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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	I.1e

Final	Model	Fitting

Total	Hydrocarbons	(THC)	(Bag	2)

No. Observations: 832

No. Vehicles: 13 (exclude Odyssey and Sienna)

No. censored measurements: 0

No. missing measurements: 0

No. measurements removed: 0

Model Type: Mixed model
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I.1e.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). 
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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). 
4

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

Model Term Notation Model

Superset SM3

1

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.

THC (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>χ2

Superset 10 195.334 226.113

SM3

2

7 201.213 224.297 5.879 3 0.118

1 A lower value indicates a better fit.

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

to the 11-term design model.
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I.1e.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).
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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).
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THC (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).
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Models fit for THC (Bag 2): (all models include an intercept term).

Model Term Notation Model

Superset SM1

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 1, etc.”

THC (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>χ2

Superset 11 190.795 224.139

SM1

2

10 192.466 223.245 1.671 1 0.196

SM4

3

7 203.136 226.220 10.670 3 0.0137

1 A lower value indicates a better fit.

2 Best fit with respect to the 16-term extended model.

3

Fit because the linear term for Aromatics appears insignificant in 

The superset model; therefore, dropped aromatics and its interactions.

Test results indicate that these three terms contribute to fit.
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THC (Bag 2): Coefficients and Tests of Effect for the Superset and Reduced Models, with respect 

to the 16-term extended model.
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