1

Assessing	the	Effect	of	Five	Gasoline	Properties	on	Exhaust	Emissions	from	LightDuty	Vehicles	certified	to	Tier-2	Standards

Analysis	of	Data	from	EPA	Phase	3

(EPAct/V2/E-89)

Appendix	H.4

Final	Model	Fitting

Carbon	Monoxide (CO)	(Bag	2)

No. Observations: 956

No. Vehicles: 15 

No. censored measurements: 0

No. missing measurements: 0

No. measurements removed: 0

Model Type: Mixed model
2

H.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). 
3

CO (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).
4

Models fit for CO (Bag 2): (all models include an intercept term).

Model Term Notation Model

Superset SM31SM4

etOH Ze ● ● ×

Arom Za ● ● ●

RVP Zr ● ● ●

T50 Z5 ● ● ●

T90 Z9 ● ● ●

etOH × etOH ZZee ● ×

T50 × T50 ZZ55 ● ×

etOH × Arom ZZea --- --- ---

etOH × RVP ZZer --- --- ---

etOH × T50 ZZe5 --- --- ---

etOH × T90 ZZe9 ● ×

1 Represents “Superset minus 3,” etc.

CO (Bag 2): Model fitting history, starting with the 8-term superset model.

Fit Parameters Test with respect to

Previous Model

Model p -2lnL BIC

1 Dev. d Pr>χ2

Superset 9 828.049 857.838

SM3 6 831.913 853.577 3.864 3 0.277

SM425 832.664 851.620 0.751 1 0.386

1 A lower value indicates a better fit.

2 Best fit with respect to the 11-term design model.
5

CO (Bag 2): Coefficients and Tests of Effect for the Superset and Reduced Models, with respect to 

the 11-term design model.
6

H.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).
7

CO (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).
8

Models fit for CO (Bag 2): (all models include an intercept term).

Model Term Notation Model

Superset SM3

1

SM4

etOH Ze ● ● ×

Arom Za ● ● ●

RVP Zr ● ● ●

T50 Z5 ● ● ●

T90 Z9 ● ● ●

etOH × etOH ZZee ● ×

T50 × T50 ZZ55 ● ×

etOH × Arom ZZea --- --- ---

etOH × RVP ZZer --- --- ---

etOH × T50 ZZe5 --- --- ---

etOH × T90 ZZe9 --- --- ---

Arom × RVP ZZar --- --- ---

Arom × T50 ZZa5 --- --- ---

Arom × T90 ZZa9 --- --- ---

T50 × T90 ZZ59 ● ● ●

RVP × T90 ZZr9 ● ×

1

denotes “Superset minus 3, etc.”

CO (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>χ2

Superset 10 826.828 859.325

SM3 7 828.868 853.241 2.040 3 0.564

SM4

2

6 829.667 851.331 0.798 1 0.37

1 A lower value indicates a better fit.

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

the 16-term extended model.