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Diagnostics on Least Squares Regression Line

Explore coefficient of determination, perform residual analysis, and identify influential observations in regression models. Learn how to assess the appropriateness of linear models and identify outliers.

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Diagnostics on Least Squares Regression Line

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  1. Lesson 4 - 3 Diagnostics on the Least- Squares Regression Line

  2. Objectives • Compute and interpret the coefficient of determination • Perform residual analysis on a regression model • Identify influential observations

  3. Vocabulary • Coefficient of determination, R2– measures the percentage of total variation in the response variable explained by the least-squares regression line. • Deviations – differences between predicted value and actual value • Total Deviation – deviation between observed value, y, and mean of y, y-bar • Explained Deviation – deviation between predicted value, y-hat, and mean of y, y-bar • Unexplained Deviation – deviation between observed value, y, and predicted value, y-hat • Influential Observation – observation that significantly affects the value of the slope

  4. Is the Linear Model Appropriate? • Patterned Residuals • If a plot of the residuals against the explanatory variable shows a discernible pattern, such as a curve, then the response and explanatory variable may not be linearly related • Variance of the Residuals Constant • If a plot of the residuals against the explanatory variable shows the spread of the residuals increasing or decreasing as the explanatory variable increases, then a strict requirement of the linear model is violated. This requirement is called constant error variance. • Influential Observations • Influential observations typically exist when the point is an outlier relative to its X-value • Outliers and Influential Observations • Remove only if there is justification to do so

  5. Deviations and Predictions • The relationship is Total Deviation = Explained + Unexplained • The larger the explained deviation, the better the model is at prediction / explanation • The larger the unexplained deviation, the worse the model is at prediction / explanation

  6. Identifying Outliers • From a scatter diagram • From a residual plot • From a boxplot

  7. TI-83 Instructions for Residuals Plot • With diagnostics turned on and explanatory variable in L1 and response variable in L2 • Press STAT, highlight CALC and select 4: LinReg (ax + b) and hit enter twice • 2nd Y= (STAT PLOT) • Select Plot 1 • Choose scatter diagram icon • XList is L1 • Ylist is RESID by putting cursor on List 2, pressing 2nd Stat and choosing the list (scroll down) entitled RESID • Press Zoom and select 9: ZoomStat

  8. TI-83 Instructions for Boxplots • With explanatory variable in L1 • Press 2nd Y= (STAT PLOT) • Select Plot 1 • Choose modified boxplot icon • XList is L1 • Press Zoom and select 9: ZoomStat

  9. Summary and Homework • Summary: • Diagnostics are very important in assessing the quality of a least-squares regression model • The coefficient of determination measures the percent of total variation explained by the model • The plot of residuals can detect nonlinear patterns, error variances that are not constant, and outliers • We must be careful when there are influential observations because they have an unusually large effect on the computation of our model parameters • Homework:pg 235 – 239; 2, 5, 9, 11-15, 29

  10. Homework Answers • 12 -- no pattern • 14 -- patterned residuals (parabolic)

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