Lecture 8 CORRELATION AND LINEAR REGRESSION
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1 Announcements CBA5 open in exam mode - deadline midnight Friday! Question 2 on this week s exercises is a prize question. The first good attempt handed in to me by 12 midday this Friday will merit a prize...
2 Assignment 2 The Minitab-based Assignment 2 which would have been set if there had been no disruption has been scrapped, and the material on Minitab is non-examinable. A revised assignment 2 is due to be set. Proposed deadline is Friday May 4th, 1pm, subject to confirmation (it won t be before that!)
3 Lecture 8 CORRELATION AND LINEAR REGRESSION
4 Introduction In this lecture we look at relationships between pairs of continuous variables. These might be
5 Introduction In this lecture we look at relationships between pairs of continuous variables. These might be height and weight
6 Introduction In this lecture we look at relationships between pairs of continuous variables. These might be height and weight market value and number of transactions
7 Introduction In this lecture we look at relationships between pairs of continuous variables. These might be height and weight market value and number of transactions temperatures and sales of ice cream
8 Introduction In this lecture we look at relationships between pairs of continuous variables. These might be height and weight market value and number of transactions temperatures and sales of ice cream We can examine the relationship between our two variables through a scatter diagram (plot one against the other).
9 Introduction In this lecture we look at relationships between pairs of continuous variables. These might be height and weight market value and number of transactions temperatures and sales of ice cream We can examine the relationship between our two variables through a scatter diagram (plot one against the other). Today we will look at how to examine such relationships more formally.
10 Example: ice cream sales The following data are monthly ice cream sales at Luigi Minchella s ice cream parlour.
11 Month Average Temp ( o C) Sales ( 000 s) January 4 73 February 4 57 March 7 81 April 8 94 May June July August September October November 7 81 December 5 80
12 Month Average Temp ( o C) Sales ( 000 s) January 4 73 February 4 57 March 7 81 April 8 94 May June July August September October November 7 81 December 5 80 Is there any relationship between temperature and sales?
13 Month Average Temp ( o C) Sales ( 000 s) January 4 73 February 4 57 March 7 81 April 8 94 May June July August September October November 7 81 December 5 80 Is there any relationship between temperature and sales? Can you DESCRIBE this relationship?
14
15 The scatter plot goes uphill, so we say there is a positive relationship between temperature and ice cream sales.
16 The scatter plot goes uphill, so we say there is a positive relationship between temperature and ice cream sales. As temperatures rise, so do ice cream sales!
17 The scatter plot goes uphill, so we say there is a positive relationship between temperature and ice cream sales. As temperatures rise, so do ice cream sales! If the scatter plot had shown a downhill slope, we would have a negative relationship.
18 The scatter plot goes uphill, so we say there is a positive relationship between temperature and ice cream sales. As temperatures rise, so do ice cream sales! If the scatter plot had shown a downhill slope, we would have a negative relationship. We could draw a straight line through the middle of most of the points. Thus, there is also a linear association present.
19 The scatter plot goes uphill, so we say there is a positive relationship between temperature and ice cream sales. As temperatures rise, so do ice cream sales! If the scatter plot had shown a downhill slope, we would have a negative relationship. We could draw a straight line through the middle of most of the points. Thus, there is also a linear association present. If we were to draw a line through the points, most points would lie close to this line. The closer the points lie to a line, the stronger the relationship.
20 Correlation We now look at how to quantify the relationship between two variables by calculating the sample correlation coefficient. This is
21 Correlation We now look at how to quantify the relationship between two variables by calculating the sample correlation coefficient. This is r = S XY SXX S YY, where
22 Correlation We now look at how to quantify the relationship between two variables by calculating the sample correlation coefficient. This is r = S XY SXX S YY, where S XY = ( xy ) n xȳ,
23 Correlation We now look at how to quantify the relationship between two variables by calculating the sample correlation coefficient. This is r = S XY SXX S YY, where S XY = S XX = ( ) xy n xȳ, ( ) x 2 n x 2,
24 Correlation We now look at how to quantify the relationship between two variables by calculating the sample correlation coefficient. This is r = S XY SXX S YY, where S XY = S XX = S YY = ( ) xy n xȳ, ( ) x 2 n x 2, ( ) y 2 nȳ 2.
25 r always lies between 1 and +1.
26 r always lies between 1 and +1. If r is close to +1, we have evidence of a strong positive (linear) association.
27 r always lies between 1 and +1. If r is close to +1, we have evidence of a strong positive (linear) association. If r is close to 1, we have evidence of a strong negative (linear) association.
28 r always lies between 1 and +1. If r is close to +1, we have evidence of a strong positive (linear) association. If r is close to 1, we have evidence of a strong negative (linear) association. If r is close to zero, there is probably no association!
29 r always lies between 1 and +1. If r is close to +1, we have evidence of a strong positive (linear) association. If r is close to 1, we have evidence of a strong negative (linear) association. If r is close to zero, there is probably no association! A correlation coefficient close to zero does not imply no relationship at all, just no linear relationship.
