Document ID: EPA-HQ-OAR-2011-0135-0635
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.2d
                              Final Model Fitting
                   Non-Methane Organic Gases (NMOG) (Bag 1)

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

I.2d.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). 

NMOG (Bag 1):  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 1-8 comprise the "superset" for final model-fitting).

     Models fit for NMOG (Bag 1): (all models include an intercept term).
Model Term
                                   Notation
                                     Model

                                       
                                   Superset
                                    SM2[1]
                                     etOH
Ze
                                      ●
                                      ●
                                     Arom
Za
                                      ●
                                      ●
                                      RVP
Zr
                                      ●
                                      ●
                                      T50
Z5
                                      ●
                                      ●
                                      T90
Z9
                                      ●
                                      x
                                 etOH x etOH
ZZee
                                      ●
                                      ●
                                  T50 x T50
ZZ55
                                      ●
                                      ●
                                 etOH x Arom
ZZea
                                      ●
                                      ●
                                  etOH x RVP
ZZer
                                      ---
                                      ---
                                  etOH x T50
ZZe5
                                      ●
                                      ●
                                  etOH x T90
ZZe9
                                      ●
                                      x
[1] denotes "Superset minus 2."
                                       
                                       
NMOG (Bag 1): 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[2]
                                                                             11
311.321
346.525

                                                                               
                                                                               
                                                                               
SM2
                                                                              9
318.083
347.871

                                                                          6.762
                                                                              2
                                                                          0.034

                                                                               

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

NMOG (Bag 1): Coefficients  and Tests of Effect for the Superset and Reduced Models, with respect to the 11-term design model. 
Effect
                                       
                                Superset Model

                           Reduced Model (Superset)

Estimate
Std.Err.
d.f.
t-value
Pr> t

                                                                       Estimate
Std.Err.
d.f.
t-value
Pr> t
Intercept
                                                                               
                                                                        -0.9520
                                                                        0.09077
                                                                             15
                                                                         -10.49
                                                                     <0.0001

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
Ze
                                                                               
                                                                        0.07981
                                                                        0.01326
                                                                            941
                                                                           6.02
                                                                     <0.0001

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
Za
                                                                               
                                                                        0.08789
                                                                        0.00929
                                                                            941
                                                                           9.46
                                                                     <0.0001

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
Zr
                                                                               
                                                                       -0.04595
                                                                        0.01053
                                                                            941
                                                                          -4.36
                                                                     <0.0001

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
Z5
                                                                               
                                                                         0.1344
                                                                        0.01329
                                                                            941
                                                                          10.12
                                                                     <0.0001

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
Z9
                                                                               
                                                                        0.01593
                                                                        0.00925
                                                                            941
                                                                           1.72
                                                                         0.0855

                        Reduced Model = Superset Model
ZZee
                                                                               
                                                                        0.04594
                                                                        0.01760
                                                                            941
                                                                           2.61
                                                                        0.00918

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
ZZ55
                                                                               
                                                                        0.07680
                                                                        0.01336
                                                                            941
                                                                           5.75
                                                                     <0.0001

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
ZZea
                                                                               
                                                                        0.01635
                                                                        0.00906
                                                                            941
                                                                           1.80
                                                                         0.0714

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
ZZer
                                                                               
                                                                              -
                                                                              -
                                                                              -
                                                                              -
                                                                              -

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
ZZe5
                                                                               
                                                                        0.04754
                                                                        0.01893
                                                                            941
                                                                           2.51
                                                                         0.0122

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
ZZe9
                                                                               
                                                                        0.01961
                                                                        0.00902
                                                                            941
                                                                           2.17
                                                                         0.0300

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
                                                                               

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               

                                                                               
                                                                         0.1224
                                                                               
                                                                               
                                                                               
                                                                               

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               

                                                                               
                                                                        0.07538
                                                                               
                                                                               
                                                                               
                                                                               

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               

I.2d.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).

NMOG (Bag 1):  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).
 

     Models fit for NMOG (Bag 1): (all models include an intercept term).
Model Term
                                   Notation
                                     Model

                                       
                                   Superset
                                    SM3[1]
                                      SM7
                                     etOH
Ze
                                      ●
                                      ●
                                      ●
                                     Arom
Za
                                      ●
                                      ●
                                      ●
                                      RVP
Zr
                                      ●
                                      ●
                                      ●
                                      T50
Z5
                                      ●
                                      ●
                                      ●
                                      T90
Z9
                                      ●
                                      ●
                                      x
                                 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
                                       

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

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

 NMOG (Bag 1): 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
                                                                             17
282.156
333.609

                                                                               
                                                                               
                                                                               
SM3[2]
                                                                             14
285.473
328.802

                                                                          3.317
                                                                              3
                                                                          0.345
SM7
                                                                             10
313.769
346.266

                                                                         28.296
                                                                              4
                                                                       0.000011

                                                                               

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

NMOG (Bag 1): Coefficients and Tests of Effect for the Superset and Reduced Models, with respect to the 16-term extended model.