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Multiple and complex regression . Extensions of simple linear regression. Multiple regression models: predictor variables are continuous Analysis of variance: predictor variables are categorical (grouping variables),
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Extensions of simple linear regression • Multiple regression models: predictor variables are continuous • Analysis of variance: predictor variables are categorical (grouping variables), • But… general linear models can include both continuous and categorical predictors
Relative abundance of C3 and C4 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, C3 grasses and C4 grasses.
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) data 73 sites across temperate central North America Response variable Predictor variables
Box 6.1 Relative abundance transformed ln(dat+1) because positively skewed
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
Detecting collinearlity • Check tolerance values • Plot the variables • Examine a matrix of correlation coefficients between predictor variables
Dealing with collinearity • Omit predictor variables if they are highly correlated with other predictor variables that remain in the model
(lnC3)= βo+ β1(lat)+ β2(long)+ β3(latxlong) After centering both lat and long
Matrix algebra approach to OLS estimation of multiple regression models • Y=βX+ε • X’Xb=XY • b=(X’X) -1 (XY)
Criteria for “best” fitting in multiple regression with p predictors.