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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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



Collinearity
Collinearity skewed

  • 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 skewed

  • Check tolerance values

  • Plot the variables

  • Examine a matrix of correlation coefficients between predictor variables


Dealing with collinearity
Dealing with collinearity skewed

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


Correlations
Correlations skewed


Multiple and complex regression

(lnC skewed3)= βo+ β1(lat)+ β2(long)+ β3(latxlong)

After centering both lat and long



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)




Multiple and complex regression

R p predictors. 2=0.48

C3

Longitude

Latitude

Model Lat + Long


Multiple and complex regression

45 Lat p predictors.

35 Lat

Model Lat * Long


Multiple and complex regression

The final forward model selection is: p predictors.

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 p predictors.

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