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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.1f

Final	Model	Fitting

Total	Hydrocarbons	(THC)	(Bag	3)

No. Observations: 956

No. Vehicles: 15 

No. censored measurements: 2

No. missing measurements: 0

No. measurements removed: 0

Model Type: Mixed model
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I.1f.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). 
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THC (Bag 3): 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 1-9 comprise the “superset” for final model-fitting).
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Models fit for THC (Bag 3): (all models include an intercept term).

Model Term Notation Model

Superset SM21SM4 SM52

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 2,” etc.

2 Not nested within SM4; test with respect to SM2.

THC (Bag 3): 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 10 527.403 559.900

SM2

2

8 530.138 557.218 2.735 2 0.255

SM4 6 556.555 534.890 4.753 2 0.0929

SM5 5 562.345 543.388 13.250 3 0.00413

1 A lower value indicates a better fit.

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

3 Not nested within SM4; test with respect to SM2.
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THC (Bag 3): Coefficients and Tests of Effect for the Superset and Reduced Models, with respect to 

the 11-term design model.
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I.1f.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 3): 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 1-9 comprise the “superset” for final model-fitting).
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NOTE: Model fitting for 16-term extended model has results identical to that for the 11-term 

design model, as presented above.