Multiple and complex regression
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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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Multiple and complex regression

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Multiple and complex regression

Multiple and complex regression


Extensions of simple linear regression

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 c 3 and c 4 plants

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.


Multiple and complex regression

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


Multiple and complex regression

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


Collinearity

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

Detecting collinearlity

  • Check tolerance values

  • Plot the variables

  • Examine a matrix of correlation coefficients between predictor variables


Dealing with collinearity

Dealing with collinearity

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


Correlations

Correlations


Multiple and complex regression

(lnC3)= βo+ β1(lat)+ β2(long)+ β3(latxlong)

After centering both lat and long


Analysis of variance

Analysis of variance


Matrix algebra approach to ols estimation of multiple regression models

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

Criteria for “best” fitting in multiple regression with p predictors.


Hierarchical partitioning and model selection

Hierarchical partitioning and model selection


Multiple and complex regression

R2=0.48

C3

Longitude

Latitude

Model Lat + Long


Multiple and complex regression

45 Lat

35 Lat

Model Lat * Long


Multiple and complex regression

The final forward model selection is:

Step: AIC=-228.67

SQRT_C3 ~ LAT + MAP + JJAMAP + DJFMAP

Df Sum of Sq RSS AIC

<none> 2.7759 -228.67

+ LONG 1 0.0209705 2.7549 -227.23

+ MAT 1 0.0001829 2.7757 -226.68

Call:

lm(formula = SQRT_C3 ~ LAT + MAP + JJAMAP + DJFMAP)

Coefficients:

(Intercept) LAT MAP JJAMAP DJFMAP

-0.7892663 0.0391180 0.0001538 -0.8573419 -0.7503936


Multiple and complex regression

The final backward selection model is

Step: AIC=-229.32

SQRT_C3 ~ LAT + JJAMAP + DJFMAP

Df Sum of Sq RSS AIC

<none> 2.8279 -229.32

- DJFMAP 1 0.26190 3.0898 -224.85

- JJAMAP 1 0.31489 3.1428 -223.61

- LAT 1 2.82772 5.6556 -180.72

Call:

lm(formula = SQRT_C3 ~ LAT + JJAMAP + DJFMAP)

Coefficients:

(Intercept) LAT JJAMAP DJFMAP

-0.53148 0.03748 -1.02823 -1.05164


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