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.3d

Final	Model	Fitting

Non-Methane	Hydrocarbons (NMHC)	(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.3d.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

NMHC (Bag 1): Number of terms (p), Goodness-of-fit (BIC) and terms included in the 35 bestfitting 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).

Rank p BIC

etO

H

AromR

V

P

T50T90etO

H × etO

H

T50 × T50etO

H × Arom

etO

H × R

VP

etO

H × T50

etO

H × T90

1 11 357.702 • • • • • • • • • •

2 9 359.339 • • • • • • • •

3 10 360.017 • • • • • • • • •

4 8 360.223 • • • • • • •

5 12 360.243 • • • • • • • • • • •

6 10 361.141 • • • • • • • • •

7 10 361.761 • • • • • • • • •

8 10 362.04 • • • • • • • • •

9 10 362.209 • • • • • • • • •

10 9 362.298 • • • • • • • •

11 9 362.635 • • • • • • • •

12 11 362.678 • • • • • • • • • •

13 9 362.892 • • • • • • • •

14 9 363.339 • • • • • • • •

15 6 363.737 • • • • •

16 8 363.753 • • • • • • •

17 11 363.849 • • • • • • • • • •

18 11 364.257 • • • • • • • • • •

19 7 364.352 • • • • • •

20 11 364.464 • • • • • • • • • •

21 5 364.584 • • • •

22 9 364.705 • • • • • • • •

23 7 364.811 • • • • • •

24 10 364.985 • • • • • • • • •

25 10 365.131 • • • • • • • • •

26 10 365.31 • • • • • • • • •

27 6 365.57 • • • • •

28 9 365.696 • • • • • • • •

29 8 365.966 • • • • • • •

30 9 366.096 • • • • • • • •

31 8 366.36 • • • • • • •

32 9 366.361 • • • • • • • •

33 8 366.716 • • • • • • •

34 8 366.841 • • • • • • •

35 8 367.009 • • • • • • •

Design Terms
5

Models fit for NMHC (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 × etOH ZZee ● ●

T50 × T50 ZZ55 ● ●

etOH × Arom ZZea ● ●

etOH × RVP ZZer ● ×

etOH × T50 ZZe5 ● ●

etOH × T90 ZZe9 ● ●

1

denotes “Superset minus 1.”

NMHC (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 12 322.331 360.243

SM2211 322.497 357.702 0.167 1 0.683

1 A lower value indicates a better fit.

2 Best fit with respect to the 11-term design model.
6

NMHC (Bag 1): Coefficients and Tests of Effect for the Superset and Reduced Models, with 

respect to the 11-term design model.

Effect Full Model (superset) Reduced Model (SM1)

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

Intercept -1.0314 0.09233 15 -11.2 0.00000 -1.0315 0.09231 15 -11.2 0.00000

Z e 0.03153 0.01341 941 2.35 0.019 0.03094 0.01333 941 2.32 0.02054

Z a 0.09476 0.00935 941 10.1 0.00000 0.09461 0.00934 941 10.1 0.00000

Z r-0.04514 0.01068 941 -4.23 0.000026 -0.04568 0.01059 941 -4.31 0.000018

Z 5 0.1376 0.01347 941 10.22 0.00000 0.13689 0.01336 941 10.25 0.00000

Z 9 0.02123 0.00935 941 2.27 0.023 0.02160 0.00931 941 2.32 0.021

ZZee 0.04746 0.01800 941 2.64 0.0085 0.04612 0.01770 941 2.61 0.0093

ZZ55 0.07579 0.01348 941 5.62 0.00000 0.07534 0.01343 941 5.61 0.00000

ZZea 0.02076 0.00914 941 2.27 0.023 0.02045 0.00911 941 2.24 0.025

ZZer 0.00381 0.00933 941 0.408 0.68

ZZe5 0.04862 0.01932 941 2.52 0.012 0.04729 0.01904 941 2.48 0.013

ZZe9 0.02384 0.00918 941 2.60 0.0095 0.02441 0.00907 941 2.69 0.0072

0.1267 0.1266

0.07623 0.07624

2

veh

2

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2

veh

2

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7

I.3d.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).
8

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

NMHC (Bag 1): Number of terms (p), Goodness-of-fit (BIC) and terms included in the 35 bestfitting candidate models (out of a total of 2,964 possible models with hierarchy). (Terms included in 

models ranked 1-8 comprise the “superset” for final model-fitting).

