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.2d

Final	Model	Fitting

Non-Methane	Organic	Gases (NMOG)	(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.2d.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 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 1-8 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 11 346.525 • • • • • • • • • •

2 10 347.069 • • • • • • • • •

3 8 347.64 • • • • • • •

4 9 347.871 • • • • • • • •

5 9 348.234 • • • • • • • •

6 10 348.532 • • • • • • • • •

7 8 348.624 • • • • • • •

8 6 348.746 • • • • •

9 12 349.101 • • • • • • • • • • •

10 7 349.135 • • • • • •

11 9 349.202 • • • • • • • •

12 7 349.698 • • • • • •

13 9 349.712 • • • • • • • •

14 11 349.733 • • • • • • • • • •

15 10 349.944 • • • • • • • • •

16 9 350.005 • • • • • • • •

17 10 350.101 • • • • • • • • •

18 8 350.178 • • • • • • •

19 9 350.301 • • • • • • • •

20 10 350.607 • • • • • • • • •

21 11 350.763 • • • • • • • • • •

22 9 350.914 • • • • • • • •

23 10 350.925 • • • • • • • • •

24 7 350.96 • • • • • •

25 7 351.056 • • • • • •

26 8 351.155 • • • • • • •

27 7 351.404 • • • • • •

28 8 351.455 • • • • • • •

29 10 351.619 • • • • • • • • •

30 8 351.751 • • • • • • •

31 8 351.87 • • • • • • •

32 8 351.992 • • • • • • •

33 9 352.154 • • • • • • • •

34 8 352.393 • • • • • • •

35 10 352.393 • • • • • • • • •

Design Terms
5

Models fit for NMOG (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 ● ×

1

denotes “Superset minus 2.”

NMOG (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

Superset211 311.321 346.525

SM2 9 318.083 347.871 6.762 2 0.034

1 A lower value indicates a better fit.

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

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

respect to the 11-term design model.

Effect Superset Model Reduced Model (Superset)

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

Intercept -0.9520 0.09077 15 -10.49 <0.0001

Ze 0.07981 0.01326 941 6.02 <0.0001

Za 0.08789 0.00929 941 9.46 <0.0001

Zr -0.04595 0.01053 941 -4.36 <0.0001

Z5 0.1344 0.01329 941 10.12 <0.0001

Z9 0.01593 0.00925 941 1.72 0.0855 Reduced Model = Superset Model

ZZee 0.04594 0.01760 941 2.61 0.00918

ZZ55 0.07680 0.01336 941 5.75 <0.0001

ZZea 0.01635 0.00906 941 1.80 0.0714

ZZer - - - - -

ZZe5 0.04754 0.01893 941 2.51 0.0122

ZZe9 0.01961 0.00902 941 2.17 0.0300

2  veh 0.1224

2  0.07538
7

I.2d.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 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 14 328.80 • • • • • • • • • • • • •

2 12 329.14 • • • • • • • • • • •

3 13 329.58 • • • • • • • • • • • •

4 15 329.62 • • • • • • • • • • • • • •

5 11 330.02 • • • • • • • • • •

6 13 330.32 • • • • • • • • • • • •

7 12 330.33 • • • • • • • • • • •

8 15 330.35 • • • • • • • • • • • • • •

9 11 330.54 • • • • • • • • • •

10 15 330.60 • • • • • • • • • • • • • •

11 13 330.73 • • • • • • • • • • • •

12 10 330.77 • • • • • • • • •

13 13 330.86 • • • • • • • • • • • •

14 14 330.87 • • • • • • • • • • • • •

15 11 330.87 • • • • • • • • • •

16 14 330.90 • • • • • • • • • • • • •

17 14 331.03 • • • • • • • • • • • • •

18 13 331.38 • • • • • • • • • • • •

19 16 331.41 • • • • • • • • • • • • • • •

20 11 331.42 • • • • • • • • • •

21 13 331.45 • • • • • • • • • • • •

22 10 331.52 • • • • • • • • •

23 14 331.52 • • • • • • • • • • • • •

24 12 331.56 • • • • • • • • • • •

25 13 331.61 • • • • • • • • • • • •

26 13 331.66 • • • • • • • • • • • •

27 12 331.69 • • • • • • • • • • •

28 13 331.75 • • • • • • • • • • • •

29 13 331.80 • • • • • • • • • • • •

30 12 331.83 • • • • • • • • • • •

31 12 331.83 • • • • • • • • • • •

32 16 331.87 • • • • • • • • • • • • • • •

33 14 331.89 • • • • • • • • • • • • •

34 16 332.06 • • • • • • • • • • • • • • •

35 9 332.34 • • • • • • • •

Design Terms Extended Terms
10

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

Model Term Notation Model

Superset SM31SM7

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 3, etc.”

NMOG (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 17 282.156 333.609

SM3

2

14 285.473 328.802 3.317 3 0.345

SM7 10 313.769 346.266 28.296 4 0.000011

1 A lower value indicates a better fit.

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

NMOG (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 (SM3)

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

Intercept -0.9508 0.09047 15 -10.5 0.00000 -0.9513 0.09042 15 -10.5 0.00000

Ze 0.1054 0.01586 941 6.65 0.00000 0.0927 0.01406 941 6.60 0.00000

Za 0.1072 0.00995 941 10.8 0.00000 0.1051 0.00988 941 10.6 0.00000

Zr-0.03611 0.01339 941 -2.70 0.0071 -0.0483 0.01064 941 -4.54 0.00001

Z5 0.1696 0.01767 941 9.60 0.00000 0.1541 0.01514 941 10.2 0.00000

Z9 0.01217 0.01025 941 1.19 0.24 0.0112 0.00961 941 1.17 0.24

ZZee 0.05186 0.01849 941 2.80 0.0051 0.0420 0.01756 941 2.39 0.017

ZZ55 0.08968 0.01508 941 5.94 0.00000 0.0787 0.01360 941 5.78 0.00000

ZZea 0.03775 0.01496 941 2.52 0.012 0.0217 0.00918 941 2.36 0.018

ZZer 0.00726 0.01014 941 0.72 0.47

ZZe5 0.04249 0.01953 941 2.18 0.030 0.0357 0.01914 941 1.87 0.062

ZZe9 0.04148 0.01199 941 3.46 0.00056 0.0476 0.01098 941 4.34 0.00002

ZZar 0.04390 0.01620 941 2.71 0.0069 0.0272 0.01263 941 2.16 0.031

ZZa5 0.01941 0.01806 941 1.07 0.28

ZZa9 0.01980 0.01014 941 1.95 0.051 0.0205 0.00996 941 2.06 0.040

ZZ59 0.05104 0.01349 941 3.78 0.00016 0.0544 0.01201 941 4.53 0.00001

ZZr9 -0.01130 0.01146 941 -0.985 0.32

0.1216 0.1215

0.07309 0.07335

2

veh

2



2

veh

2

