Chapter 19 Sir Migo Mendoza

Transcription

1 The Linear Regression Chapter 19 Sir Migo Mendoza

2 Linear Regression and the Line of Best Fit Lesson 19.1 Sir Migo Mendoza

3 Question: Once we have a Linear Relationship, what can we do with it?

4 Something to think about What would happen if we wanted to use the relationship based on our sample data to try to predict values for other people in the population?

5 Something to thiuk about For example we wanted to estimate a person's weight, as closely as possible, based on their height? Fortunately for us, we have another correlational tool, the LINEAR REGRESSION; that allow us to do exactly that.

6 Something to think about If we had a dataset of height and weight data and wound up with an r value of 1.00, we would be able to state, with 100% accuracy, either a person's height based on knowing his weight or his weight based on knowing his height. Unfortunately, we rarely, if ever, have that degree of accuracy.

7 Something to thiuk about At the same time, unless the r value is zero, we know there is some degree of relationship between height and weight. Knowing that, and depending on the value of r, we should be able to predict, with some accuracy, one of the variables based on knowing the other variable. The key to doing so is a thorough understanding of the line of best fit.

8 Question: At this point, what do we know about the line of best fit?

9 Something to thiuk about Up to this point we have referred to it as a line that shows the trend of a data distribution, but know we need to know what exactly it is, how to compute it, and how to use it.

10 The Regression Equation Lesson 19.2 Sir Migo Mendoza

11 The Regression Equation: It is known as the formula in Algebra for plotting a line on a graph.

12 Regression Equation:

13 y It is the value we are trying to predict. For example, if we wanted to predict weight based on height, we would change our equation to read:

14 y

15 x It represents the value of the predictor variable. In this case, we know a person's height, so let's enter that into our equation:

16 x

17 b It represents the slope. This tells us how steep or flat the best-fit line is. We can enter it into our equation:

18 b

19 a It is the point on the y-axis where the line of best fit crosses for a given value of x; we call this the intercept.

20 a

21 Note: Since we do not know the value for weight but we do know the value for height, all we need to get this equation to work is to determine how to compute the intercept and the slope.

22 Computing the Slope Lesson 19.3 Sir Migo Mendoza

23 Question: What is a slope of a line?

24 Slope of a Line It is the change in the y value on a graph based on every incremental change in the x value on the same graph.

25 Graphs In order for us to a better conceptual understanding of the slope of a line, let's take a look at the following graphs:

26 Figure 19.1 Positive Slope

27 Interpretation: You can see the line labeled A starts at the zero and runs up, to the right, at a 45-degree angle (i.e., a positive slope). In this case, as x goes up 1 point, so does y. For example, you can see the line passes through the points (x = 1, y = 1), (x = 2, y = 2), and so on.

28 Figure 19.2 Negative Slope

29 Graphs In order for us to a better conceptual understanding of the slope of a line, let's take a look at the following graphs:

30 Interpretation: In the above figure, we have the opposite. The slope is negative; for each value that x goes up, the value for y goes down. This means that there is a negative correlation.

31 Figure 19.2 Zero Slope

32 Interpretation: You can see the slope is equal to zero; as the value of x goes up, the value of y stays the same.

33 Note: It stands to reason then that if we want to use regression to estimate one value based on another, we need to know the slope of the line.

34 Formula for Computing the Slope of the Line of Best Fit:

35 Example 19.1 Let's go back to our example where we want to predict a weight based on a height.

36 Table 19.1 Values Needed to Compute the Slope

37 Task 1: Using the formula, determine the slope.

38 Answer: The line of best fit is 3.24.

39 Computing the Intercept Lesson 19.4 Sir Migo Mendoza

40 Question: What do you remember about the intercept of a line?

41 Intercept It is the value of y when the value of x is zero.

42 Understanding the Intercept Let's use the following figure to help us better understand the concept of intercept.

43 Figure 19.4 Intercept of Zero

44 Interpretation: Here, when x is zero, so is y; this means our intercept is zero.

45 Figure 19.5 Intercept of 6

46 Interpretation: Here, when x is zero, y (i.e., our intercept) is 6.

47 Figure 19.6 Intercept of 3 with Zero Slope

48 Interpretation: The slope in the figure is 3; when x is zero, the line of best fit crosses the y-axis at 3.

49 Formula for Computing the Intercept of the Line of Best Fit:

50 Task 2: Using the formula, determine the intercept of the line of best fit.

51 Answer: The intercept of the line of best fit is

52 Note: Notice that the value of our intercept is negative; that is perfectly normal and happens from time to time.

53 Task 3: Using it, we have everything we need for our regression formula so let's pick a value of x (i.e., the height), 65 for example, and predict what our y value (i.e., the weight) should be.

