1. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Chapter 8: Working with Possible Relationships

2. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Objectives Be able to discuss whether a possible relationship exists Calculate and comment on the correlation coefficient Describe a simple relationship using an equation Make a forecast Be aware of multiple regression

3. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Plotting the data We need to decide whether a relationship exists between two variables of interest The variable we are trying to explain (plotted on the y axis) is referred to as the dependent variable The variable being used for explanation (plotted on the x axis) is referred to as the independent variable

4. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage In this example, we are looking at the impact of advertising on sales

5. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage In this case: We would talk about a positive relationship as the points lie on an upward sloping line Of interest is the correspondence of high sales with high levels of advertising We need to assess how good the relationship is

6. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Correlation Measures the strength of the linear relationship between two variables Correlations takes a value between –1 and +1 A correlation of +1 indicates a perfect positive correlation, -1 a perfect negative correlation and 0 no correlation

7. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage A correlation measure of +1

8. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage A correlation measure of -1

9. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage A correlation measure of 0 suggesting no relation

10. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage A correlation measure of 0 confirming no linear relationship

11. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Pearson’s correlation coefficient or just the correlation coefficient (r)

12. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Interpretation: r =+1 – perfect positive correlation r =-1 – perfect negative correlation r = 0 – no correlation r close to +1 – high degree of positive correlation r close to -1 – high degree of negative correlation r close to 0 – little correlation

13. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage xyxyx2x2 y2y2 26012043,600 5100 470 690 380 xx yy  xy x2x2 y2y2 The data used for the scatter graphs (n=5)

14. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage The summary statistics required:  x = 20,  y = 400,  xy = 1,680  x 2 = 90,  y 2 = 33,000 and n =5 Be sure that you can do these calculations

15. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage By substitution: By doing the sums you will find that r = 0.8

16. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage We need to give some interpretation r fairly close to +1 Reasonably high degree of positive correlation Low values of advertising tend to be associated with low sales High values of advertising tend to be associated with high sales

17. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Regression Having established that a relationship is likely we then attempt to describe the relationship with an equation But we do need to be aware of spurious correlations: Indirect connection (third common factor) Coincidence

18. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage xyxyx2x2 y2y2 26012043,600 5100 470 690 380 xx yy  xy x2x2 y2y2 We again need the summary statistics (n=5)

19. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage The summary statistics required:  x = 20,  y = 400,  xy = 1,680  x 2 = 90 and n =5 The equation of a straight line is given by y = a + bx and we need to find a and b

20. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage We find a and b using the following:

21. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage By substitution:

22. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Again by substitution (using the value b = 8)

23. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage This gives the equation y = 48 + 8x Wherea is the intercept and b is the gradient The interpretation of b = 8, is that each unit increase advertising produces an 8 unit increase in sales. But note – real business relationship are usually more complex than this.

24. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage The equation is given by y = 48 + 8x When advertising is £2500 (let x = 2.5) then y = 48 + 8 x 2.5 = 68 The sales forecast is £68000 Making a Forecast These figures are within the current range of the data (called interpolation) and we would feel comfortable with such a forecast.

25. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage These figures are outside the current range of the data (called extrapolation) and we would have concerns about such a forecast. When advertising is £8000 (let x = 8) then y = 48 + 8 x 8 = 112 The sales forecast is therefore £68000

26. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Multiple Regression It is likely that the variable of interest to us e.g. sales, is best explained by a number of factors e.g. price, advertising, perceived quality. To allow for a number of factors the predictive equation would take the form:

27. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage When building a multiple regression model two or more (2 have been shown) predictor variables can be used Multiple regression is available on Excel but the you do need to be careful with the interpretation of the outcomes.

28. Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Thomson Learning 2004 Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN 1-86152-991-0 © Cengage Conclusions Correlation provides a measure of how well a straight line will fit a scatter of data Regression gives the equation of a line that will best fit (using the least squares criteria) the data We can forecast from the regression line but need to be aware of the limitations of such forecasting methods