Linear regression

Linear regression is still a frequently used approximation for explaining linear dependencies between phenomena. It is often the first out-of-shelf method in various fields, such as econometrics, biology, engineering, and social sciences, due to its simplicity and interpretability. Linear regression is a statistical model that estimates the linear relationship between a scalar response, often referred to as the dependent variable, and one or more explanatory variables, also known as independent variables or predictors. Linear regression models can be classified into two main types:

  • Simple Linear Regression: This model involves a single explanatory variable. It aims to find the best-fitting straight line through the data points, which minimizes the sum of the squared differences between the observed values and the values predicted by the model.
  • Multiple Linear Regression: This model includes two or more explanatory variables. It extends the simple linear regression model to account for multiple factors that might influence the dependent variable, providing a more comprehensive understanding of the relationships within the data.

Linear regression relies on several key assumptions to produce valid results:

  • Linearity: The relationship between the dependent and independent variables is linear.
  • Independence: The observations are independent of each other.
  • Homoscedasticity: The variance of the errors is constant across all levels of the independent variables.
  • Normality: The errors are normally distributed.

The most common method for estimating the parameters (β) of a linear regression model is the Ordinary Least Squares (OLS) method. OLS estimates the parameters by minimizing the sum of the squared differences between the observed responses and the responses predicted by the linear model. Once the parameters are estimated, they can be interpreted to understand the effect of each independent variable on the dependent variable.

For example, in a multiple linear regression model, a coefficient β represents the expected change in the dependent variable for a one-unit change in the ith independent variable, holding all other variables constant.

Linear regression is widely used for:

  • Predictive Analysis: Forecasting future values based on past trends. Inferential Statistics: Testing hypotheses and understanding relationships between variables.
  • Econometrics: Analyzing economic data to understand relationships between economic indicators.
  • Engineering: Modeling relationships between physical measurements and outcomes.
  • Medical Research: Understanding the impact of various factors on health outcomes.

Despite its popularity, linear regression has limitations:

  • Assumption Violations: The model’s assumptions may not hold in real-world data, leading to biased or inaccurate results.
  • Outliers: Sensitive to outliers, which can disproportionately affect the model.
  • Multicollinearity: High correlation between independent variables can make it difficult to isolate the effect of each variable.

In summary, linear regression remains a foundational technique in statistical modeling and data analysis, valued for its simplicity, ease of interpretation, and wide applicability across various domains.