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Quantitative Methods

Quantitative Methods. Regression. 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?

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Quantitative Methods

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  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

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