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Multiple and complex regression. Extensions of simple linear regression Multiple regression models: predictor variables are continuous Analysis of variance:

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Presentation on theme: "Multiple and complex regression. Extensions of simple linear regression Multiple regression models: predictor variables are continuous Analysis of variance:"— Presentation transcript:

1 Multiple and complex regression

2 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

3 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.

4 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

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

6 Comparing l 10 vs ln

7 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

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

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

10

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

12 R 2 =0.514

13 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

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

15 The forward selection is

16 The backward selection is

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

18 Hierarchical partitioning and model selection No predModelr2r2 Adjr 2 AIC (R)AIC 1 Lon Lat Lon + Lat Long +Lat + Lon x Lat


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