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.2e

Final	Model	Fitting

Non-Methane	Organic	Gases (NMOG)	(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
2

I.2e.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

NMOG (Bag 2): 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 1-8 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 9 776.93 • • • • • • • •

2 8 777.286 • • • • • • •

3 8 777.584 • • • • • • •

4 6 777.605 • • • • •

5 10 777.832 • • • • • • • • •

6 9 777.881 • • • • • • • •

7 10 777.894 • • • • • • • • •

8 7 777.94 • • • • • •

9 9 778.301 • • • • • • • •

10 9 778.384 • • • • • • • •

11 8 778.426 • • • • • • •

12 9 778.57 • • • • • • • •

13 8 778.607 • • • • • • •

14 9 778.677 • • • • • • • •

15 9 778.785 • • • • • • • •

16 8 779.082 • • • • • • •

17 10 779.09 • • • • • • • • •

18 11 779.235 • • • • • • • • • •

19 10 779.274 • • • • • • • • •

20 10 779.362 • • • • • • • • •

21 10 779.566 • • • • • • • • •

22 9 779.642 • • • • • • • •

23 9 779.794 • • • • • • • •

24 10 779.909 • • • • • • • • •

25 9 780.024 • • • • • • • •

26 11 780.253 • • • • • • • • • •

27 10 780.321 • • • • • • • • •

28 11 780.336 • • • • • • • • • •

29 10 780.446 • • • • • • • • •

30 11 780.56 • • • • • • • • • •

31 10 780.817 • • • • • • • • •

32 12 781.731 • • • • • • • • • • •

33 11 781.927 • • • • • • • • • •

34 7 782.237 • • • • • •

35 8 782.744 • • • • • • •

Design Terms
5

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

Model Term Notation Model

Superset SM21SM5 SM42SM3a3SM3b4

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 SM5; formed by dropping etOH×Arom and etOH×T50 from SM2.

3 Not nested within SM5 or SM4; formed by dropping etOH×Arom from SM2.

4Not nested within SM5, SM4 or SM3a; formed by dropping etOH×T50 from SM2.

NMOG (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 11 745.891 779.235

SM2

2

9 748.716 776.930 2.825 2 0.244

SM5 6 757.086 777.605 8.370 3 0.0390

SM437 754.855 777.940 6.140 2 0.0464

SM3a38 751.636 777.286 2.920 1 0.0875

SM3b38 751.934 777.584 3.219 1 0.0728

1 A lower value indicates a better fit.

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

3 Not nested within previous model; test with respect to SM2.
6

NMOG (Bag 2): 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 (SM2)

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

Intercept -5.2370 0.2562 13 -20.4 0.00000 -5.2360 0.2560 13 -20.4 0.00000

Z e 0.02684 0.01675 819 1.60 0.11 0.02673 0.01676 819 1.59 0.11

Z a 0.03630 0.01306 819 2.78 0.0056 0.03634 0.01308 819 2.78 0.0056

Z r-0.04775 0.01489 819 -3.21 0.0014 -0.04786 0.01475 819 -3.24 0.0012

Z 5 0.04777 0.01810 819 2.64 0.0085 0.04915 0.01807 819 2.72 0.0067

Z 9 0.07197 0.01314 819 5.48 0.00000 0.07252 0.01295 819 5.60 0.00000

ZZee --- --- --- --- --- --- --- --- --- ---

ZZ55 0.05379 0.01451 819 3.71 0.00022 0.05349 0.01452 819 3.68 0.00025

ZZea 0.02087 0.01271 819 1.64 0.10 0.02171 0.01269 819 1.71 0.088

ZZer -0.01429 0.01291 819 -1.11 0.27

ZZe5 0.02592 0.01439 819 1.80 0.072 0.02586 0.01440 819 1.80 0.073

ZZe9 -0.01393 0.01292 819 -1.08 0.28

0.8514 0.8502

0.1306 0.1310

2

veh

2



2

veh

2


7

I.2e.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

NMOG (Bag 2): 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 1-5 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 11 775.72 • • • • • • • • • •