54 Task 3: Determine the weight when height is 65.

55 Answer: For a person who is 65 inches tall, we are predicting a weight of

56 Figure 19.7 Regression Plot for Height and Weight In order to verify this figure, look at the point on the scatterplot in Figure 19.7 where 65 inches and 145 inches intersect.

57 Figure 19.7 Regression Plot for Height and Weight

58 Interpretation: When you find that point, you'll see that it is very near the line of best fit. This means our regression formula is a pretty good estimator of weight based on height.

59 Interpretation: I know most of you are surprised when we get to this point. After all of the computations, they want to know why our regression formula did not exactly predict the person's weight based on their height.

60 The Reason The correlation coefficient between height and weight is less than Because of that, error is introduced into the process, just as it was in the other statistical tests we have used. These verified by three things:

61 First: We can look at the scatterplot and see that all of the values do not fall directly onto the line of best fit.

62 Second: We can see that the standard errors for both the independent variable and the slope are greater than zero.

63 Third: We can compute a coefficient of determination by squaring Pearson's r value from the correlation This coefficient tells us that of the change in the criterion variable (i.e., weight) is caused by the predictor variable (i.e., height). The remaining is either due to error or other variables not considered in the equation.

64 Example 19.2 To better understand what we have mentioned above, let's imagine we had a dataset where there was a perfect correlation between the independent and dependent variables. In Table 19.2, you can see that, for every 1-inch difference in height, there is a 5-pound increase in weight.

65 Table 19.2 Height and Weight Data

66 Table 19.3 Pearson Correlation Coefficients between Height and Weight

67 Note: If we computed Pearson's r for the two values, as shown in Table 19.3, this would mean it would be exactly 1.00, indicating a perfect positive linear relationship.

68 Table 19.4 Inferential Statistics from the Linear Regression

69 Note: If we look at Table 19.4, we first see that our standard error is zero; given that, we do not expect error to interfere with our ability to use the regression procedure to accurately use the predictor variable to predict the criterion variable. We can also see our intercept (i.e., constant) and our slope (i.e., height). In this case if we know a person's height, we can accurately state their weight.

70 Task: If everything we have said up to this point is true, then we should be able to use the slope and height, along with a given value of the predictor variable, to accurately predict a criterion variable. Let's use a height of 64 and enter all of these data into our Regression Equation. What is weight when height is 64?

71 Answer: As we can see, for height of 64 inches, we are expecting the weight to be 120; we can verify by looking back at the table.

72 Task: Compute the Coefficient of Determination.

73 Answer: The coefficient of determination is 100%. This tells us 100% of the change in the criterion variable is caused by the predictor variable.

74 Figure 19.8 Scatterplot of the Height and Weight Data

75 Interpretation: As I said, however, this perfect prediction can only come true if there is a perfect linear relationship between the predictor and criterion variables. This perfect relationship, of course, happens very rarely. In the vast majority of cases, we can use our regression equation to help predict values, but we have to be aware of the error inherent in the process.

76 Let s Practice Chapter 19 Sir Migo Mendoza

77 Direction: Answer the following questions.

78 The Case of More Is Better Still You are to speak to a group of potential high school dropouts. At this point, you tell potential dropouts that students who drop out of school earlier tend to get married earlier and have more children. The following tables show the output of a regression procedure using highest level of education attained and average salary in the workplace.

79 Question 1: What would you tell the students about Pearson's r?

80 Question 2: What is the coefficient of determination? What does it tell us?

81 Question 3: In comparison to the overall average, what would you say to someone who was thinking about dropping out in the eighth grade?

82 Question 4: Generally speaking, in which direction would the line of best fit flow?

83 Question 5: What would you say to the students, in terms of living in poverty?

84 Table 19.5 Descriptive Statistics for the Number of Children and Highest Year of Education

85 Table 19.6 Small Negative Pearson Correlation Coefficient between Number of Children and Highest Year of Education

86 Table 19.7 Inferential Statistics from the Linear Regression