Rank p BIC

etO

H

AromR

VP

T50T90etO

H × etO

H

T50 × T50etO

H × Arom

etO

H × R

VP

etO

H × T50

etO

H × T90

Arom × R

VP

Arom × T50Arom × T90T50 × T90R

VP × T90

1 14 341.37 • • • • • • • • • • • • •

2 12 341.85 • • • • • • • • • • •

3 15 342.09 • • • • • • • • • • • • • •

4 13 342.11 • • • • • • • • • • • •

5 11 342.77 • • • • • • • • • •

6 15 342.83 • • • • • • • • • • • • • •

7 12 342.89 • • • • • • • • • • •

8 13 342.92 • • • • • • • • • • • •

9 15 343.19 • • • • • • • • • • • • • •

10 14 343.31 • • • • • • • • • • • • •

11 14 343.39 • • • • • • • • • • • • •

12 13 343.51 • • • • • • • • • • • •

13 14 343.52 • • • • • • • • • • • • •

14 13 343.57 • • • • • • • • • • • •

15 11 343.78 • • • • • • • • • •

16 13 343.85 • • • • • • • • • • • •

17 16 343.90 • • • • • • • • • • • • • • •

18 14 343.99 • • • • • • • • • • • • •

19 13 344.08 • • • • • • • • • • • •

20 13 344.13 • • • • • • • • • • • •

21 12 344.24 • • • • • • • • • • •

22 16 344.30 • • • • • • • • • • • • • • •

23 14 344.43 • • • • • • • • • • • • •

24 13 344.48 • • • • • • • • • • • •

25 13 344.49 • • • • • • • • • • • •

26 12 344.52 • • • • • • • • • • •

27 16 344.57 • • • • • • • • • • • • • • •

28 11 345.02 • • • • • • • • • •

29 15 345.05 • • • • • • • • • • • • • •

30 14 345.12 • • • • • • • • • • • • •

31 11 345.21 • • • • • • • • • •

32 10 345.28 • • • • • • • • •

33 12 345.31 • • • • • • • • • • •

34 13 345.40 • • • • • • • • • • • •

35 10 345.44 • • • • • • • • •

Design Terms Extended Terms
10

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

Model Term Notation Model

Superset SM21

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 Represents “Superset minus 2,” etc.

NMHC (Bag 1): 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 16 295.558 344.303

SM2

2

14 298.043 341.372 2.485 2 0.289

1 A lower value indicates a better fit.

2 Best fit with respect to the 16-term extended model.
11

NMHC (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 -1.0304 0.09200 15 -11.2 0.00000 -1.0308 0.09197 15 -11.2 0.00000

Z e 0.05238 0.01514 941 3.46 0.00057 0.04439 0.01415 941 3.14 0.0018

Z a 0.1125 0.00998 941 11.3 0.00000 0.11119 0.00994 941 11.2 0.00000

Z r-0.03708 0.01333 941 -2.78 0.0055 -0.04765 0.01070 941 -4.45 0.000010

Z 5 0.1683 0.01710 941 9.84 0.00000 0.15733 0.01524 941 10.3 0.00000

Z 9 0.01811 0.01032 941 1.76 0.079 0.01682 0.00967 941 1.74 0.082

ZZee 0.04814 0.01801 941 2.67 0.0076 0.04274 0.01768 941 2.42 0.016

ZZ55 0.08384 0.01428 941 5.87 0.00000 0.07769 0.01369 941 5.68 0.00000

ZZea 0.04042 0.01492 941 2.71 0.0069 0.02594 0.00924 941 2.81 0.0051

ZZer 0.00718 0.01020 941 0.703 0.48

ZZe5 0.03943 0.01938 941 2.03 0.042 0.03579 0.01926 941 1.86 0.063

ZZe9 0.04967 0.01110 941 4.47 0.00001 0.05125 0.01105 941 4.64 0.00000

ZZar 0.04215 0.01599 941 2.64 0.00854 0.02820 0.01272 941 2.22 0.027

ZZa5 0.02025 0.01818 941 1.11 0.26569

ZZa9 0.01889 0.01016 941 1.86 0.06315 0.02068 0.01002 941 2.06 0.039

ZZ59 0.05480 0.01223 941 4.48 0.00001 0.05225 0.01209 941 4.32 0.00002

ZZr9 --- --- --- --- --- --- --- --- --- ---

0.1258 0.1257

0.07410 0.07430

2

veh

2

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2

veh

2

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