Regression and Prediction (Multiple Linear Regression)

Multiple linear regression is a fundamental statistical method in data science used to examine the relationship between one numerical outcome variable and two or more predictor variables. This topic introduces students to the multiple linear regression model, expressed as Y = B0 + B1(X1) + B2(X2) + ... + BP(XP) + e, where Y represents the outcome, X1, ..., XP represent the predictors, B0 is the intercept, B1, ..., BP are the regression coefficients, and e represents random error. Students will learn how multiple predictors can be incorporated into a single model to explain variations in an outcome and improve predictions. Practical examples, such as predicting house prices from floor area, location, and building age or examining health outcomes based on age, body mass index, and lifestyle factors, will illustrate the applications of multiple linear regression in real-world problems.

The material covers the estimation and interpretation of regression coefficients using the ordinary least squares (OLS) method, which estimates model parameters by minimizing the sum of squared residuals. Students will learn how to interpret each coefficient while holding the other predictors constant, evaluate statistical significance using hypothesis tests and p-values, and construct confidence intervals for regression coefficients. The topic also introduces measures of model performance, including the coefficient of determination R^2, adjusted R^2, and residual standard error. Students will explore the concept of multicollinearity, which occurs when predictor variables are strongly correlated, and learn how it can affect coefficient estimates and their interpretation. Practical exercises using R and Python will help students fit multiple regression models, interpret statistical outputs, and compare models containing different sets of predictors.

Furthermore, students will learn to assess the assumptions and predictive performance of multiple linear regression models. Key assumptions include linearity, independence of errors, constant error variance, and approximate normality of residuals for conventional statistical inference. Students will use residual plots, variance inflation factors (VIF), and other diagnostic tools to identify potential problems, including multicollinearity, outliers, and influential observations. The material will also introduce the distinction between explaining relationships and making predictions, emphasizing the importance of evaluating predictive performance using appropriate measures and validation techniques, such as train-test splitting. By the end of this topic, students are expected to develop, evaluate, and interpret multiple linear regression models using R and Python, while recognizing the limitations of their models and understanding that statistical associations do not necessarily imply causation.