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	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
2

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). 
3

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 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).
5

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

Model Term Notation Model

Superset SM11

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 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>χ2

Superset 12 225.176 263.089

SM1211 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.
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THC (Bag 1): Coefficients and Tests of Effect for the Superset and Reduced Models, with respect 

to the 11-term design model.
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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).
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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).
9

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

Model Term Notation Model

Superset SM21SM6

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 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>χ2

Superset 16 196.798 245.543

SM2214 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.
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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 ----- ----- ----- ----- ----- ----- ----- ----- ----- -----

2  veh 0.1317 0.1317

2  0.06667 0.06687