The Coefficient of Determination
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1 The Coefficient of Determination Lecture 46 Section 13.9 Robb T. Koether Hampden-Sydney College Tue, Apr 13, 2010 Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
2 Outline 1 The Regression Identity 2 Sums of Squares on the TI-83 3 Explaining Variation 4 TI-83 - The Coefficient of Determination 5 Assignment Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
3 Outline 1 The Regression Identity 2 Sums of Squares on the TI-83 3 Explaining Variation 4 TI-83 - The Coefficient of Determination 5 Assignment Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
4 Explaining the Variation in y Statisticians use regression models to explain y. More specifically, through the model they use variation in x to explain variation in y. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
5 Explaining the Variation in y For example, why do some people weigh more than other people? One explanation is that some people weigh more than others because they are taller. That is, there is variation in weight because their is variation in height and because weight and height are correlated. But that is only a partial explanation. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
6 Explaining the Variation in y Statisticians want to quantify how much of the variation in y is explained by the variation in x. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
7 The Regression Identity As always, variation is measure by calculating a sum of squared deviations. There are three different deviations that we can measure. Deviations of y from y (variation in the data). Deviations of ŷ from y (variation in the model). Deviations of y from ŷ (difference between the data and the model). Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
8 The Regression Identity Variation in the data (Total sum of squares): SST = (y y) 2. Variation in the model (Regression sum of squares): SSR = (ŷ y) 2. Residues (Sum of squared Errors): SSE = (y ŷ) 2. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
9 Example - SST, SSR, and SSE The following data represent the heights and weights of 10 adult males. Height (x) Weight (y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
10 Example - SST, SSR, and SSE The regression line is ŷ = x. The model predicts, for example, that if a person is 70 inches tall, he will weigh 180 pounds. The model also predicts that a person will weigh an additional 7 pounds for each additional inch of height. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
11 Example - SST, SSR, and SSE Compute the predicted weight: Y 1 (L 1 ) L 3. Height (x) Weight (y) Pred. Wgt. (ŷ) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
12 Example - SST, SSR, and SSE The regression line Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
13 Example - SST, SSR, and SSE The deviations of y from y Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
14 Example - SST, SSR, and SSE The deviations of ŷ from y Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
15 Example - SST, SSR, and SSE The deviations of y from ŷ Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
16 Example Compute SST. x y y y (y y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
17 Example Compute SST: L 2 -y. x y y y (y y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
18 Example Compute SST: Ans 2. x y y y (y y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
19 Example Compute SST: sum(ans). x y y y (y y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
20 Example Compute SSR. x y ŷ ŷ y (ŷ y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
21 Example Compute SSR: Y 1 (L 1 ) L 3. x y ŷ ŷ y (ŷ y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
22 Example Compute SSR: L 3 -y. x y ŷ ŷ y (ŷ y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
23 Example Compute SSR: Ans 2. x y ŷ ŷ y (ŷ y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
24 Example Compute SSR: sum(ans). x y ŷ ŷ y (ŷ y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
25 Example Compute SSE. x y ŷ y ŷ (y ŷ) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
26 Example Compute SSE: Y 1 (L 1 ) L 3. x y ŷ y ŷ (y ŷ) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
27 Example Compute SSE: L 2 -L 3 L 4. x y ŷ y ŷ (y ŷ) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
28 Example Compute SSE: Ans 2. x y ŷ y ŷ (y ŷ) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
29 Example Compute SSE: sum(ans). x y ŷ y ŷ (y ŷ) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
30 Example We have now found that SSR = SSE = SST = We see that SSR + SSE = SST. This is called the regression identity. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
31 Outline 1 The Regression Identity 2 Sums of Squares on the TI-83 3 Explaining Variation 4 TI-83 - The Coefficient of Determination 5 Assignment Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
32 TI-83 - Finding SSR, SSE, and SST TI-83 SSR, SSE, and SST Put the x values into L 1 and the y values into L 2. Use LinReg(a+bx) L 1,L 2,Y 1. Enter Y 1 (L 1 ) L 3. To get SSR, evaluate sum((l 3 -y) 2 ). To get SSE, evaluate sum((l 2 -L 3 ) 2 ). To get SST, evaluate sum((l 2 -y) 2 ). Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
33 Outline 1 The Regression Identity 2 Sums of Squares on the TI-83 3 Explaining Variation 4 TI-83 - The Coefficient of Determination 5 Assignment Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
34 Explaining Variation One goal of regression is to explain the variation in y. For example, if y were weight, how would we explain the variation in weight? That is, why do some people weigh more than others? A partial answer is that some people weigh more because they are taller. That is, an explanatory variable is height x. What are some other partial answers? Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
35 Explaining Variation How much of the variation in weight is explained by variation in height? The total variation in weight is SST. The linear model (the regression line) explains some of the variation. The model predicts the variation SSR. The remainder is SSE, the variation not predicted by the model. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
36 Explaining Variation Statisticians consider the predicted variation SSR to be the amount of variation in y that is explained by the model. The residual variation SSE is the remaining variation in y that is not explained by the model. It all checks out because SST = SSR + SSE. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
37 Variation Explained by the Model The regression line Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
38 Variation Explained by the Model The total variation in y (SST) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
39 Variation Explained by the Model The variation in y that is explained by the model (SSR) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
40 Variation Explained by the Model The variation in y that is unexplained by the model (SSE) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
41 Explaining Variation It can be shown that and, therefore, r 2 = SSR SST 1 r 2 = SSE SST. Therefore, r 2 is the proportion of variation in y that is explained by the model. It is called the coefficient of determination. 1 r 2 is the proportion that is not explained by the model. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
42 Outline 1 The Regression Identity 2 Sums of Squares on the TI-83 3 Explaining Variation 4 TI-83 - The Coefficient of Determination 5 Assignment Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
43 TI-83 - Coefficient of Determination TI-83 Coefficient of Determination To calculate r 2 on the TI-83, follow the procedure that produces the regression line and r. In the same window, the TI-83 reports the value of r 2. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
44 TI-83 - Finding SSR, SSE, and SST Practice The data on the next slide represent crude oil prices a (x) vs. gasoline prices b (y). Draw the scatter plot. Find the equation of the regression line. Perform the residual analysis. Find the correlation coefficient. Find the coefficient of determination. Compute SST, SSR, and SSE. a b Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
45 TI-83 - Finding SSR, SSE, and SST Practice Date Crude Oil Date Gasoline Jan Jan Jan Jan Jan Feb Feb Feb Feb Feb Feb Feb Feb Mar Mar Mar Mar Mar Mar Mar Mar Mar Apr Apr Find SST, SSR, and SSE. Find r 2 and interpret the value. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
46 Outline 1 The Regression Identity 2 Sums of Squares on the TI-83 3 Explaining Variation 4 TI-83 - The Coefficient of Determination 5 Assignment Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
47 Assignment Homework Read Section 13.9, pages Work the practice problem on the previous slide. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
48 Answers to Even-Numbered Exercises Answers to Even-Numbered Exercises SST = , SSR = , SSE = r 2 = About 65.44% of the variation in gas prices is due to variation in oil prices. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Tue, Apr 13, / 48
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