What is Linear Regression?

Linear regression stands as a fundamental and widely utilized form of predictive analysis. It primarily seeks to address two critical questions: Firstly, how effectively can a set of predictor variables forecast an outcome (dependent or criterion) variable? Secondly, which specific variables emerge as significant predictors of the outcome variable, and how do their beta estimates—reflecting both magnitude and direction—affect this outcome? Linear regression employs these estimates to describe the dynamics between one dependent variable and one or more independent variables. The most straightforward regression model, featuring one dependent and one independent variable, is encapsulated by the equation y = c + b*x, where:

  • y represents the predicted score of the dependent variable,
  • c is the constant,
  • b denotes the regression coefficient, and
  • x is the score on the independent variable.

Naming the Variables

The dependent variable in a regression analysis may be referred to by various terms, including outcome variable, criterion variable, endogenous variable, or regressand. Similarly, independent variables are also known as exogenous variables, predictor variables, or regressors.

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Key Applications of Regression Analysis

  1. Determining the Strength of Predictors: This involves assessing the influence of independent variables on a dependent variable. Common inquiries include examining the relationship between variables such as dose and effect, sales and marketing expenditure, or age and income.
  2. Forecasting Effects: Regression helps predict how changes in independent variables impact the dependent variable. A typical question might be, “What is the expected increase in sales revenue for every additional $1000 spent on marketing?”
  3. Trend Forecasting: It is used for predicting future trends and values, providing point estimates. For instance, one might ask, “What will the price of gold be in 6 months?”

Types of Linear Regression

Simple linear regression
Involves one dependent variable (interval or ratio) and one independent variable (interval or ratio or dichotomous).

Multiple linear regression
Features one dependent variable (interval or ratio) and two or more independent variables (interval or ratio or dichotomous).

Logistic regression
Deals with one dependent variable (dichotomous) and two or more independent variables (interval or ratio or dichotomous).

Ordinal regression
Comprises one dependent variable (ordinal) and one or more independent variables (nominal or dichotomous).

Multinomial regression
Includes one dependent variable (nominal) and one or more independent variables (interval or ratio or dichotomous).

Model Selection and Fitting

Choosing the appropriate model for analysis necessitates careful consideration of model fitting. Adding independent variables to a linear regression model invariably increases the explained variance (often expressed as R²). However, overfitting—a scenario where too many variables compromise the model’s generalizability—can occur. The principle of Occam’s razor aptly advises that a simpler model is generally preferable over a complex one. Statistically, incorporating a large number of variables may lead to some achieving statistical significance merely by chance.

To Reference this Page: Statistics Solutions. (2024). What is Linear Regression . Retrieved from here.

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Assumptions of a Linear Regression

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