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Multiple and complex regression

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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

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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.

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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

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

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Comparing l 10 vs ln

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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

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

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

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

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R 2 =0.514

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Analysis of variance Source of variation SSdfMS RegressionΣ(y hat -Y) 2 p p ResidualΣ(y obs -y hat ) 2 n-p-1Σ(y obs -y hat ) 2 n-p-1 TotalΣ(y obs -Y) 2 n-1

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Matrix algebra approach to OLS estimation of multiple regression models Y=βX+ε XXb=XY b=(XX) -1 (XY)

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The forward selection is

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The backward selection is

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Criteria for best fitting in multiple regression with p predictors. CriterionFormula r2r2 Adjusted r 2 Akaike Information Criteria AIC

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Hierarchical partitioning and model selection No predModelr2r2 Adjr 2 AIC (R)AIC 1 Lon 0.00005-0.01449.179-165.10 1 Lat 0.46190.4543.942-204.44 2 Lon + Lat 0.46710.45195.220-201.20 3 Long +Lat + Lon x Lat 0.51370.49260.437-209.69

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Multiple Linear Regression Response Variable: Y Explanatory Variables: X 1,...,X k Model (Extension of Simple Regression): E(Y) = + 1 X 1 + + k.

Multiple Linear Regression Response Variable: Y Explanatory Variables: X 1,...,X k Model (Extension of Simple Regression): E(Y) = + 1 X 1 + + k.

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