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Multiple and complex regression

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Multiple and complex 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

- 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)

73 sites across temperate central North America

Response variable

Predictor variables

Relative abundance transformed ln(dat+1) because positively 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

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

- 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

- Y=βX+ε
- X’Xb=XY
- b=(X’X) -1 (XY)

R2=0.48

C3

Longitude

Latitude

Model Lat + Long

45 Lat

35 Lat

Model Lat * Long

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

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