1. LBSRE1021 Data Interpretation Lecture 11 Correlation and Regression

2. Example Data DayOutput (TONS)Cost £000 12358 21750 32454 43564 51040 61643 71542 82450 91853 103062

3. The scatter diagram of the data would appear as below:

4. Alternatively a negative correlation would appear as below:

5. Alternatively data with no correlation may appear as below:

6. Correlation Scale -10+1 Perfect negative No correlation Perfect positive correlation correlation

7. Pearson’s product moment correlation coefficient (r) r = n ∑ xy - ∑x ∑y √ [n ∑x - (∑x)] [n ∑y - (∑y)] xyxyx y 23581334529 3364 1750850289 2500 24541296576 2916 ∑ 212 516114525000 27242

8. Pearson’s product moment correlation coefficient (r) (2) r = 10 x 11452 – 212 * 516 √ [10 x 5000 – (212)] [10 x 27242 – (516)] = 5128 √ 5056 x 6164 = 0.9186

9. Linear Regression Need to establish a ‘line of best fit’ The ‘freehand method’ has many drawbacks. In some sense we need the ‘best fit’ to the data. To obtain this we do not use crude graphical techniques. We identify the ‘line of best fit’ or ‘least squares line.’

10. Linear Regression (2)

11. Linear Regression (3) The equation of this line is Y =30.10 +1.014X But how is this obtained? The scattered points illustrate the actual data, while the least squares line is an estimate of Y for a given value of X. Notice the distance between the scattered points and the line; this will give you some idea of how good a fit the line is.

12. Linear Regression (4) How do we determine the least squares line? Simply we need to determine the intercept (a) and the (b) gradient. The formula is therefore Y = a + bx You need to apply a little calculus (we will omit that process here) to develop standard equations.

13. Linear Regression Equations b = n ∑ xy - ∑ x ∑ y n ∑ x - (∑ x) b = 10 x 11452 – 212 x 516 10 x 5000 – 44944 b = 1.0142405

14. Linear Regression Equations (2) And a = y – b.x a = 51.6 – 1.0142405 x 21.2 a = 30.098101 Rounding these values a little: Y = 30.10 + 1.014X

15. Coefficient of Determination The coefficient of determination measures the proportion of the variation in the dependent variable (y) explained by the variation in the independent variable (x). It is reported as r - the square of the product moment correlation coefficient.

16. Coefficient of Determination (2) For our previous example: r = 0.9186 = 0.844 This means that 84.4% of the variation in cost is dependent upon output volume. Alternatively, 15.6% of variation is not explained.

17. Summary Correlation is measured on a scale from -1 to +1 using Pearson’s product moment correlation coefficient (r). Linear regression identifies the line of ‘best fit’ using the formula Y = a + bx The coefficient of determination (r) measures the extent to which the dependent variable is explained by the independent variable.

18. Exam Question – May 2008 Q. 7. The data below shows annual company income (£m) against year of trading. YearIncome (£m) 120 223 326 428 535 A regression of income on year gives the following results: r = 0.974, r squared = 0.948, intercept = 11.4, slope = 3.5 a. Explain each of the results above (1 mark each). b. Use the results above to make a forecast for company income for year 6 (4marks). c. What assumption is made in making this forecast? (2marks).