Advanced Statistics Course

Lesson 1 • Probability Theory Fundamentals

Introduces axioms of probability, conditional probability, and independence. These concepts underpin all inferential and Bayesian methods covered later.

Lesson 2 • Descriptive Statistics Essentials

Covers central tendency, dispersion, and shape metrics for summarising distributions. Provides the numerical vocabulary used throughout every later chapter.

Lesson 3 • Types of Data and Measurement

Distinguishes nominal, ordinal, interval, and ratio scales and their analytic implications. Anchors all subsequent modelling choices to correct data classification.

Lesson 4 • Exploratory Data Analysis Techniques

Applies visual and numerical EDA to detect patterns, outliers, and relationships before modelling. Develops the diagnostic habit required for advanced model validation.

Lesson 1 • Non-Parametric Alternatives

Introduces rank-based tests for non-normal or ordinal data. Equips learners to handle assumption violations encountered in applied datasets.

Lesson 2 • Parametric Tests for Means

Applies z-tests, one-sample and two-sample t-tests, and paired t-tests to mean comparisons. Connects distributional assumptions from Chapter 1 to practical test selection.

Lesson 3 • Point and Interval Estimation

Covers maximum likelihood and method-of-moments estimators plus confidence interval construction. Establishes the estimation logic that hypothesis tests formalise.

Lesson 4 • Hypothesis Testing Framework

Formalises null and alternative hypotheses, Type I/II errors, and decision rules. Provides the logical scaffold for every significance test in later chapters.

Lesson 1 • Principles of Experimental Design

Covers randomisation, blocking, replication, and factorial efficiency for study planning. Ensures learners can design experiments that yield unconfounded causal estimates.

Lesson 2 • Factorial and Two-Way ANOVA

Analyses main effects and interactions in two-factor designs. Introduces interaction plots as a diagnostic tool for complex experimental outcomes.

Lesson 3 • Repeated Measures and Mixed Designs

Handles within-subject factors and sphericity assumptions in repeated-measures ANOVA. Prepares learners for longitudinal and crossover experimental structures.

Lesson 4 • One-Way ANOVA Mechanics

Decomposes total variance into between-group and within-group components via the F-ratio. Builds directly on t-test logic while introducing the ANOVA table structure.

Lesson 1 • Multiple Linear Regression

Extends OLS to multiple predictors, addressing multicollinearity and partial effects. Connects ANOVA decomposition from Chapter 3 to the regression F-test.

Lesson 2 • Variable Selection and Regularisation

Compares stepwise, AIC/BIC, ridge, and lasso selection strategies for parsimonious models. Bridges classical regression to the penalised methods used in machine learning.

Lesson 3 • Regression Diagnostics

Evaluates linearity, homoscedasticity, normality, and independence of residuals. Diagnostic mastery prevents invalid inference from violated OLS assumptions.

Lesson 4 • Simple Linear Regression

Estimates slope and intercept via OLS, tests coefficients, and measures fit with R-squared. Establishes the regression framework extended throughout this chapter.

Lesson 1 • Binary Logistic Regression

Models binary outcomes using the logit link, interpreting odds ratios and predicted probabilities. Directly applicable to classification and risk-factor analysis tasks.

Lesson 2 • Multinomial and Ordinal Regression

Handles outcomes with more than two unordered or ordered categories. Extends logistic regression logic to polytomous response variables.

Lesson 3 • Poisson and Negative Binomial Models

Models count and rate outcomes, addressing overdispersion with negative binomial alternatives. Equips learners to analyse event-frequency data common in health and operations.

Lesson 4 • GLM Framework and Link Functions

Unifies exponential family distributions, link functions, and deviance as a generalisation of OLS. Provides the theoretical scaffold for all GLM variants in this chapter.

Lesson 1 • Exploratory Factor Analysis

Identifies latent constructs underlying observed correlations using factor extraction and rotation. Widely used in psychometrics, survey research, and scale development.

Lesson 2 • Discriminant Analysis

Classifies observations into predefined groups by maximising between-group separation. Complements logistic regression as a parametric classification alternative.

Lesson 3 • Principal Component Analysis

Decomposes covariance matrices into orthogonal components that capture maximum variance. Provides the dimension-reduction foundation for factor analysis and clustering.

Lesson 4 • Multivariate Analysis of Variance

Tests group differences across multiple dependent variables simultaneously using MANOVA. Extends ANOVA logic while controlling experiment-wise error across outcomes.

Lesson 1 • Bayesian Model Comparison

Uses Bayes factors, WAIC, and LOO-CV to select among competing Bayesian models. Provides a principled alternative to frequentist AIC/BIC model selection.

Lesson 2 • Bayesian Inference Foundations

Formalises prior, likelihood, and posterior distributions within Bayes' theorem. Contrasts frequentist and Bayesian interpretations of probability and uncertainty.

Lesson 3 • Bayesian Regression and Hierarchical Models

Applies Bayesian estimation to regression and introduces partial pooling via hierarchical priors. Handles grouped data and small-sample problems more flexibly than classical methods.

Lesson 4 • Markov Chain Monte Carlo Methods

Implements Metropolis-Hastings and Gibbs sampling for posterior approximation. Enables Bayesian inference in models without analytical posteriors.

Lesson 1 • Cross-Validation and Resampling

Estimates out-of-sample predictive performance via k-fold CV, LOOCV, and bootstrap. Prevents overfitting and provides honest model evaluation metrics.

Lesson 2 • Linear Mixed-Effects Models

Partitions variance into fixed and random effects for clustered and longitudinal data. Extends repeated-measures ANOVA with continuous time and unbalanced structures.

Lesson 3 • Generalised Additive Models

Fits smooth non-linear functions of predictors using splines within a GLM framework. Bridges parametric regression and flexible machine learning approaches.

Lesson 4 • Model Deployment and Reporting

Translates statistical models into reproducible reports and decision-ready outputs. Covers uncertainty communication, sensitivity analysis, and stakeholder presentation.