Linear regression

Definition

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. This technique is foundational in data science as it helps in understanding trends, making predictions, and quantifying relationships among variables.

5 Must Know Facts For Your Next Test

1. In linear regression, the goal is to minimize the sum of the squared residuals to find the best-fitting line through the data points.
2. The coefficients in a linear regression model represent the change in the dependent variable for a one-unit change in an independent variable, assuming all other variables are held constant.
3. Linear regression assumes that there is a linear relationship between the independent and dependent variables, which can be visually assessed with scatterplots.
4. Goodness-of-fit measures, such as R-squared, indicate how well the model explains the variability in the dependent variable, with higher values suggesting a better fit.
5. Assumptions of linear regression include linearity, independence of errors, homoscedasticity (constant variance of errors), and normality of residuals.

Review Questions

• How does linear regression help in understanding relationships between variables?
  • Linear regression helps in understanding relationships by quantifying how changes in independent variables affect a dependent variable. By fitting a linear equation to observed data, it allows us to see trends and make predictions based on those relationships. The coefficients derived from the model give clear insights into the strength and direction of these relationships.
• What role do residuals play in assessing the performance of a linear regression model?
  • Residuals are crucial for evaluating how well a linear regression model fits the data. They indicate the differences between actual observed values and those predicted by the model. Analyzing residuals helps identify patterns that suggest whether the assumptions of linear regression are met or if adjustments are needed for better accuracy.
• Evaluate the impact of violating key assumptions of linear regression on model results and interpretations.
  • Violating key assumptions of linear regression, such as linearity, independence, or homoscedasticity, can significantly distort model results and interpretations. For instance, if there’s a non-linear relationship, applying linear regression could lead to inaccurate predictions and misleading conclusions. Additionally, if residuals are not independent, it may indicate issues like autocorrelation, leading to underestimated standard errors and inflated significance levels. Therefore, checking these assumptions is essential for reliable analysis and decision-making.

Related terms

Dependent Variable:

The variable being predicted or explained in a regression model; it is influenced by one or more independent variables.

Independent Variable: The variable(s) used to predict or explain changes in the dependent variable; also known as predictors or features.

Residuals:

The differences between observed values and the values predicted by the regression model; they are used to assess the accuracy of the model.