Regression Residuals and Analysis

Overview

This lecture continues the discussion on correlation and regression, focusing on the concept of residuals, their calculation, and how to interpret residual plots and histograms.

Review of Regression Concepts

• The regression line predicts Y from X using a formula with a slope and y-intercept.
• The correlation coefficient (r), R squared, slope, and y-intercept are key statistics in regression analysis.

Understanding Residuals

• A residual is the vertical distance between an actual data point and the predicted value on the regression line.
• Residual = Actual Y value − Predicted Y value (Y − Ŷ).
• Positive residuals indicate points above the regression line; negative residuals indicate points below.

Calculating Predicted Values and Residuals

• Plug the X value into the regression formula to get the predicted Y value (Ŷ).
• Calculate residuals for each data point by subtracting predicted Y from actual Y.
• Example: For X = 17, plug into Y = 26.4 + 18.06X to get predicted Y and then find the residual.

Interpreting Residual Plots

• A residual plot graphs residuals against the X values; Y-axis shows residuals.
• Residual plots help visualize how far each point is from the regression line.
• Evenly spread residuals suggest a good fit; clusters or patterns may indicate problems.

Histograms of Residuals

• Histograms display the frequency of residual values.
• Ideally, residual histograms should be bell-shaped (normal); skewed histograms may signal issues with model fit.
• The highest bar should be close to zero for a good fit.

Standard Deviation of Residuals

• The standard deviation of residual errors measures the average distance of points from the regression line.
• Formula: Square each residual, sum them, divide by (n−2), then take the square root.
• Standard deviation of residuals uses the same units as the Y variable.

Key Terms & Definitions

• Residual — The vertical distance between an actual data point and its predicted value from the regression line.
• Predicted Y value (Ŷ) — The Y value calculated from the regression equation for a given X.
• Residual Plot — A graph plotting residuals on the Y-axis against X values.
• Standard Deviation of Residuals — Measures the typical prediction error, calculated as the square root of the average squared residuals.

Action Items / Next Steps

• Practice calculating predicted Y values and residuals for given data.
• Analyze residual plots and histograms for model fit.
• Review the formula for the standard deviation of residual errors.