1. Quantitative Methods Regression

2. Regression Examples for linear regression • Do more brightly coloured birds have more parasites? • How should we estimate merchantable volume of wood from the height of a living tree? • How is pest infestation late in the season affected by the concentration of insecticide applied early in the season?

3. Regression Similarities to analysis of variance

4. Regression Geometry y Y M x

5. Regression Geometry y Y M x

6. Regression Geometry y Y M x

7. Regression Geometry y Y M x

8. Regression Geometry y Y M x

9. Regression Geometry y Y M x

10. Regression Geometry y Y M F1 x

11. Regression Geometry y Y M F1 x

12. Regression Geometry y Y M F1 x Sum of squares of residuals = Squared distance from Y to F1

13. Regression Geometry y Y M x

14. Regression Geometry y Y M F1 F2 F3 x

15. Regression Geometry y Y M F1 F2 F3 x

16. Regression

17. Regression Geometry

18. Regression Geometry

19. Regression Geometry

20. Regression Geometry

21. Regression Geometry

22. Regression Minitab commands

23. Regression Minitab commands

24. Regression Minitab commands

25. Regression Minitab commands Minitab Supplement is in a PDF file in the same directory as the dataset.

26. Regression Regression Output

27. Regression Confidence intervals and t-tests

28. Regression Confidence intervals and t-tests estimate ± tcrit Standard Error of estimate Coef ± tcrit (on 29 DF)  SECoef 1.5433 ± 2.0452  0.3839 = (0.758, 2.328)

29. Regression Confidence intervals and t-tests

30. vs Regression Confidence intervals and t-tests t = distance between estimate and hypothesised value, in units of standard error

31. Regression Confidence intervals and t-tests

32. Regression Confidence intervals and t-tests

33. Regression Regression output

34. Regression SS and DF again

35. Regression Regression output

36. Regression Extreme residuals

37. Regression Outliers

38. Regression Regression output

39. Regression Four possible outcomes Low p-value: significant High p-value: non-significant Low R-sq High R-sq

40. Regression Difference from analysis of variance Continuous vs Categorical • Continuously varying • Values have meaning as numbers • Values are ordered • Interpolation makes sense • Examples: • height • concentration • duration • Discrete values • Values are just “names” that define subsets • Values are unordered • Interpolation is meaningless • Examples • drug • breed of sheep • sex

41. Regression Why linear? • Not because relationships are linear • Good simple starting point - cf recipes • Approximation to a smoothly varying curve

42. Regression Last words… • Regression is a powerful and simple tool, very commonly used in biology • Regression and ANOVA have deep similarities • Learn the numerical skills of calculating confidence intervals and testing for non-zero slopes. Next week: Models, parameters and GLMs Read Chapter 3