
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 G.4
                              Final Model Fitting
                       Oxides of Nitrogen (NOx) (Bag 2)


No. Observations:			893
No. Vehicles:		  		  14	(drop Cobalt)   
No. censored measurements:		    0
No. missing measurements:		    0
No. measurements removed:		    0
Model Type:				Mixed model







G.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). 


NOx (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 1-10 comprise the "superset" for final model-fitting).


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

                                       
                                   Superset
                                    SM4[1]
                                      SM6
                                     etOH
Ze
                                      ●
                                      ●
                                      ●
                                     Arom
Za
                                      ●
                                      ●
                                      ●
                                      RVP
Zr
                                      ●
                                      ●
                                      x
                                      T50
Z5
                                      ●
                                      x
                                       
                                      T90
Z9
                                      ●
                                      x
                                       
                                 etOH x etOH
ZZee
                                      ●
                                      x
                                       
                                  T50 x T50
ZZ55
                                      ---
                                      ---
                                      ---
                                 etOH x Arom
ZZea
                                      ●
                                      x
                                       
                                  etOH x RVP
ZZer
                                      ●
                                      ●
                                      x
                                  etOH x T50
ZZe5
                                      ---
                                      ---
                                      ---
                                  etOH x T90
ZZe9
                                      ---
                                      ---
                                      ---
[1] denotes "Superset minus 4," etc.
                                       
                                       
 NOx (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
                                                                              9
1089.365
1118.395

                                                                               
                                                                               
                                                                               
SM4
                                                                              5
1089.700
1108.173

                                                                          0.335
                                                                              4
                                                                          0.987
SM6[2]
                                                                              3
1091.971
1105.166

                                                                          2.271
                                                                               
                                                                          0.321

                                                                               



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











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













G.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).

NOx (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 1-8 comprise the "superset" for final model-fitting). 


NOTE: the model-fitting process for the extended model gives identical results to those from the design model, as presented above.