30 Example: ice cream sales The easiest way to calculate r is to draw up a table!
31 Example: ice cream sales The easiest way to calculate r is to draw up a table! Ü Ý Ü 2 Ý 2 ÜÝ 4 73
32 Example: ice cream sales The easiest way to calculate r is to draw up a table! Ü Ý Ü 2 Ý 2 ÜÝ
33 Example: ice cream sales The easiest way to calculate r is to draw up a table! Ü Ý Ü 2 Ý 2 ÜÝ
34 Example: ice cream sales The easiest way to calculate r is to draw up a table! Ü Ý Ü 2 Ý 2 ÜÝ
35 Example: ice cream sales The easiest way to calculate r is to draw up a table! Ü Ý Ü 2 Ý 2 ÜÝ
36 Example: ice cream sales The easiest way to calculate r is to draw up a table! Ü Ý Ü 2 Ý 2 ÜÝ
37 Example: ice cream sales The easiest way to calculate r is to draw up a table! Ü Ý Ü 2 Ý 2 ÜÝ
38 Example: ice cream sales The easiest way to calculate r is to draw up a table! Ü Ý Ü 2 Ý 2 ÜÝ
39 Example: ice cream sales The easiest way to calculate r is to draw up a table! Ü Ý Ü 2 Ý 2 ÜÝ
40 Example: ice cream sales The easiest way to calculate r is to draw up a table! Ü Ý Ü 2 Ý 2 ÜÝ
41 Notice in the formulae for S XY, S XX and S YY we need the sample means x and ȳ. Thus,
42 Notice in the formulae for S XY, S XX and S YY we need the sample means x and ȳ. Thus, x = = 10 and
43 Notice in the formulae for S XY, S XX and S YY we need the sample means x and ȳ. Thus, x = = 10 and ȳ = = 100.
44 Similarly,
45 Similarly, S XY = ( xy ) n xȳ = = 1362,
46 Similarly, S XY = ( xy ) n xȳ = = 1362, ( ) S XX = x 2 n x 2 = = 250 and
47 Similarly, S XY = ( xy ) n xȳ = = 1362, ( ) S XX = x 2 n x 2 = = 250 and ( ) S YY = y 2 nȳ 2 = = 7674.
48 Thus,
49 Thus, r = S XY SXX S YY
50 Thus, r = = S XY SXX S YY
51 Thus, r = = S XY SXX S YY = (to 3 decimal places).
52 A few points to remember...
53 A few points to remember... If your calculated correlation coefficient does not lie between 1 and +1, you ve done something wrong!
54 A few points to remember... If your calculated correlation coefficient does not lie between 1 and +1, you ve done something wrong! Check that your correlation coefficient agrees with your plot
55 A few points to remember... If your calculated correlation coefficient does not lie between 1 and +1, you ve done something wrong! Check that your correlation coefficient agrees with your plot an uphill slope ties in with a positive correlation coefficient;
56 A few points to remember... If your calculated correlation coefficient does not lie between 1 and +1, you ve done something wrong! Check that your correlation coefficient agrees with your plot an uphill slope ties in with a positive correlation coefficient; a downhill slope ties in with a negative correlation coefficient;
57 A few points to remember... If your calculated correlation coefficient does not lie between 1 and +1, you ve done something wrong! Check that your correlation coefficient agrees with your plot an uphill slope ties in with a positive correlation coefficient; a downhill slope ties in with a negative correlation coefficient; a random scattering of points indicates a correlation coefficient close to zero;
58 A few points to remember... If your calculated correlation coefficient does not lie between 1 and +1, you ve done something wrong! Check that your correlation coefficient agrees with your plot an uphill slope ties in with a positive correlation coefficient; a downhill slope ties in with a negative correlation coefficient; a random scattering of points indicates a correlation coefficient close to zero; the closer the points lie to a straight line (either uphill or downhill ), the closer to either +1 or 1 the correlation coefficient will be!
59 Simple linear regression A correlation analysis helps to establish whether or not there is a linear relationship between two variables. However, it doesn t allow us to use this linear relationship.
60 Simple linear regression A correlation analysis helps to establish whether or not there is a linear relationship between two variables. However, it doesn t allow us to use this linear relationship. Regression analysis allows us to use the linear relationship between two variables. For example, with a regression analysis, we can predict the value of one variable given the value of another.
61 Simple linear regression A correlation analysis helps to establish whether or not there is a linear relationship between two variables. However, it doesn t allow us to use this linear relationship. Regression analysis allows us to use the linear relationship between two variables. For example, with a regression analysis, we can predict the value of one variable given the value of another. To perform a regression analysis, we must assume that
62 Simple linear regression A correlation analysis helps to establish whether or not there is a linear relationship between two variables. However, it doesn t allow us to use this linear relationship. Regression analysis allows us to use the linear relationship between two variables. For example, with a regression analysis, we can predict the value of one variable given the value of another. To perform a regression analysis, we must assume that the scatter plot of the two variables (roughly) shows a straight line, and
63 Simple linear regression A correlation analysis helps to establish whether or not there is a linear relationship between two variables. However, it doesn t allow us to use this linear relationship. Regression analysis allows us to use the linear relationship between two variables. For example, with a regression analysis, we can predict the value of one variable given the value of another. To perform a regression analysis, we must assume that the scatter plot of the two variables (roughly) shows a straight line, and the spread in the Y direction is roughly constant with X.
