1. Linear regression models

2. Purposes: To describe the linear relationship between two continuous variables, the response variable (y- axis) and a single predictor variable (x-axis) To determine how much of the variation in Y can be explained by the linear relationship with X and how much of this relationship remains unexplained To predict new values of Y from new values of X

3. The linear regression model is: ß 0 = population intercept (when x i =0) ß 1 = population slope, measures the change in Y per unit change in X ε i = random or unexplained error associated with the i th observation.

4. Linear relationship Y X ß0ß0 ß1ß1 1.0

5. Linear models approximate non-linear functions over a limited domain extrapolation interpolation

6. μ yi = β o + β 1 *x i + є i x1x1 x2x2 μ y1 μ y2 yiyi y i pred y i –y pred = residual xixi Fitting data to a linear model

7. The residual The residual sum of squares

8. The “best fit” estimates are the values that minimize the residual sum of squares (RSS) between each observed value and the predicted value of the model

9. Sum of squares Sun of cross products

10. Variance Covariance

11. Least-squares parameter estimates where

12. To solve the intercept

13. Variance of the error of regression

14. Variance components and Coefficient of determination

15. Coefficient of determination

16. Product-moment correlation coefficient

17. ANOVA table for regression SourceDegrees of freedom Sum of squaresMean square Expected mean square F ratio Regression 1 Residual n-2 Total n-1

18. Publication form of ANOVA table for regression Source Sum of Squaresdf Mean SquareFSig. Regression 11.4791 21.0440.00035 Residual 8.18215.545 Total 19.66116

19. Variance of estimated intercept

20. Variance of the slope estimator

21. Variance of the fitted value

22. Regression

23. Assumptions of regression The linear model correctly describes the functional relationship between X and Y The X variable is measured without error For a given value of X, the sampled Y values are independent with normally distributed errors Variances are constant along the regression line

24. Residual plot for species-area relationship

25. The influence function

26. Logistic regression

28. Height vs. survival in Hypericum cumulicola

29. Multiple regression

30. Relative abundance of C 3 and C 4 plants Paruelo & Lauenroth (1996) Geographic distribution and the effects of climate variables on the relative abundance of a number of plant functional types (PFTs): shrubs, forbs, succulents, C 3 grasses and C 4 grasses.

31. data Relative abundance of PTFs (based on cover, biomass, and primary production) for each site Longitude Latitude Mean annual temperature Mean annual precipitation Winter (%) precipitation Summer (%) precipitation Biomes (grassland, shrubland) 73 sites across temperate central North America Response variablePredictor variables

32. Box 6.1 Relative abundance transformed ln(dat+1) because positively skewed

33. Collinearity Causes computational problems because it makes the determinant of the matrix of X-variables close to zero and matrix inversion basically involves dividing by the determinant (very sensitive to small differences in the numbers) Standard errors of the estimated regression slopes are inflated

34. Detecting collinearlity Check tolerance values Plot the variables Examine a matrix of correlation coefficients between predictor variables

35. Dealing with collinearity Omit predictor variables if they are highly correlated with other predictor variables that remain in the model

37. Additive model (log 10 C 3 )= β o + β 1 (lat)+ β 2 (long)+ β 3 (map)+ β 4 (mat)+ β 5 (JJAmap)+ β 6 (DJFmap) (lnC 3 )= β o + β 1 (lat)+ β 2 (long)+ β 3 (map)+ β 4 (mat)+ β 5 (JJAmap)+ β 6 (DJFmap) Adding 0.1

38. Additive model (log 10 C 3 )= β o + β 1 (lat)+ β 2 (long)+ β 3 (map)+ β 4 (mat)+ β 5 (JJAmap)+ β 6 (DJFmap) (lnC 3 )= β o + β 1 (lat)+ β 2 (long)+ β 3 (map)+ β 4 (mat)+ β 5 (JJAmap)+ β 6 (DJFmap) Adding 0.1 Adding 1

39. (lnC 3 )= β o + β 1 (lat)+ β 2 (long)+ β 3 (latxlong) After centering both lat and long

40. If we omit the interaction and refit the model, the partial regression slope for latitude changes The collinearity problems disappear

42. R 2 =0.514

43. Matrix algebra approach to OLS estimation of multiple regression models Y=bX+ε X’Xb=X’Y b=(X’X) -1 (X’Y)

44. The forward selection is

45. The backward selection is

46. Adjusted r 2

47. Akaike information criteria

48. Bayesian information criteria

49. Hierarchical partitioning and model selection No predmodelr2r2 Adjr 2 CpCp AICSchwarz BIC 1 Lon 0.00005-0.014072.89-159.90-155.32 1 Lat 0.46180.45427.37-205.12-200.54 1 Lon x Lat 0.0062-0.007872.01-160.35-155.77 2 Lon + Lat 0.46710.45198.61-203.85-196.98 3 Long +Lat + Lon x Lat 0.51370.49264-208.53-199.67