Chapter 4 Section 3

1. Chapter 4Section 3 The Coefficient of Determination

2. 1 Chapter 4 – Section 3 • Learning objectives • Compute and interpret the coefficient of determination

3. Chapter 4 – Section 3 • We want to measure how well our model explains the relationship • How much does our model explain? • How much does our model improve our prediction? • We have a value of x … we use our model to predict the value of y … how accurate are we? • We want to measure how well our model explains the relationship • How much does our model explain? • How much does our model improve our prediction? • We have a value of x … we use our model to predict the value of y … how accurate are we? • We analyze our prediction by examining the errors, or deviations

4. Chapter 4 – Section 3 • Step 1 • We are asked to predict the value of y for a specific x but we are given no information at all • Step 1 • We are asked to predict the value of y for a specific x but we are given no information at all • What would we guess? • Maybe we would guess a value of 0 • Step 1 • We are asked to predict the value of y for a specific x but we are given no information at all • What would we guess? • Maybe we would guess a value of 0 • We would have an error of Residual = Observed – Predicted = y – 0 = y

5. Chapter 4 – Section 3 • Step 2 • We are asked again to predict the value of y for a specific x, but now we are given a list of observed values of y • Step 2 • We are asked again to predict the value of y for a specific x, but now we are given a list of observed values of y • What would we guess? • We would guess a value of y • Step 2 • We are asked again to predict the value of y for a specific x, but now we are given a list of observed values of y • What would we guess? • We would guess a value of y • We would have an error of Residual = Observed – Predicted = y – y

6. Chapter 4 – Section 3 • Step 3 • We are asked again to predict the value of y for a specific x, and now we are given our linear model and the value of x • Step 3 • We are asked again to predict the value of y for a specific x, and now we are given our linear model and the value of x • What would we guess? • We would guess a value of y = b1x + b0 • Step 3 • We are asked again to predict the value of y for a specific x, and now we are given our linear model and the value of x • What would we guess? • We would guess a value of y = b1x + b0 • We would have a remaining error of Residual = Observed – Predicted = y – y

7. Chapter 4 – Section 3 • We began with y – y or the totaldeviation • We began with y – y or the totaldeviation • Our model reduces this to y – y or the unexplaineddeviation • We began with y – y or the totaldeviation • Our model reduces this to y – y or the unexplaineddeviation • The amount of reduction y – y is the explaineddeviation

8. Chapter 4 – Section 3 • The relationship is • The relationship is • 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

9. Chapter 4 – Section 3 • Instead of straight deviations, we use variations Variation = Deviation2 • Instead of straight deviations, we use variations Variation = Deviation2 • It is also true that • Instead of straight deviations, we use variations Variation = Deviation2 • It is also true that • A measure of the explanatory power of the model is the proportion of variation that is explained

10. Chapter 4 – Section 3 • The coefficientofdetermination or R2 measures the percent of the variation that is explained • The coefficientofdetermination or R2 measures the percent of the variation that is explained • The coefficient of determination R2 • Varies between 0 and 1 • The coefficientofdetermination or R2 measures the percent of the variation that is explained • The coefficient of determination R2 • Varies between 0 and 1 • A value of R2 close to 0 (i.e. 0% explained) indicates a model with very little explanatory power • The coefficientofdetermination or R2 measures the percent of the variation that is explained • The coefficient of determination R2 • Varies between 0 and 1 • A value of R2 close to 0 (i.e. 0% explained) indicates a model with very little explanatory power • A value of R2 close to 1 (i.e. 100% explained) indicates a model with much explanatory power

11. Chapter 4 – Section 3 • The coefficient of determination is related to the correlation coefficient R2 = r2 • The coefficient of determination is related to the correlation coefficient R2 = r2 • The two exponents have different meanings • R2 is the coefficient of determination • r2 is the square of the correlation coefficient • You should use technology (a calculator or software) to compute R2

12. Chapter 4 – Section 3 • The relationship R2 = r2 gives us guidance about values of R2 • r around ± 0.8  very strong relationships, soR2 around 0.65  very strong relationships • r around ± 0.5  moderately strong relationships, soR2 around 0.25  moderately strong relationships • r around ± 0.1  weak relationships, soR2 around 0.01  weak relationships • These are very rough guidelines for assessing R2

13. Summary: Chapter 4 – Section 3 • The coefficient of determination measures the percent of total variation explained by the model • The total variation measures the deviation between the values of y and its mean • The total variation consists of the explained variation part (which the regression model accounts for) and the unexplained variation part (which the regression model does not account for)