Linear model

Linear model

A linear model is a type of statistical model that assumes a linear relationship between the input variables (independent variables) and the single output variable (dependent variable). Linear models are widely used in various fields such as economics, biology, engineering, and social sciences due to their simplicity and interpretability.

Types of Linear Models

Linear models can be categorized into several types, including:

• Simple linear regression: This model describes the relationship between two variables by fitting a linear equation to observed data. One variable is considered to be an explanatory variable, and the other is considered to be a dependent variable.
• Multiple linear regression: This model extends simple linear regression by using multiple explanatory variables to predict the dependent variable.
• Generalized linear model: This model generalizes linear regression by allowing the dependent variable to have a non-normal distribution.

Mathematical Formulation

The general form of a linear model can be expressed as:

\[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_p x_p + \epsilon \]

where:

• \( y \) is the dependent variable.
• \( \beta_0 \) is the intercept.
• \( \beta_1, \beta_2, \ldots, \beta_p \) are the coefficients of the independent variables \( x_1, x_2, \ldots, x_p \).
• \( \epsilon \) is the error term.

Assumptions

Linear models rely on several key assumptions:

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

Applications

Linear models are used in various applications, including:

• Predictive modeling: Forecasting future values based on historical data.
• Hypothesis testing: Testing the relationship between variables.
• Data analysis: Understanding the impact of one or more independent variables on a dependent variable.

Related Pages

• Regression analysis
• Statistical model
• Predictive modeling
• Hypothesis testing
• Data analysis

See Also

• Simple linear regression
• Multiple linear regression
• Generalized linear model
• Statistical model
• Predictive modeling