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.4n
                              Final Model Fitting
                             Methane (CH4) (Bag 2)


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








I.4n.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). 














CH4 (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 1-6 comprise the "superset" for final model-fitting).

      Models fit for CH4 (Bag 2): (all models include an intercept term).
Model Term
                                   Notation
                                     Model

                                       
                                   Superset
                                    SM2[1]
                                     etOH
Ze
                                      ●
                                      ●
                                     Arom
Za
                                      ●
                                      ●
                                      RVP
Zr
                                      ●
                                      ●
                                      T50
Z5
                                      ●
                                      ●
                                      T90
Z9
                                      ●
                                      ●
                                 etOH x etOH
ZZee
                                      ---
                                      ---
                                  T50 x T50
ZZ55
                                      ●
                                      ●
                                 etOH x Arom
ZZea
                                      ●
                                      ●
                                  etOH x RVP
ZZer
                                      ●
                                      x
                                  etOH x T50
ZZe5
                                      ---
                                      ---
                                  etOH x T90
ZZe9
                                      ●
                                      x
                                       

                                       
                                       
[1] Represents "Superset minus 2," etc.

 CH4 (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
-691.760
-659.263

                                                                               
                                                                               
                                                                               
SM1[2]
                                                                              8
-687.712
-660.632

                                                                          4.048
                                                                              2
                                                                          0.132

                                                                               



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












CH4 (Bag 2): Coefficients and Tests of Effect for the Superset and Reduced Models, with respect to the 11-term design model. 












I.4n.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).















CH4 (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 CH4 (Bag 2): (all models include an intercept term).
Model Term
                                   Notation
                                     Model

                                       
                                   Superset
                                      SM1
                                     etOH
Ze
                                      ●
                                      ●
                                     Arom
Za
                                      ●
                                      ●
                                      RVP
Zr
                                      ●
                                      ●
                                      T50
Z5
                                      ●
                                      ●
                                      T90
Z9
                                      ●
                                      ●
                                 etOH x etOH
ZZee
                                      ---
                                      ---
                                  T50 x T50
ZZ55
                                      ●
                                      ●
                                 etOH x Arom
ZZea
                                      ●
                                      ●
                                  etOH x RVP
ZZer
                                      ●
                                      x
                                  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
                                      ●
                                      ●
                                       

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

 CH4 (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
-700.789
-665.585

                                                                               
                                                                               
                                                                               
SM1[2]
                                                                             10
-699.537
-667.041

                                                                          1.252
                                                                              1
                                                                          0.263

                                                                               



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











CH4 (Bag 2): Coefficients  and Tests of Effect for the Superset and Reduced Models, with respect to the 16-term extended model. 








