Document ID: EPA-HQ-OAR-2011-0135-0629
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 I.1e
                              Final Model Fitting
                       Total Hydrocarbons (THC) (Bag 2)

No. Observations:			832
No. Vehicles:		  		  13   (exclude Odyssey and Sienna)
No. censored measurements:		    0
No. missing measurements:		    0
No. measurements removed:		    0
Model Type:				Mixed model

I.1e.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). 

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). 

THC (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 THC (Bag 2): (all models include an intercept term).
Model Term
                                   Notation
                                     Model

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

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

 THC (Bag 2): Model fitting history, starting with the 10-term superset model.
                                Fit Parameters

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

                                                                           Dev.
                                                                              d
                                                                    Pr>χ[2]
Superset
                                                                             10
195.334
226.113

                                                                               
                                                                               
                                                                               
SM3[2]
                                                                              7
201.213
224.297

                                                                          5.879
                                                                              3
                                                                          0.118

                                                                               

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

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

I.1e.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).

 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).

THC (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 THC (Bag 2): (all models include an intercept term).
Model Term
                                   Notation
                                     Model

                                       
                                   Superset
                                    SM1[1]
                                      SM4
                                     etOH
Ze
                                      ●
                                      ●
                                      ●
                                     Arom
Za
                                      ●
                                      ●
                                      x
                                      RVP
Zr
                                      ●
                                      ●
                                      ●
                                      T50
Z5
                                      ●
                                      ●
                                      ●
                                      T90
Z9
                                      ●
                                      ●
                                      ●
                                 etOH x etOH
ZZee
                                      ---
                                       
                                       
                                  T50 x T50
ZZ55
                                      ●
                                      ●
                                      ●
                                 etOH x Arom
ZZea
                                      ●
                                      ●
                                      x
                                  etOH x RVP
ZZer
                                      ●
                                      ●
                                      ●
                                  etOH x T50
ZZe5
                                      ---
                                       
                                       
                                  etOH x T90
ZZe9
                                      ---
                                       
                                       
                                       

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

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

THC (Bag 2): Model fitting history, starting with the 15-term superset model.
                                Fit Parameters

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

                                                                           Dev.
                                                                              d
                                                                    Pr>χ[2]
Superset
                                                                             11
190.795
224.139

                                                                               
                                                                               
                                                                               
SM1[2]
                                                                             10
192.466
223.245

                                                                          1.671
                                                                              1
                                                                          0.196
SM4[3]
                                                                              7
203.136
226.220

                                                                         10.670
                                                                              3
                                                                         0.0137

                                                                               

                                                                               
                                                                               
                                                                               
[1] A lower value indicates a better fit.
[2] Best fit with respect to the 16-term extended model.
[3] Fit because the linear term for Aromatics appears insignificant in 
The superset model; therefore, dropped aromatics and its interactions.
Test results indicate that these three terms contribute to fit.

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