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
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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
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DEPENDENT VARIABLE: RESPONSE |
MINIMUM TOLERANCE FOR ENTRY INTO MODEL = 0.010000 |
Initial
model
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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
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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
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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
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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
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SOURCE | SS | DF | MS | F | P |
REGRESSION | 0.97 | 2 | 0.49 | 23.6 | 0.0000 |
RESIDUAL | 0.72 | 1 | 35 | 0.02 |