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.3e

Final	Model	Fitting

Non-Methane	Hydrocarbons (NMHC)	(Bag	2)

No. Observations: 832

No. Vehicles: 13 (drop Odyssey and Siena) 

No. censored measurements: 0

No. missing measurements: 0

No. measurements removed: 0

Model Type: Mixed model
2

I.3e.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 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 1-6 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

V

P

etO

H × T50

etO

H × T90

1 6 987.73 • • • • •

2 7 990.10 • • • • • •

3 8 990.28 • • • • • • •

4 8 990.75 • • • • • • •

5 8 990.83 • • • • • • •

6 9 990.94 • • • • • • • •

7 9 991.28 • • • • • • • •

8 9 991.56 • • • • • • • •

9 8 991.76 • • • • • • •

10 8 991.77 • • • • • • •

11 9 991.91 • • • • • • • •

12 10 992.01 • • • • • • • • •

13 9 992.15 • • • • • • • •

14 9 992.30 • • • • • • • •

15 9 992.33 • • • • • • • •

16 5 992.43 • • • •

17 10 992.49 • • • • • • • • •

18 9 992.83 • • • • • • • •

19 10 992.98 • • • • • • • • •

20 10 993.13 • • • • • • • • •

21 9 993.15 • • • • • • • •

22 10 993.23 • • • • • • • • •

23 9 993.38 • • • • • • • •

24 10 993.49 • • • • • • • • •

25 10 993.73 • • • • • • • • •

26 11 993.89 • • • • • • • • • •

27 10 994.09 • • • • • • • • •

28 10 994.27 • • • • • • • • •

29 11 994.41 • • • • • • • • • •

30 11 994.69 • • • • • • • • • •

31 7 994.71 • • • • • •

32 10 994.75 • • • • • • • • •

33 11 994.90 • • • • • • • • • •

34 6 994.96 • • • • •

35 5 995.41 • • • •

Design Terms
5

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

Model Term Notation Model

Superset SM31SM4

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 3,” etc.

NMHC (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 10 961.229 992.008

SM3 7 967.013 990.098 5.785 3 0.123

SM426 967.211 987.730 0.197 1 0.657

1 A lower value indicates a better fit.

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

NMHC (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 (SM4)

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

Intercept -5.3253 0.2738 13 -19.45 0.00000 -5.3253 0.2734 13 -19.48 0.00000

Z e 0.009380 0.01907 819 0.492 0.62

Z a 0.03825 0.01488 819 2.57 0.010 0.03987 0.01476 819 2.70 0.0071

Z r-0.05762 0.01685 819 -3.42 0.00066 -0.05881 0.01555 819 -3.78 0.00017

Z 5 0.04836 0.02061 819 2.35 0.019 0.04548 0.01567 819 2.90 0.0038

Z 9 0.08329 0.01480 819 5.63 0.00000 0.08202 0.01478 819 5.55 0.00000

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

ZZ55 0.05627 0.01653 819 3.41 0.00069 0.04774 0.01487 819 3.21 0.00137

ZZea 0.02106 0.01447 819 1.46 0.15

ZZer -0.01776 0.01453 819 -1.22 0.22

ZZe5 0.02222 0.01638 819 1.36 0.18

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

0.9722 0.9691

0.1695 0.1708

2

veh

2



2

veh

2


7

I.3e.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 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 1-7 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 6 987.73 • • • • •

2 7 989.24 • • • • • •

3 7 989.3 • • • • • •

4 11 989.38 • • • • • • • • • •

5 7 989.61 • • • • • •

6 10 989.8 • • • • • • • • •

7 7 989.89 • • • • • •

8 12 990.06 • • • • • • • • • • •

9 7 990.1 • • • • • •

10 8 990.28 • • • • • • •

11 7 990.29 • • • • • •

12 8 990.75 • • • • • • •

13 11 990.8 • • • • • • • • • •

14 8 990.83 • • • • • • •

15 10 990.85 • • • • • • • • •

16 11 990.86 • • • • • • • • • •

17 9 990.94 • • • • • • • •

18 9 990.96 • • • • • • • •

19 8 990.99 • • • • • • •

20 9 991.19 • • • • • • • •

21 8 991.19 • • • • • • •

22 12 991.21 • • • • • • • • • • •

23 8 991.28 • • • • • • •

24 9 991.28 • • • • • • • •

25 11 991.44 • • • • • • • • • •

26 11 991.47 • • • • • • • • • •

27 8 991.49 • • • • • • •

28 13 991.52 • • • • • • • • • • • •

29 12 991.56 • • • • • • • • • • •

30 9 991.56 • • • • • • • •

31 8 991.59 • • • • • • •

32 8 991.63 • • • • • • •

33 8 991.63 • • • • • • •

34 8 991.63 • • • • • • •

35 9 991.66 • • • • • • • •

Design Terms Extended Terms
10

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

Model Term Notation Model

Superset SM41SM8

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 4,” etc.

NMHC (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 15 952.581 996.185

SM4

2

11 956.038 989.382 3.457 4 0.484

SM8 7 967.210 990.295 11.172 4 0.025

1 A lower value indicates a better fit.

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

NMHC (Bag 2): 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 (SM4)

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

Intercept -5.3250 0.2740 13 -19.4 0.00000 -5.3256 0.2739 13 -19.4 0.00000

Z e 0.02885 0.02218 819 1.30 0.19 0.01334 0.01909 819 0.70 0.48

Z a 0.04286 0.01594 819 2.69 0.0073 0.03743 0.01483 819 2.52 0.012

Z r-0.03307 0.02098 819 -1.58 0.12 -0.03909 0.01865 819 -2.10 0.036

Z 5 0.07846 0.02673 819 2.93 0.0034 0.05566 0.02080 819 2.68 0.0076

Z 9 0.09945 0.01660 819 5.99 0.00000 0.09799 0.01610 819 6.09 0.00000

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

ZZ55 0.06524 0.01810 819 3.60 0.00 0.05909 0.01652 819 3.58 0.00037

ZZea 0.06090 0.02371 819 2.57 0.010 0.05087 0.01946 819 2.61 0.0091

ZZer -0.03344 0.01630 819 -2.05 0.041 -0.03153 0.01569 819 -2.01 0.045

ZZe5 0.02558 0.01745 819 1.47 0.14 0.02865 0.01657 819 1.73 0.084

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

ZZar 0.01733 0.02569 819 0.67 0.50

ZZa5 0.06319 0.02923 819 2.16 0.031 0.05429 0.02379 819 2.28 0.023

ZZa9 0.02290 0.01620 819 1.41 0.16

ZZ59 0.00490 0.01743 819 0.28 0.78

ZZr9 -0.01760 0.01679 819 -1.05 0.29

0.9734 0.9723

0.1677 0.1684

2

veh

2



2

veh

2

