Linear Regression Analysis
Linear regression analysis is a statistical method used to model the relationship between one or more independent variables (predictors) and a continuous dependent variable. It aims to identify and quantify the linear relationship between the independent variables and the dependent variable, allowing researchers to make predictions or infer the effect of changes in the independent variables on the dependent variable.
When to Use Linear Regression Analysis:
Linear regression analysis is appropriate when:
There is a hypothesized linear relationship between one or more independent variables and a continuous dependent variable.
The goal is to predict or estimate the value of the dependent variable based on the values of the independent variables.
The assumptions of linear regression are met, including linearity, independence of observations, homoscedasticity, and normality of residuals.
Assumptions and Data Requirements:
Before conducting linear regression analysis, several assumptions must be met:
Linearity: The relationship between the independent variables and the dependent variable should be linear.
Independence of Observations: The observations should be independent of each other.
Homoscedasticity: The residuals (errors) should have constant variance across all levels of the independent variables.
Normality of Residuals: The residuals should be approximately normally distributed around zero.
Additionally, the data required for linear regression analysis should consist of continuous variables obtained from a random sample.
Interpreting Linear Regression:
The linear regression model estimates the relationship between the independent variables (predictors) and the dependent variable in terms of a regression equation, typically in the form of Y = β₀ + β₁X₁ + β₂X₂ + ... + ε.
The coefficients (β₁, β₂, etc.) represent the estimated change in the dependent variable for a one-unit change in the corresponding independent variable, holding other variables constant.
The intercept (β₀) represents the predicted value of the dependent variable when all independent variables are equal to zero.
The residuals (ε) represent the differences between the observed values of the dependent variable and the values predicted by the regression model.
Sample Situation with Sample Data:
Suppose a researcher wants to predict students' exam scores based on their study hours and attendance in classes. The researcher collects data on study hours, class attendance, and exam scores for a sample of students.