64
65 Look at the scatter plot of ice cream sales against temperature.
66 Look at the scatter plot of ice cream sales against temperature. A line of best fit can be drawn through the data;
67 Look at the scatter plot of ice cream sales against temperature. A line of best fit can be drawn through the data; This line could then be used to make predictions of ice cream sales based on temperature.
68 Look at the scatter plot of ice cream sales against temperature. A line of best fit can be drawn through the data; This line could then be used to make predictions of ice cream sales based on temperature. However, everyone s line is bound to be slightly different!
69 Look at the scatter plot of ice cream sales against temperature. A line of best fit can be drawn through the data; This line could then be used to make predictions of ice cream sales based on temperature. However, everyone s line is bound to be slightly different! And so everyone s predictions will be slightly different!
70 Look at the scatter plot of ice cream sales against temperature. A line of best fit can be drawn through the data; This line could then be used to make predictions of ice cream sales based on temperature. However, everyone s line is bound to be slightly different! And so everyone s predictions will be slightly different! The aim of regression analysis is to find the best line which goes through the data.
71 Look at the scatter plot of ice cream sales against temperature. A line of best fit can be drawn through the data; This line could then be used to make predictions of ice cream sales based on temperature. However, everyone s line is bound to be slightly different! And so everyone s predictions will be slightly different! The aim of regression analysis is to find the best line which goes through the data. At this point, we should remind ourselves about the equation for a straight line and how to draw it!
72 The regression equation
73 The regression equation The simple linear regression model is given by Y = α+βx +ǫ, where
74 The regression equation The simple linear regression model is given by Y = α+βx +ǫ, where Y is the response variable and
75 The regression equation The simple linear regression model is given by Y = α+βx +ǫ, where Y is the response variable and X is the explanatory variable.
76 The regression equation The simple linear regression model is given by where Y = α+βx +ǫ, Y is the response variable and X is the explanatory variable. ǫ is the random error taken to be zero on average, with constant variance.
77 The regression equation The simple linear regression model is given by where Y = α+βx +ǫ, Y is the response variable and X is the explanatory variable. ǫ is the random error taken to be zero on average, with constant variance. α represents the intercept of the regression line (the point where the line cuts the Y axis),
78 The regression equation The simple linear regression model is given by where Y = α+βx +ǫ, Y is the response variable and X is the explanatory variable. ǫ is the random error taken to be zero on average, with constant variance. α represents the intercept of the regression line (the point where the line cuts the Y axis), β represents the slope of the regression line (i.e. how steep the line is), and
79 The regression equation The simple linear regression model is given by where Y = α+βx +ǫ, Y is the response variable and X is the explanatory variable. ǫ is the random error taken to be zero on average, with constant variance. α represents the intercept of the regression line (the point where the line cuts the Y axis), β represents the slope of the regression line (i.e. how steep the line is), and We need to estimate α and β from our sample data. But how?
80 Remember, the aim of regression analysis is to find a line which goes through the middle of the data in the scatter plot, closer to the points than any other line;
81 Remember, the aim of regression analysis is to find a line which goes through the middle of the data in the scatter plot, closer to the points than any other line; So the best line will minimise any gaps between the line and the data!
82 Remember, the aim of regression analysis is to find a line which goes through the middle of the data in the scatter plot, closer to the points than any other line; So the best line will minimise any gaps between the line and the data! It turns out that the values of α and β which give this best line are ˆβ = S XY S XX and
83 Remember, the aim of regression analysis is to find a line which goes through the middle of the data in the scatter plot, closer to the points than any other line; So the best line will minimise any gaps between the line and the data! It turns out that the values of α and β which give this best line are ˆβ = S XY S XX and ˆα = ȳ ˆβ x.
84 Example: ice cream sales For the ice cream sales data, we can find the best line using the formulae on the previous slide! Thus
85 Example: ice cream sales For the ice cream sales data, we can find the best line using the formulae on the previous slide! Thus and ˆβ = S XY S XX = = 5.448,
86 Example: ice cream sales For the ice cream sales data, we can find the best line using the formulae on the previous slide! Thus and ˆβ = S XY S XX = = 5.448, ˆα = ȳ ˆβ x = =
87 Thus, the regression equation is
88 Thus, the regression equation is Ý = Ü
89
90 Making predictions We can use our estimated regression equation to make predictions of ice cream sales based on average temperature.
91 For example: Predict monthly sales if the average temperature is 10 o C.
92 For example: Predict monthly sales if the average temperature is 10 o C. We can use take a reading from our graph, or, more accurately, use our regression equation! y = x = = = 100, i.e.
93 For example: Predict monthly sales if the average temperature is 10 o C. We can use take a reading from our graph, or, more accurately, use our regression equation! y = x = = = 100, i.e. if the average temperature is 10 o C, we can expect sales of 100,000. i.e.
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