Document ID: EPA-HQ-OAR-2011-0135-0623
Agency: epa
Document Type: Supporting & Related Material
Title: 
Posted Date: 2013-04-17T04:00Z

Assessing the Effect of Five Gasoline Properties on Exhaust Emissions from Light-Duty Vehicles certified to Tier-2 Standards
                                       
                       Analysis of Data from EPA Phase 3
                                       
                                       
                                (EPAct/V2/E-89)

                                 Appendix H.4
                              Final Model Fitting
                         Carbon Monoxide (CO) (Bag 2)

No. Observations:			956
No. Vehicles:		  		  15   
No. censored measurements:		    0
No. missing measurements:		    0
No. measurements removed:		    0
Model Type:				Mixed model

H.4.1   Model fitting with respect to the 11-term design model.
Design Model (11-terms):  Bayesian Information Criterion (BIC) vs. number of terms in model (p) for all possible models respecting hierarchy. 

Design Model (11-terms):  Bayesian Information Criterion (BIC) vs. number of terms in model (p) for all possible models respecting hierarchy (CLOSEUP of previous figure). 

CO (Bag 2):  Number of terms (p), Goodness-of-fit (BIC) and terms included in the 35 best-fitting candidate models (out of a total of 294 possible models with hierarchy).  (Terms included in models ranked 1-5 comprise the "superset" for final model-fitting).

      Models fit for CO (Bag 2): (all models include an intercept term).
Model Term
                                   Notation
                                     Model

                                       
                                   Superset
                                    SM3[1]
                                      SM4
                                     etOH
Ze
                                      ●
                                      ●
                                      x
                                     Arom
Za
                                      ●
                                      ●
                                      ●
                                      RVP
Zr
                                      ●
                                      ●
                                      ●
                                      T50
Z5
                                      ●
                                      ●
                                      ●
                                      T90
Z9
                                      ●
                                      ●
                                      ●
                                 etOH x etOH
ZZee
                                      ●
                                      x
                                       
                                  T50 x T50
ZZ55
                                      ●
                                      x
                                       
                                 etOH x Arom
ZZea
                                      ---
                                      ---
                                      ---
                                  etOH x RVP
ZZer
                                      ---
                                      ---
                                      ---
                                  etOH x T50
ZZe5
                                      ---
                                      ---
                                      ---
                                  etOH x T90
ZZe9
                                      ●
                                      x
                                       
                                       

                                       
                                       
                                       
[1] Represents "Superset minus 3," etc.

  CO (Bag 2): Model fitting history, starting with the 8-term superset model.
                                Fit Parameters

                             Test with respect to
                                Previous Model
Model
                                                                              p
-2lnL
BIC[1]

                                                                           Dev.
                                                                              d
                                                                    Pr>χ[2]
Superset
                                                                              9
828.049
857.838

                                                                               
                                                                               
                                                                               
SM3
                                                                              6
831.913
853.577

                                                                          3.864
                                                                              3
                                                                          0.277
SM4[2]
                                                                              5
832.664
851.620

                                                                          0.751
                                                                              1
                                                                          0.386

                                                                               

                                                                               
                                                                               
                                                                               
[1] A lower value indicates a better fit.
[2] Best fit with respect to the 11-term design model.

CO (Bag 2): Coefficients  and Tests of Effect for the Superset and Reduced Models, with respect to the 11-term design model. 

H.4.2	Model Fitting with respect to the 16-term extended Model.
Extended Model (16-terms):  Bayesian Information Criterion (BIC) vs. number of terms in model (p) for all possible models respecting hierarchy. 

Extended Model (16-terms):  Bayesian Information Criterion (BIC) vs. number of terms in model (p) for subset of models respecting hierarchy (CLOSEUP of previous figure).

CO (Bag 2):  Number of terms (p), Goodness-of-fit (BIC) and terms included in the 35 best-fitting candidate models (out of a total of 2,964 possible models with hierarchy).  (Terms included in models ranked 1-5 comprise the "superset" for final model-fitting).
 

      Models fit for CO (Bag 2): (all models include an intercept term).
Model Term
                                   Notation
                                     Model

                                       
                                   Superset
                                    SM3[1]
                                      SM4
                                     etOH
Ze
                                      ●
                                      ●
                                      x
                                     Arom
Za
                                      ●
                                      ●
                                      ●
                                      RVP
Zr
                                      ●
                                      ●
                                      ●
                                      T50
Z5
                                      ●
                                      ●
                                      ●
                                      T90
Z9
                                      ●
                                      ●
                                      ●
                                 etOH x etOH
ZZee
                                      ●
                                      x
                                       
                                  T50 x T50
ZZ55
                                      ●
                                      x
                                       
                                 etOH x Arom
ZZea
                                      ---
                                      ---
                                      ---
                                  etOH x RVP
ZZer
                                      ---
                                      ---
                                      ---
                                  etOH x T50
ZZe5
                                      ---
                                      ---
                                      ---
                                  etOH x T90
ZZe9
                                      ---
                                      ---
                                      ---
                                       

                                       
                                       
                                       
                                  Arom x RVP
ZZar
                                      ---
                                      ---
                                      ---
                                  Arom x T50
ZZa5
                                      ---
                                      ---
                                      ---
                                  Arom x T90
ZZa9
                                      ---
                                      ---
                                      ---
                                  T50 x T90
ZZ59
                                      ●
                                      ●
                                      ●
                                  RVP x T90
ZZr9
                                      ●
                                      x
                                       

[1] denotes "Superset minus 3, etc."

CO (Bag 2): Model fitting history, starting with the 9-term superset model.
                                Fit Parameters

                             Test with respect to
                                Previous Model
Model
                                                                              p
-2lnL
BIC[1]

                                                                           Dev.
                                                                              d
                                                                    Pr>χ[2]
Superset
                                                                             10
826.828
859.325

                                                                               
                                                                               
                                                                               
SM3
                                                                              7
828.868
853.241

                                                                          2.040
                                                                              3
                                                                          0.564
SM4[2]
                                                                              6
829.667
851.331

                                                                          0.798
                                                                              1
                                                                           0.37

                                                                               

                                                                               
                                                                               
                                                                               
[1] A lower value indicates a better fit.
[2] Best fit with respect to the 16-term extended model.

CO (Bag 2): Coefficients  and Tests of Effect for the Superset and Reduced Models, with respect to the 16-term extended model.