Document ID: EPA-HQ-OAR-2011-0135-0628
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.1d
                              Final Model Fitting
                       Total Hydrocarbons (THC) (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.1d.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 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 1-5 comprise the "superset" for final model-fitting).
      Models fit for THC (Bag 1): (all models include an intercept term).
Model Term
                                   Notation
                                     Model

                                       
                                   Superset
                                    SM1[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
                                      ●
                                      ●
                                       

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

 THC (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
                                                                             12
225.176
263.089

                                                                               
                                                                               
                                                                               
SM1[2]
                                                                             11
225.426
260.631

                                                                          0.250
                                                                              1
                                                                           0.62

                                                                               

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

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

I.1d.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 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 1-5 comprise the "superset" for final model-fitting).
 

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

                                       
                                   Superset
                                    SM2[1]
                                      SM6
                                     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
                                       
                                       
                                       

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

THC (Bag 1): 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
                                                                             16
196.798
245.543

                                                                               
                                                                               
                                                                               
SM2[2]
                                                                             14
199.598
242.927

                                                                          2.800
                                                                              2
                                                                          0.247
SM6
                                                                             10
229.637
262.134

                                                                         30.039
                                                                              4
                                                                      0.0000048

                                                                               

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

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

                              Reduced Model (SM2)

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

                                                                       Estimate
Std.Err.
d.f.
t-value
Pr> t
Intercept
                                                                               
                                                                        -0.8653
                                                                        0.09410
                                                                             15
                                                                          -9.20
                                                                     <0.0001

                                                                        -0.8658
                                                                        0.09407
                                                                             15
                                                                          -9.20
                                                                     <0.0001
Ze
                                                                               
                                                                        0.07607
                                                                        0.01436
                                                                            941
                                                                           5.30
                                                                     <0.0001

                                                                        0.06794
                                                                        0.01342
                                                                            941
                                                                           5.06
                                                                     <0.0001
Za
                                                                               
                                                                        0.08474
                                                                       0.009469
                                                                            941
                                                                           8.95
                                                                     <0.0001

                                                                        0.08344
                                                                        0.00943
                                                                            941
                                                                           8.85
                                                                     <0.0001
Zr
                                                                               
                                                                       -0.03584
                                                                        0.01264
                                                                            941
                                                                          -2.83
                                                                         0.0047

                                                                       -0.04669
                                                                        0.01016
                                                                            941
                                                                          -4.60
                                                                     <0.0001
Z5
                                                                               
                                                                         0.1602
                                                                        0.01622
                                                                            941
                                                                           9.88
                                                                     <0.0001

                                                                         0.1490
                                                                        0.01446
                                                                            941
                                                                          10.30
                                                                     <0.0001
Z9
                                                                               
                                                                        0.01582
                                                                       0.009786
                                                                            941
                                                                           1.62
                                                                           0.11

                                                                        0.01434
                                   0.009179
                                                                            941
                                                                           1.56
                                                                           0.12
ZZee
                                                                               
                                                                        0.04604
                                                                        0.01708
                                                                            941
                                                                           2.69
                                                                         0.0072

                                                                        0.04065
                                                                        0.01677
                                                                            941
                                                                           2.22
                                                                          0.016
ZZ55
                                                                               
                                                                         0.0818
                                                                        0.01354
                                                                            941
                                                                           6.04
                                                                     <0.0001

                                                                        0.07555
                                                                        0.01299
                                                                            941
                                                                           5.82
                                                                     <0.0001
ZZea
                                                                               
                                                                        0.03790
                                                                        0.01415
                                                                            941
                                                                           2.68
                                                                         0.0075

                                                                        0.02297
                                                                       0.008769
                                                                            941
                                                                           2.62
                                                                         0.0090
ZZer
                                                                               
                                                                       0.006772
                                                                       0.009678
                                                                            941
                                                                          0.700
                                                                           0.48

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
ZZe5
                                                                               
                                                                        0.03790
                                                                        0.01839
                                                                            941
                                                                           2.06
                                                                          0.040

                                                                        0.03426
                                                                        0.01827
                                                                            941
                                                                           1.87
                                                                          0.061
ZZe9
                                                                               
                                                                        0.04627
                                                                        0.01053
                                                                            941
                                                                           4.39
                                                                     <0.0001

                                                                        0.04782
                                                                        0.01048
                                                                            941
                                                                           4.56
                                                                     <0.0001

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
                                                                               

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
ZZar
                                                                               
                                                                        0.03977
                                                                        0.01517
                                                                            941
                                                                           2.62
                                                                         0.0089

                                                                        0.02553
                                                                        0.01206
                                                                            941
                                                                           2.12
                                                                          0.035
ZZa5
                                                                               
                                                                        0.02101
                                                                        0.01724
                                                                            941
                                                                           1.22
                                                                           0.22

                                                                               
                                                                               
                                                                               
                                                                               
                                                                               
ZZa9
                                                                               
                                                                        0.02087
                                                                       0.009634
                                                                            941
                                                                           2.17
                                                                          0.031

                                                                        0.02260
                                                                       0.009506
                                                                            941
                                                                           2.38
                                                                          0.018
ZZ59
                                                                               
                                                                        0.05255
                                                                        0.01161
                                                                            941
                                                                           4.53
                                                                     <0.0001

                                                                        0.05004
                                                                        0.01147
                                                                            941
                                                                           4.36
                                                                     <0.0001
ZZr9
                                                                               
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                                                                         0.1317
                                                                               
                                                                               
                                                                               
                                                                               

                                                                         0.1317
                                                                               
                                                                               
                                                                               
                                                                               

                                                                               
                                                                        0.06667
                                                                               
                                                                               
                                                                               
                                                                               

                                                                        0.06687