The Coefficient of Determination

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1 The Coefficient of Determination Lecture 46 Section 13.9 Robb T. Koether Hampden-Sydney College Wed, Apr 17, 2012 Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

2 Outline 1 The Regression Identity 2 Sums of Squares on the TI-83 3 TI-83 - The Coefficient of Determination 4 Assignment Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

3 Outline 1 The Regression Identity 2 Sums of Squares on the TI-83 3 TI-83 - The Coefficient of Determination 4 Assignment Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

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 Wed, Apr 17, / 37

5 Explaining the Variation in y For example, why do some people weigh more than other people? Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

6 Explaining the Variation in y For example, why do some people weigh more than other people? That is, why is there variability in weight? Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

7 Explaining the Variation in y For example, why do some people weigh more than other people? That is, why is there variability in weight? One explanation is that some people weigh more than others because they are taller. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

8 Explaining the Variation in y For example, why do some people weigh more than other people? That is, why is there variability in weight? One explanation is that some people weigh more than others because they are taller. That is, because there is variability in height. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

9 Explaining the Variation in y For example, why do some people weigh more than other people? That is, why is there variability in weight? One explanation is that some people weigh more than others because they are taller. That is, because there is variability in height. But that is only a partial explanation. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

10 Explaining the Variation in y Statisticians want to quantify how much of the variation in y is explained by the variation in x. We measure variation in y by the standard deviation SSY s y = n 1. However, the key component is SSY. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

11 The Regression Identity There are three different variations that we can measure. How the y values in the data deviate from their mean. How the y values in the model deviate from their mean. How the y values in the data deviate from the y values in the model. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

12 The Regression Identity There are three different variations that we can measure. That is, compare y to y. How the y values in the model deviate from their mean. How the y values in the data deviate from the y values in the model. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

13 The Regression Identity There are three different variations that we can measure. That is, compare y to y. That is, compare ŷ to y. How the y values in the data deviate from the y values in the model. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

14 The Regression Identity There are three different variations that we can measure. That is, compare y to y. That is, compare ŷ to y. That is, compare y to ŷ. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

15 The Regression Identity Variation in the data (Total sum of squares): SST = (y y) 2. (This is the same as SSY.) 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 Wed, Apr 17, / 37

16 The Regression Identity SST is the observed variation in y. SSR is the predicted variation in y (according to the model). SSE is the unexplained variation in y (not explained by the model). Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

17 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 Wed, Apr 17, / 37

18 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 Wed, Apr 17, / 37

19 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 Wed, Apr 17, / 37

20 Example - SST, SSR, and SSE The regression line Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

21 Example - SST, SSR, and SSE The deviations of y from y Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

22 Example - SST, SSR, and SSE The deviations of ŷ from y Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

23 Example - SST, SSR, and SSE The deviations of y from ŷ Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

24 Example Compute SST. x y y y (y y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

25 Example Compute SST: L 2 -y. x y y y (y y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

26 Example Compute SST: Ans 2. x y y y (y y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

27 Example Compute SST: sum(ans). x y y y (y y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

28 Example Compute SSR. x y ŷ ŷ y (ŷ y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

29 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 Wed, Apr 17, / 37

30 Example Compute SSR: L 3 -y. x y ŷ ŷ y (ŷ y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

31 Example Compute SSR: Ans 2. x y ŷ ŷ y (ŷ y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

32 Example Compute SSR: sum(ans). x y ŷ ŷ y (ŷ y) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

33 Example Compute SSE. x y ŷ y ŷ (y ŷ) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

34 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 Wed, Apr 17, / 37

35 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 Wed, Apr 17, / 37

36 Example Compute SSE: Ans 2. x y ŷ y ŷ (y ŷ) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

37 Example Compute SSE: sum(ans). x y ŷ y ŷ (y ŷ) Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

38 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 Wed, Apr 17, / 37

39 Explaining Variation It follows that the total variation in y (SST) can be separated into two parts: SSR = the part explained by the model. SSE = the part unexplained by the model. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

40 Explaining Variation Furthermore, 1 = SSR SST + SSE SST. So, the fraction SSR SST can be interpreted as the proportion of SST that is explained by the model. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

41 Outline 1 The Regression Identity 2 Sums of Squares on the TI-83 3 TI-83 - The Coefficient of Determination 4 Assignment Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

42 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 Wed, Apr 17, / 37

43 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 Wed, Apr 17, / 37

44 Outline 1 The Regression Identity 2 Sums of Squares on the TI-83 3 TI-83 - The Coefficient of Determination 4 Assignment Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

45 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 Wed, Apr 17, / 37

46 Coefficient of Determination Example (Coefficient of Determination) In the height-weight example, we found that r = Then r 2 = So, 82% of variation in weight can be explained by variation in height. The remaining 18% of variation in weight is due to some other factor. Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

47 Outline 1 The Regression Identity 2 Sums of Squares on the TI-83 3 TI-83 - The Coefficient of Determination 4 Assignment Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

48 Assignment Homework Read Section 13.9, pages Exercises 35(a), 36(f), Robb T. Koether (Hampden-Sydney College) The Coefficient of Determination Wed, Apr 17, / 37

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