Multiple and complex regression

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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 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.

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data
73 sites across temperate central North America
Response variable
Predictor variables

• 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)

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Relative abundance transformed ln(dat1) because
positively skewed
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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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Correlations
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(lnC3) ßo ß1(lat) ß2(long) ß3(latxlong)
After centering both lat and long
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Analysis of variance
Source of variation SS df MS
Regression S(yhat-Y)2 p S(yhat-Y)2 p
Residual S(yobs-yhat)2 n-p-1 S(yobs-yhat)2 n-p-1
Total S(yobs-Y)2 n-1
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Matrix algebra approach to OLS estimation of
multiple regression models

• YßXe
• XXbXY
• b(XX) -1 (XY)

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Criteria for best fitting in multiple
regression with p predictors.
Criterion Formula
r2
Adjusted r2
Akaike Information Criteria AIC
Akaike Information Criteria AIC
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Hierarchical partitioning and model selection
No pred Model r2 Adjr2 P AIC (R)
1 Lon 0.0006 -0.013 0.84 30.15
1 Lat 0.47 0.46 gt0.001 -16.16
2 Lon Lat 0.48 0.46 gt0.001 -15.25
3 Long Lat Lon x Lat 0.54 0.52 gt0.001 -22.55
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R20.48
C3
Longitude
Latitude
Model Lat Long
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45 Lat
35 Lat
Model Lat Long
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The final forward model selection is
Step AIC-228.67 SQRT_C3 LAT MAP JJAMAP
DJFMAP Df Sum of Sq RSS AIC ltnonegt
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
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The final backward selection model is
Step AIC-229.32 SQRT_C3 LAT JJAMAP
DJFMAP Df Sum of Sq RSS
AIC ltnonegt 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