Ecological Archives E084-073-A3

Michael H. Graham. 2003. Confronting multicollinearity in ecological multiple regression. Ecology 84:2809–2815.

Appendix C. Principal components analysis (PCA) and SYSTAT output for backwards stepwise principal components regression. PC1-4 are the saved principal components scores after the PCA was done on the original explanatory variables (same as used in Appendix A). The PCA was run using FACTOR:PRINCIPAL COMPONENTS under the STATS menu in SYSTAT 5.2 for Macintosh. The PCA utilized the correlation matrix with no factor rotations. Backwards stepwise multiple regression was then run on the transformed data using MGLH:REGRESSION under the STATS menu in SYSTAT 5.2 for Macintosh. Response was the dependent variable (same as used in Appendix A); CONSTANT, PC1, PC2, PC3, and PC4 were the dependent variables; Stepwise was set to custom, with a backwards step order, and P = 0.15 to remove. Only standard SYSTAT output is presented.

Principal components analysis of OD, BD, LTD, and W
         
LATENT ROOTS (EIGENVALUES)
         
         
PC1 PC2 PC3 PC4  
         
2.56537 0.80552 0.37082 0.25829  
         
         
COMPONENT LOADINGS
         
VARIABLE PC1 PC2 PC3 PC4
         
OD 0.87771 0.26037 -0.09692 -0.39044
         
BD 0.87347 0.16099 -0.35373 0.29328
         
LTD -0.54211 0.83786 0.04084 0.04939
         
W 0.85917 0.09901 0.484399 0.13188
         
         
VARIANCE EXPLAINED BY COMPONENTS
         
PC1 PC2 PC3 PC4  
         
2.56537 0.80552 0.37082 0.25829  
         
         
PERCENT OF TOTAL VARIANCE EXPLAINED
         
PC1 PC2 PC3 PC4  
         
64.1342 20.1380 9.2706 6.4572  

 

Backwards stepwise variable selection
 
DEPENDENT VARIABLE: RESPONSE
 
MINIMUM TOLERANCE FOR ENTRY INTO MODEL = 0.010000

 

Initial model
             
STEP #0; R = 0.775; RSQUARE = 0.601
             
IN            
             
---            
             
VARIABLE COEFF SE STD COEFF TOLER F P
             
1 CONST            
             
2 PC1 0.1571 0.0235 0.73 1.0 45.0 0.0000
             
3 PC2 0.0267 0.0235 0.12 1.0 1.29 0.2651
             
4 PC3 0.0219 0.0235 0.10 1.0 0.87 0.3564
             
5 PC4 -0.0398 0.0235 -0.19 1.0 2.86 0.1003
             
             
OUT PART. CORR          
             
---
             
none            

 

1st variable removal
             
SREP #1; R = 0.768; RSQUARE = 0.590
             
TERM REMOVED: PC3
             
IN
             
---
             
VARIABLE COEFF SE STD COEF TOLER F P
             
1 CONST            
             
2 PC1 0.1571 0.0235 0.73 1.0 45.0 0.0000
             
3 PC2 0.0267 0.0235 0.12 1.0 1.29 0.2640
             
5 PC4 -0.0398 0.0235 -0.19 1.0 2.87 0.0994
             
             
OUT PART. CORR          
             
---
             
4 PC3 0.161 N/A N/A 1.0 0.87 0.3564

 

2nd variable removal
             
STEP #2; R = 0.758; RSQUARE = 0.574
             
TERM REMOVED: PC2
             
IN
             
---
             
VARIABLE COEFF SE STD COEF TOLER F P
             
1 CONST            
             
2 PC1 0.1571 0.0235 0.73 1.0 44.0 0.0000
             
5 PC4 -0.0398 0.0236 -0.19 1.0 2.85 0.1005
             
             
OUT PART. CORR          
             
---
             
3 PC2 0.19 N/A N/A 1.0 1.29 0.2640
             
4 PC3 0.158 N/A N/A 1.0 0.87 0.3582
             
             
THE SUBSET MODEL INCLUDES THE FOLLOWING PREDICTORS:
             
CONSTANT
             
PC1
 
PC4

 

Parameterization of final model
             
DEP VAR: RESPONSE N: 38; MULTIPLE R: 0.758 SQUARED MULTIPLE R: 0.574; ADJUSTED SQUARED MULTIPLE R: 0.550 STANDARD ERROR OF ESTIMATE: 0.143
             
VARIABLE COEFF SE STD COEF TOLER F P
             
CONST 3.2498 0.0233 0.00 N/A 140.0 0.0000
             
PC1 0.1571 0.024 0.74 1.0 6.65 0.0000
             
PC4 -0.0398 0.024 -0.19 1.0 -1.69 0.1005

 

Analysis of variance
           
SOURCE SS DF MS F P
           
REGRESSION 0.97 2 0.49 23.6 0.0000
           
RESIDUAL 0.72 1 35 0.02  

 



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