Multiple and complex regression
Sponsored Links
This presentation is the property of its rightful owner.
1 / 22

Multiple and complex regression PowerPoint PPT Presentation


  • 99 Views
  • Uploaded on
  • Presentation posted in: General

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),

Download Presentation

Multiple and complex regression

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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),

  • But… general linear models can include both continuous and categorical predictors


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.


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


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


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

  • Check tolerance values

  • Plot the variables

  • Examine a matrix of correlation coefficients between predictor variables


Dealing with collinearity

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


Correlations


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

After centering both lat and long


Analysis of variance


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.


Hierarchical partitioning and model selection


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


  • Login