1. Quantitative Methods Regression

2. 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. x y M Y Regression Geometry

5. x y M Y Regression Geometry

6. x y M Y Regression Geometry

7. x y M Y Regression Geometry

8. x y M Y Regression Geometry

9. x y M Y Regression Geometry

10. x y M Y F1F1 Regression Geometry

11. x y M Y F1F1 Regression Geometry

12. x y M Y Sum of squares of residuals = Squared distance from Y to F 1 F1F1 Regression Geometry

13. x y M Y Regression Geometry

14. M Y F1F1 F2F2 F3F3 x y Regression Geometry

15. M Y F1F1 F2F2 F3F3 x y Regression Geometry

16. Regression Geometry

17. Regression Geometry

18. Regression Minitab commands

19. Regression Minitab commands

20. Regression Minitab commands

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

22. Regression Regression Output

23. Regression Regression Output

24. Regression Regression Output

25. Regression Confidence intervals and t-tests

26. Regression estimate ± t crit  Standard Error of estimate Coef ± t crit (on 29 DF)  SECoef 1.5433 ± 2.0452  0.3839 = (0.758, 2.328) Confidence intervals and t-tests t crit is always on Error degrees of freedom

27. Regression Confidence intervals and t-tests

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

29. Regression Confidence intervals and t-tests

30. Regression Confidence intervals and t-tests

31. Regression Regression output

32. Regression Regression output

33. Regression Extreme residuals

34. Regression Outliers

35. Regression Regression output

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

37. 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

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

39. 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.

40. Regression Last words… Next week: Models, parameters and GLMs Read Chapter 3 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.