Linear Models

Linear models are a foundational class of statistical models used to describe the relationship between a dependent (or response) variable and one or more independent (or explanatory) variables. The core assumption is that this relationship can be approximated by a straight line, meaning the dependent variable is represented as a linear combination of the predictor variables plus an error term. By fitting a model to observed data, statisticians can estimate the magnitude and direction of each predictor's effect, test hypotheses about these relationships, and make predictions for new outcomes, making it a cornerstone of both inferential and predictive statistics.

1. Introduction to Linear Relationships
   1. Core Concepts of Statistical Modeling
      1. Definition of a Statistical Model
      2. Purposes of Statistical Modeling
      3. Dependent vs. Independent Variables
         1. Identifying Dependent Variables
         2. Identifying Independent Variables
         3. Direction of Causality
      4. Parameters vs. Statistics
         1. Definition of Parameters
         2. Definition of Statistics
         3. Estimating Parameters with Statistics
      5. Population vs. Sample
         1. Definition of Population
         2. Definition of Sample
         3. Sampling Methods
         4. Sampling Error
   2. Correlation and Covariance
      1. Definition of Covariance
         1. Calculation of Covariance
         2. Interpretation of Covariance
      2. Pearson Correlation Coefficient
         1. Formula for Pearson Correlation
         2. Properties of Pearson Correlation
         3. Range and Interpretation
      3. Interpreting Correlation
         1. Strength and Direction
         2. Scatterplot Visualization
         3. Limitations of Correlation
      4. Correlation vs. Causation
         1. Spurious Correlation
         2. Confounding Variables
         3. Establishing Causality
   3. The Straight Line Equation
      1. General Form of a Linear Equation
         1. Slope-Intercept Form
         2. Point-Slope Form
         3. Standard Form
      2. Slope and Intercept
         1. Definition of Slope
         2. Interpretation of Slope
         3. Definition of Intercept
         4. Interpretation of Intercept
      3. Representing Relationships Graphically
         1. Plotting Data Points
         2. Drawing the Regression Line
         3. Visualizing Fit and Residuals