2 12 776.56 • • • • • • • • • • •

3 9 776.93 • • • • • • • •

4 10 777.25 • • • • • • • • •

5 8 777.29 • • • • • • •

6 12 777.49 • • • • • • • • • • •

7 11 777.50 • • • • • • • • • •

8 12 777.52 • • • • • • • • • • •

9 10 777.54 • • • • • • • • •

10 8 777.58 • • • • • • •

11 6 777.61 • • • • •

12 10 777.61 • • • • • • • • •

13 12 777.64 • • • • • • • • • • •

14 10 777.83 • • • • • • • • •

15 9 777.88 • • • • • • • •

16 10 777.89 • • • • • • • • •

17 7 777.94 • • • • • •

18 9 778.06 • • • • • • • •

19 12 778.07 • • • • • • • • • • •

20 11 778.08 • • • • • • • • • •

21 9 778.12 • • • • • • • •

22 10 778.12 • • • • • • • • •

23 13 778.16 • • • • • • • • • • • •

24 12 778.21 • • • • • • • • • • •

25 11 778.30 • • • • • • • • • •

26 9 778.30 • • • • • • • •

27 9 778.38 • • • • • • • •

28 7 778.40 • • • • • •

29 8 778.43 • • • • • • •

30 13 778.53 • • • • • • • • • • • •

31 10 778.56 • • • • • • • • •

32 9 778.57 • • • • • • • •

33 8 778.61 • • • • • • •

34 12 778.62 • • • • • • • • • • •

35 8 778.65 • • • • • • •

Design Terms Extended Terms
10

Models fit for NMOG (Bag 2): (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 --- ---

Arom × RVP ZZar --- ---

Arom × T50 ZZa5 ● ●

Arom × T90 ZZa9 ● ×

T50 × T90 ZZ59 --- ---

RVP × T90 ZZr9 --- ---

1

denotes “Superset minus 1, etc.”

NMOG (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 12 740.652 776.561

SM1 11 742.372 775.716 1.720 1 0.190

1 A lower value indicates a better fit.

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

NMOG (Bag 2): NMOGefficients and Tests of Effect for the Superset and Reduced Models, with 

respect to the 16-term extended 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 -5.2367 0.2562 13 -20.44 0.00000 -5.2369 0.2561 13 -20.45 0.00000

Ze 0.03387 0.01709 819 1.98 0.048 0.02947 0.01677 819 1.76 0.079

Za 0.03531 0.01302 819 2.71 0.007 0.03540 0.01303 819 2.72 0.0067

Zr-0.03371 0.01637 819 -2.06 0.040 -0.03415 0.01639 819 -2.08 0.0375

Z5 0.06201 0.01940 819 3.20 0.001 0.05338 0.01827 819 2.92 0.0036

Z9 0.08917 0.01429 819 6.24 0.000 0.08637 0.01414 819 6.11 0.00000

ZZee --- --- --- --- --- --- --- --- --- ---

ZZ55 0.05562 0.01452 819 3.83 0.000 0.05652 0.01451 819 3.89 0.00011

ZZea 0.04671 0.01711 819 2.73 0.006 0.04544 0.01710 819 2.66 0.0080

ZZer -0.03051 0.01391 819 -2.19 0.029 -0.02795 0.01379 819 -2.03 0.043

ZZe5 0.03208 0.01458 819 2.20 0.028 0.03080 0.01456 819 2.12 0.035

ZZe9 --- --- --- --- --- --- --- --- --- ---

ZZar --- --- --- --- --- --- --- --- --- ---

ZZa5 0.04751 0.02095 819 2.27 0.024 0.04528 0.02090 819 2.17 0.031

ZZa9 0.01833 0.01397 819 1.31 0.190

ZZ59 --- --- --- --- --- --- --- --- --- ---

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

0.8516 0.8507

0.1298 0.1300

2 veh

2 

2 veh

2 