hazards Model

• Model assumptions
• Hazard ratios and covariate effects

4.4 Parametric Survival Models

• Exponential, Weibull, and other distributions
• Model fitting and comparison

5. Experimental Design

5.1 Principles of Experimental Design

• Randomization, replication, blocking
• Control of confounding variables

5.2 Factorial Designs

• Full factorial and fractional factorial designs
• Interaction effects

5.3 Designing Efficient Experiments

• Sample size considerations
• Ethical considerations in experimentation

6. Nonparametric Statistics

6.1 Overview and Applications

• When to use nonparametric methods

6.2 Wilcoxon Rank-Sum Test

• Purpose and implementation

6.3 Kruskal-Wallis Test

• Extension of Wilcoxon test for multiple groups

6.4 Spearman's Rank Correlation Coefficient

• Measuring monotonic relationships

7. Spatial Statistics

7.1 Introduction to Spatial Data

• Types of spatial data
• Spatial data structures

7.2 Spatial Autocorrelation

• Moran’s I and Geary’s C statistics
• Interpretation and testing

7.3 Kriging

• Concept and types (ordinary, universal)
• Variogram modeling

7.4 Spatial Regression Models

• Spatial lag and spatial error models
• Applications and diagnostics

8. Machine Learning in Statistics

8.1 Overview of Machine Learning Techniques

• Supervised vs. unsupervised learning

8.2 Supervised Learning Algorithms

• Linear regression
• Decision trees
• Model evaluation metrics

8.3 Unsupervised Learning Algorithms

• Clustering methods (k-means, hierarchical clustering)
• Dimensionality reduction techniques (PCA revisited, t-SNE)

8.4 Integration of Machine Learning and Statistical Methods

• Model interpretability
• Handling big data and complex models